Central product of D8 and Q8 | 32 (50) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (4) Subgroup rank of a group (4) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (1) |

Central product of D8 and Z4 | 16 (13) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (1) |

Central product of UT(3,Z) and Z identifying center with 2Z | | Nilpotency class (2) Derived length (2) Frattini length (2) Hirsch length of a polycyclic group (3) Polycyclic breadth (3) |

Cyclic group:Z2 | 2 (1) | Underlying prime of p-group (2) Order of a group (2) Prime-base logarithm of order (1) Max-length of a group (1) Chief length (1) Composition length (1) Exponent of a group (2) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) Number of conjugacy classes (2) Number of subgroups (2) Number of conjugacy classes of subgroups (2) Number of equivalence classes under real conjugacy (2) Number of equivalence classes under rational conjugacy (2) |

Cyclic group:Z3 | 3 (1) | Underlying prime of p-group (3) Order of a group (3) Prime-base logarithm of order (1) Max-length of a group (1) Chief length (1) Composition length (1) Exponent of a group (3) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Cyclic group:Z4 | 4 (1) | Underlying prime of p-group (2) Order of a group (4) Prime-base logarithm of order (2) Max-length of a group (2) Chief length (2) Composition length (2) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (1) Subgroup rank of a group (1) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Nilpotency class (1) Derived length (1) Fitting length (1) |

Dicyclic group:Dic20 | 20 (1) | Order of a group (20) Exponent of a group (10) Frattini length (2) Derived length (2) Minimum size of generating set (2) Subgroup rank of a group (2) |

Dihedral group:D10 | 10 (1) | Order of a group (10) Exponent of a group (10) Fitting length (2) Frattini length (1) Derived length (2) |

Dihedral group:D16 | 16 (7) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Number of conjugacy classes (7) Number of equivalence classes under rational conjugacy (6) Number of conjugacy classes of rational elements (5) |

Dihedral group:D20 | 20 (4) | Order of group (20) Order of a group (10) Frattini length (1) Fitting length (1) Derived length (2) |

Dihedral group:D8 | 8 (3) | Underlying prime of p-group (2) Order of a group (8) Prime-base logarithm of order (3) Max-length of a group (3) Chief length (3) Composition length (3) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Fitting length (1) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (1) Number of conjugacy classes (5) Number of equivalence classes under real conjugacy (5) Number of conjugacy classes of real elements (5) Number of equivalence classes under rational conjugacy (5) Number of conjugacy classes of rational elements (5) Number of subgroups (10) Number of conjugacy classes of subgroups (8) Number of normal subgroups (6) Number of automorphism classes of subgroups (6) Number of characteristic subgroups (4) |

Direct product of D8 and Z2 | 16 (11) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Fitting length (1) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (2) |

Direct product of D8 and Z3 | 24 (10) | Order of a group (24) Exponent of a group (12) Frattini length (2) Fitting length (1) Derived length (2) Nilpotency class (2) |

Direct product of Z4 and Z4 | 16 (2) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

Faithful semidirect product of E8 and Z4 | 32 (6) | Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (3) Derived length (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) |

Fundamental group of Klein bottle | | |

General affine group:GA(1,5) | 20 (3) | Order of a group (20) Exponent of a group (20) Frattini length (1) Fitting length (2) Derived length (2) Minimum size of generating set (2) Subgroup rank of a group (2) |

General affine group:GA(1,7) | 42 (1) | Order of a group (42) Exponent of a group (42) Frattini length (1) Fitting length (2) Derived length (2) Subgroup rank of a group (2) Minimum size of generating set (2) |

Generalized dihedral group for direct product of Z4 and Z4 | 32 (34) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (2) |

Generalized quaternion group:Q16 | 16 (9) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Number of conjugacy classes (7) Number of equivalence classes under rational conjugacy (6) Number of conjugacy classes of rational elements (5) |

Holomorph of Z9 | 54 (6) | Order of a group (54) Exponent of a group (18) Frattini length (2) Fitting length (2) Derived length (2) Minimum size of generating set (2) Subgroup rank of a group (2) |

Infinite dihedral group | | Derived length (2) Fitting length (2) Frattini length (1) Subgroup rank of a group (2) |

Inner holomorph of D8 | 32 (49) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (4) Subgroup rank of a group (4) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (1) |

Klein four-group | 4 (2) | Underlying prime of p-group (2) Order of a group (4) Prime-base logarithm of order (2) Max-length of a group (2) Chief length (2) Composition length (2) Exponent of a group (2) Prime-base logarithm of exponent (1) Frattini length (1) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Nilpotency class (1) Derived length (1) Fitting length (1) |

M16 | 16 (6) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (2) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

M64 | 64 (51) | Underlying prime of p-group (2) Order of a group (64) Prime-base logarithm of order (6) Max-length of a group (6) Chief length (6) Composition length (6) Exponent of a group (32) Prime-base logarithm of exponent (5) Nilpotency class (2) Derived length (2) Frattini length (5) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

Nontrivial semidirect product of Z4 and Z4 | 16 (4) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) Number of subgroups (15) Number of conjugacy classes of subgroups (13) Number of conjugacy classes (10) |

Quaternion group | 8 (4) | Underlying prime of p-group (2) Order of a group (8) Prime-base logarithm of order (3) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (1) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Number of conjugacy classes (5) Number of equivalence classes under real conjugacy (5) Number of conjugacy classes of real elements (5) Number of equivalence classes under rational conjugacy (5) Number of conjugacy classes of rational elements (5) Number of subgroups (6) Number of conjugacy classes of subgroups (6) Number of normal subgroups (6) Number of automorphism classes of subgroups (4) |

Semidihedral group:SD16 | 16 (8) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (1) Characteristic rank of a p-group (1) Number of conjugacy classes (7) Number of equivalence classes under rational conjugacy (6) Number of conjugacy classes of rational elements (5) |

Semidirect product of Z8 and Z4 of M-type | 32 (4) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (2) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

SmallGroup(16,3) | 16 (3) | Underlying prime of p-group (2) Order of a group (16) Prime-base logarithm of order (4) Max-length of a group (4) Chief length (4) Composition length (4) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) Number of conjugacy classes (10) Number of conjugacy classes of subgroups (17) |

SmallGroup(32,10) | 32 (10) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

SmallGroup(32,15) | 32 (15) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (2) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

SmallGroup(32,2) | 32 (2) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Frattini length (2) Nilpotency class (2) Derived length (2) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) |

SmallGroup(32,24) | 32 (24) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) |

SmallGroup(32,30) | 32 (30) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) |

SmallGroup(32,31) | 32 (31) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (4) Prime-base logarithm of exponent (2) Nilpotency class (2) Derived length (2) Frattini length (2) Fitting length (1) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (2) |

SmallGroup(32,44) | 32 (44) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Max-length of a group (5) Chief length (5) Composition length (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (3) Subgroup rank of a group (3) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

SmallGroup(32,5) | 32 (5) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (2) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (3) |

SmallGroup(32,7) | 32 (7) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (3) Characteristic rank of a p-group (2) |

SmallGroup(32,9) | 32 (9) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (3) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |

Symmetric group:S3 | 6 (1) | Order of a group (6) Exponent of a group (6) Derived length (2) Minimum size of generating set (2) Subgroup rank of a group (2) Max-length of a group (2) Number of conjugacy classes (3) Number of equivalence classes under real conjugacy (3) Number of conjugacy classes of real elements (3) Number of equivalence classes under rational conjugacy (3) Number of conjugacy classes of rational elements (3) Number of subgroups (6) Number of conjugacy classes of subgroups (4) Number of normal subgroups (3) Number of automorphism classes of subgroups (4) |

Unitriangular matrix group:UT(3,Z) | | Nilpotency class (2) Derived length (2) Frattini length (2) Hirsch length of a polycyclic group (3) Polycyclic breadth (3) |

Wreath product of Z4 and Z2 | 32 (11) | Underlying prime of p-group (2) Order of a group (32) Prime-base logarithm of order (5) Exponent of a group (8) Prime-base logarithm of exponent (3) Nilpotency class (3) Derived length (2) Frattini length (3) Minimum size of generating set (2) Subgroup rank of a group (3) Rank of a p-group (2) Normal rank of a p-group (2) Characteristic rank of a p-group (2) |