# Subgroup structure of groups of order 24

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 24.

View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 24

To understand these in a broader context, see subgroup structure of groups of order 3.2^n | subgroup structure of groups of order 2^3.3^n

## Numerical information on counts of subgroups by order

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)

Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)

Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order

Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugateMINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Note that, by Lagrange's theorem, the order of any subgroup must divide the order of the group. Thus, the order of any proper nontrivial subgroup is one of the numbers 2,4,8,3,6,12.

Here are some observations on the number of subgroups of each order:

- Congruence condition on number of subgroups of given prime power order: In particular:
- The number of subgroups of order 2 is congruent to 1 mod 2 (i.e., it is odd), and the number of conjugacy classes is positive.
- The number of subgroups of order 4 is congruent to 1 mod 2 (i.e., it is odd), and the number of conjugacy classes is positive.
- The number of subgroups of order 8 is congruent to 1 mod 2 (i.e., it is odd), and the number of conjugacy classes is positive.
- The number of subgroups of order 3 is congruent to 1 mod 3, and the number of conjugacy classes is positive.

- By the fact that Sylow implies order-conjugate, we obtain that Sylow number equals index of Sylow normalizer, and in particular, divides the index of the Sylow subgroup. Combined with (1), we get the following:
- The number of 2-Sylow subgroups (subgroups of order 8) is either 1 or 3, and there is a unique conjugacy class of such subgroups.
- The number of 3-Sylow subgroups (subgroups of order 3) is either 1 or 4, and there is a unique conjugacy class of such subgroups.

- In the case of a finite nilpotent group, the number of subgroups of a given order is the product of the number of subgroups of order equal to each of its maximal prime power divisors, in the corresponding Sylow subgroup. In particular, we get (number of subgroups of order 3) = 1, (number of subgroups of order 6) = (number of subgroups of order 2), (number of subgroups of order 12) = (number of subgroups of order 4), and (number of subgroups of order 8) = 1.
- In the special case of a finite abelian group, we have (number of subgroups of order 3) = (number of subgroups of order 8) = 1, and (number of subgroups of order 2) = (number of subgroups of order 4) = (number of subgroups of order 6) = (number of subgroups of order 12). This is because subgroup lattice and quotient lattice of finite abelian group are isomorphic.
- Finite supersolvable implies subgroups of all orders dividing the group order: For any finite supersolvable group, there are subgroups of every possible order, i.e., there are proper nontrivial subgroups of orders 2,3,4,6,8,12. All finite nilpotent groups are supersolvable.
- In particular, by (2), the normalizer of the 3-Sylow subgroup is nontrivial (order either 6 or 24). Therefore, there exists an element of order 2 normalizing a 3-Sylow subgroup, and so we obtain that there must exist a subgroup of order 6.

### Table of number of subgroups

Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Supersolvable? | Derived length | Number of subgroups of order 2 (must be odd by (1)) | Number of subgroups of order 4 (must be odd by (1)) | Number of subgroups of order 8 (must be 1 or 3 by (2)) | Number of subgroups of order 3 (must be 1 or 4 by (2)) | Number of subgroups of order 6 (must be positive by (5)) | Number of subgroups of order 12 | Total number of subgroups (includes trivial subgroup and whole group) |
---|---|---|---|---|---|---|---|---|---|---|---|

nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | Yes | 2 | 1 | 1 | 3 | 1 | 1 | 1 | 10 |

cyclic group:Z24 | 2 | 1 | Yes | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

special linear group:SL(2,3) | 3 | not nilpotent | No | 3 | 1 | 3 | 1 | 4 | 4 | 0 | 15 |

dicyclic group:Dic24 | 4 | not nilpotent | Yes | 2 | 1 | 7 | 3 | 1 | 1 | 3 | 18 |

direct product of S3 and Z4 | 5 | not nilpotent | Yes | 2 | 7 | 7 | 3 | 1 | 3 | 3 | 26 |

dihedral group:D24 | 6 | not nilpotent | Yes | 2 | 13 | 7 | 3 | 1 | 3 | 3 | 32 |

direct product of Dic12 and Z2 | 7 | not nilpotent | Yes | 2 | 3 | 7 | 3 | 1 | 3 | 3 | 22 |

semidirect product of Z3 and D8 with action kernel V4 | 8 | not nilpotent | Yes | 2 | 9 | 7 | 3 | 1 | 5 | 3 | 30 |

direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | Yes | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

direct product of D8 and Z3 | 10 | 2 | Yes | 2 | 5 | 3 | 1 | 1 | 5 | 3 | 20 |

direct product of Q8 and Z3 | 11 | 2 | Yes | 2 | 1 | 3 | 1 | 1 | 1 | 3 | 12 |

symmetric group:S4 | 12 | not nilpotent | No | 3 | 9 | 7 | 3 | 4 | 4 | 1 | 30 |

direct product of A4 and Z2 | 13 | not nilpotent | No | 2 | 7 | 7 | 1 | 4 | 4 | 1 | 26 |

direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | Yes | 2 | 15 | 19 | 3 | 1 | 7 | 7 | 54 |

direct product of E8 and Z3 | 15 | 1 | Yes | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 32 |

Possibility set | -- | 1, 2 if nilpotent | Yes, No | 1, 2, 3 | 1, 3, 5, 7, 9, 13, 15 | 1, 3, 7, 19 | 1, 3 | 1, 4 | 1, 3, 4, 5, 7 | 0, 1, 3, 7 |

### Table of number of conjugacy classes of subgroups

Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Supersolvable? | Derived length | Number of conjugacy classes of subgroups of order 2 (must be positive) | Number of conjugacy classes of subgroups of order 4 (must be positive) | Number of conjugacy classes of subgroups of order 8 (must be 1) | Number of conjugacy classes of subgroups of order 3 (must be 1) | Number of conjugacy classes of subgroups of order 6 (must be positive by (5)) | Number of conjugacy classes of subgroups of order 12 | Total number of conjugacy classes of subgroups (includes trivial subgroup and whole group |
---|---|---|---|---|---|---|---|---|---|---|---|

nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | Yes | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

cyclic group:Z24 | 2 | 1 | Yes | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

special linear group:SL(2,3) | 3 | not nilpotent | No | 3 | 1 | 1 | 1 | 1 | 1 | 0 | 7 |

dicyclic group:Dic24 | 4 | not nilpotent | Yes | 2 | 1 | 3 | 1 | 1 | 1 | 3 | 12 |

direct product of S3 and Z4 | 5 | not nilpotent | Yes | 2 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

dihedral group:D24 | 6 | not nilpotent | Yes | 2 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

direct product of Dic12 and Z2 | 7 | not nilpotent | Yes | 2 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

semidirect product of Z3 and D8 with action kernel V4 | 8 | not nilpotent | Yes | 2 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | Yes | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

direct product of D8 and Z3 | 10 | 2 | Yes | 2 | 3 | 3 | 1 | 1 | 3 | 3 | 16 |

direct product of Q8 and Z3 | 11 | 2 | Yes | 2 | 1 | 3 | 1 | 1 | 1 | 3 | 12 |

symmetric group:S4 | 12 | not nilpotent | No | 3 | 2 | 3 | 1 | 1 | 1 | 1 | 11 |

direct product of A4 and Z2 | 13 | not nilpotent | No | 2 | 3 | 2 | 1 | 1 | 2 | 1 | 12 |

direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | Yes | 2 | 7 | 7 | 1 | 1 | 7 | 7 | 32 |

direct product of E8 and Z3 | 15 | 1 | Yes | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 32 |

Possibility set | -- | 1, 2 if nilpotent | Yes, No | 1, 2, 3 | 1, 2, 3, 7 | 1, 2, 3, 7 | 1 | 1 | 1, 2, 3, 7 | 0, 1, 3, 7 | 7, 8, 11, 12, 16, 32 |

### Table of number of isomorphism classes of subgroups

The number of isomorphism classes is bounded from above both by the number of conjugacy classes *and* the number of isomorphism classes of all groups of that order. Also, if subgroups of the order exist, the number of isomorphism classes is at least 1. This restricts the number of isomorphism classes fairly narrowly.

Group | Second part of GAP ID (ID is (24,second part)) | Nilpotency class | Supersolvable? | Derived length | Number of isomorphism classes of subgroups of order 2 (must be 1) | Number of isomorphism classes of subgroups of order 4 (must be 1 or 2) | Number of isomorphism classes of subgroups of order 8 (must be 1) | Number of isomorphism classes of subgroups of order 3 (must be 1) | Number of isomorphism classes of subgroups of order 6 (must be 1 or 2) | Number of isomorphism classes of subgroups of order 12 | Total number of isomorphism classes of subgroups |
---|---|---|---|---|---|---|---|---|---|---|---|

nontrivial semidirect product of Z3 and Z8 | 1 | not nilpotent | Yes | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

cyclic group:Z24 | 2 | 1 | Yes | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

special linear group:SL(2,3) | 3 | not nilpotent | No | 3 | 1 | 1 | 1 | 1 | 1 | 0 | 7 |

dicyclic group:Dic24 | 4 | not nilpotent | Yes | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

direct product of S3 and Z4 | 5 | not nilpotent | Yes | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 11 |

dihedral group:D24 | 6 | not nilpotent | Yes | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 11 |

direct product of Dic12 and Z2 | 7 | not nilpotent | Yes | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 13 |

semidirect product of Z3 and D8 with action kernel V4 | 8 | not nilpotent | Yes | 2 | 1 | 2 | 1 | 1 | 2 | 3 | 12 |

direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | Yes | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 10 |

direct product of D8 and Z3 | 10 | 2 | Yes | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 10 |

direct product of Q8 and Z3 | 11 | 2 | Yes | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

symmetric group:S4 | 12 | not nilpotent | No | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 9 |

direct product of A4 and Z2 | 13 | not nilpotent | No | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | not nilpotent | Yes | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 10 |

direct product of E8 and Z3 | 15 | 1 | Yes | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 |

Possibility set | -- | 1, 2 if nilpotent | Yes, No | 1, 2, 3 | 1 | 1, 2 | 1 | 1 | 1, 2 | 0, 1, 2, 3 | 7, 8, 9, 10, 11, 12, 13 |

## Sylow subgroups

### 2-Sylow subgroups

Here is the occurrence summary:

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

### 3-Sylow subgroups

Note that the 3-Sylow subgroup is isomorphic to cyclic group:Z3 in all cases. By the congruence condition on Sylow numbers as well as the divisibility condition on Sylow numbers, the only possibilities for the number of 3-Sylow subgroups is 1 or 4. In the former case, we have a normal Sylow subgroup. In the latter case, the normalizer of the Sylow subgroup has order 6, and is thus either cyclic group:Z6 or symmetric group:S3.

Group | Second part of GAP ID (ID is (24,second part)) | Number of 3-Sylow subgroups | Normalizer of Sylow subgroup |
---|---|---|---|

nontrivial semidirect product of Z3 and Z8 | 1 | 1 | whole group |

cyclic group:Z24 | 2 | 1 | whole group |

special linear group:SL(2,3) | 3 | 4 | cyclic group:Z6 |

dicyclic group:Dic24 | 4 | 1 | whole group |

direct product of S3 and Z4 | 5 | 1 | whole group |

dihedral group:D24 | 6 | 1 | whole group |

direct product of Dic12 and Z2 | 7 | 1 | whole group |

semidirect product of Z3 and D8 with action kernel V4 | 8 | 1 | whole group |

direct product of Z6 and Z4 (also, direct product of Z12 and Z2) | 9 | 1 | whole group |

direct product of D8 and Z3 | 10 | 1 | whole group |

direct product of Q8 and Z3 | 11 | 1 | whole group |

symmetric group:S4 | 12 | 4 | symmetric group:S3 |

direct product of A4 and Z2 | 13 | 4 | cyclic group:Z6 |

direct product of D12 and Z2 (also direct product of S3 and V4) | 14 | 1 | whole group |

direct product of E8 and Z3 | 15 | 1 | whole group |