# Abelian subgroup structure of groups of order 2^n

From Groupprops

This article gives specific information, namely, abelian subgroup structure, about a family of groups, namely: groups of order 2^n.

View abelian subgroup structure of group families | View other specific information about groups of order 2^n

## Abelian subgroup structure by order of groups

Number of groups of order | Information on groups | Information on abelian subgroup structure | ||
---|---|---|---|---|

0 | 1 | 1 | trivial group | -- |

1 | 2 | 1 | cyclic group:Z2 | subgroup structure of cyclic group:Z2 |

2 | 4 | 2 | groups of order 4 | subgroup structure of groups of order 4 |

3 | 8 | 5 | groups of order 8 | subgroup structure of groups of order 8 |

4 | 16 | 14 | groups of order 16 | abelian subgroup structure of groups of order 16 |

5 | 32 | 51 | groups of order 32 | abelian subgroup structure of groups of order 32 |

6 | 64 | 267 | groups of order 64 | abelian subgroup structure of groups of order 64 |

7 | 128 | 2328 | groups of order 128 | abelian subgroup structure of groups of order 128 |

8 | 256 | 56092 | groups of order 256 | abelian subgroup structure of groups of order 256 |

9 | 512 | 10494213 | groups of order 512 | abelian subgroup structure of groups of order 512 |

## Summary on existence, congruence, and replacement

### Summary on existence

Note that if there exists an abelian normal subgroup of a particular order, there exist abelian normal subgroups of all smaller orders, because any normal subgroup contains normal subgroups of all orders dividing its order.

Largest for which every group of order contains an abelian subgroup of order | Corresponding value | Largest for which every group of order contains an abelian normal subgroup of order | Corresponding value | ||
---|---|---|---|---|---|

0 | 1 | 0 | 1 | 0 | 1 |

1 | 2 | 1 | 2 | 1 | 2 |

2 | 4 | 2 | 4 | 2 | 4 |

3 | 8 | 2 | 4 | 2 | 4 |

4 | 16 | 3 | 8 | 3 | 8 |

5 | 32 | 3 | 8 | 3 | 8 |

6 | 64 | 4 | 16 | 4 | 16 |

7 | 128 | 4 | 16 | 4 | 16 |

### Summary on congruence condition

Below are values of and , the values in the rows and the values in the columns. A "Yes" indicates that in any group of order , the number of abelian subgroups of order is either 0 or odd. A "No" indicates that there exists a group of order where the number of abelian subgroups of order is a nonzero even number.

0 | 1 | Yes | ||||||||

1 | 2 | Yes | Yes | |||||||

2 | 4 | Yes | Yes | Yes | ||||||

3 | 8 | Yes | Yes | Yes | Yes | |||||

4 | 16 | Yes | Yes | Yes | Yes | Yes | ||||

5 | 32 | Yes | Yes | Yes | Yes | Yes | Yes | |||

6 | 64 | Yes | Yes | Yes | Yes | Yes | Yes | Yes | ||

7 | 128 | Yes | Yes | Yes | Yes | Yes | Yes (?) | Yes | Yes | |

8 | 256 | Yes | Yes | Yes | Yes | Yes | Yes (?) | No | Yes | Yes |