# Supergroups of groups of order 8

This article gives specific information, namely, supergroups, about a family of groups, namely: groups of order 8.

View supergroups of group families | View supergroups of groups of a particular order |View other specific information about groups of order 8

Group | GAP ID second part | Hall-Senior number | Linear representation theory page |
---|---|---|---|

cyclic group:Z8 | 1 | 3 | supergroups of cyclic group:Z8 |

direct product of Z4 and Z2 | 2 | 2 | supergroups of direct product of Z4 and Z2 |

dihedral group:D8 | 3 | 4 | supergroups of dihedral group:D8 |

quaternion group | 4 | 5 | supergroups of quaternion group |

elementary abelian group:E8 | 5 | 1 | supergroups of elementary abelian group:E8 |

By Lagrange's theorem, any group that contains a group of order 8 must be a group of order a multiple of 8. Conversely, because Sylow subgroups exist and prime power order implies subgroups of all orders dividing the group order, any group whose order is a multiple of 8 must contain at least one subgroup of order 8.

## Direct products

We list here some of the direct products of groups of order 8 with other groups:

## Groups whose order has 8 as the largest prime divisor

Because Sylow subgroups exist and Sylow implies order-conjugate, any group whose order is of the form with odd must have a unique isomorphism class of subgroup of order 8. Below is given some information about groups of such orders.

### Case of order 24

`Further information: groups of order 24`

### Case of order 40

`Further information: groups of order 40`

### Case of order 56

`Further information: groups of order 56`

### Case of order 72

`Further information: groups of order 72`