# 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 in direct product Order Second part of GAP ID Order of products Cyclic group:Z8 Direct product of Z4 and Z2 Dihedral group:D8 Quaternion group Elementary abelian group:E8
cyclic group:Z2 2 1 16 direct product of Z8 and Z2 (ID: (16,5)) direct product of Z4 and V4 (ID: (16,10)) direct product of D8 and Z2 (ID: (16,11)) direct product of Q8 and Z2 (ID: (16,12)) elementary abelian group:E16 (ID: (16,14))
cyclic group:Z3 3 1 24 cyclic group:Z24 (ID: (24,2)) direct product of Z6 and Z4 (ID: (24,9)) direct product of D8 and Z3 (ID: (24,10)) direct product of Q8 and Z3 (ID: (24,11)) direct product of E8 and Z3 (ID: (24,15))
cyclic group:Z4 4 1 32 cyclic group:Z32 (ID: (32,1)) direct product of Z4 and Z4 and Z2 (ID: (32,21)) direct product of D8 and Z4 (ID: (32,25)) direct product of Q8 and Z4 (ID: (32,26)) direct product of E8 and Z4 (ID: (32,45))
Klein four-group 4 2 32 direct product of Z8 and V4 (ID: (32,36)) direct product of E8 and Z4 (ID: (32,45)) direct product of D8 and V4 (ID: (32,46)) direct product of Q8 and V4 (ID: (32,47)) elementary abelian group:E32 (ID: (32,51))
cyclic group:Z5 5 1 40 cyclic group:Z40 (ID: (40,2)) direct product of Z10 and Z4 direct product of D8 and Z5 direct product of Q8 and Z5 direct product of E8 and Z5

## 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 $8m$ with $m$ 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