Supergroups of groups of order 8

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