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.
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