# Subgroups of order 4 in groups of order 8

This article contains summary information about all subgroups of order 4 inside groups of order 8. See also groups of order 8 | groups of order 4 | subgroup structure of groups of order 8 | supergroups of groups of order 4

This article gives information on the occurrence of groups of order 4 as subgroups inside groups of order 8.

There are two groups of order 4: cyclic group:Z4 and Klein four-group. There are five groups of order 8: cyclic group:Z8, direct product of Z4 and Z2, dihedral group:D8, quaternion group, and elementary abelian group:E8.

## List of all subgroups

We make some preliminary observation: Any subgroup of order 4 in a group of order 8 must have index two. We have index two implies normal, so the subgroup is normal and the quotient group is isomorphic to cyclic group:Z2.

In the table below, we describe, for each isomorphism class of group of order 4 and group of order 8, all the possible automorphism classes of ways in which the group of order 4 occurs as a subgroup of the group of order 8. It turns out that in all cases, there is at most one automorphism class, i.e., all subgroups of order 4 in groups of order 8 are isomorph-automorphic subgroups (this feature is due to the small order, and does not carry over to higher orders).

Group GAP ID 2nd part Hall-Senior number Hall-Senior symbol Nilpotency class Occurrences of cyclic group:Z4 Number of subgroups in each automorphism class Occurrences of Klein four-group Number of subgroups in each automorphism class
cyclic group:Z8 1 3 $(3)$ 1 Z4 in Z8 1 -- --
direct product of Z4 and Z2 2 2 $(21)$ 1 Z4 in direct product of Z4 and Z2 2 first omega subgroup of direct product of Z4 and Z2 1
dihedral group:D8 3 4 $8\Gamma_2a_1$ 2 cyclic maximal subgroup of dihedral group:D8 1 Klein four-subgroups of dihedral group:D8 2
quaternion group 4 5 $8\Gamma_2a_2$ 2 cyclic maximal subgroups of quaternion group 3 -- --
elementary abelian group:E8 5 1 $(1^3)$ 1 -- -- V4 in E8 7

## Cyclic subgroups of order four

As noted above, there is at most one automorphism class for each isomorphism class of group of order 8 and subgroup of order 4. This feature is specific to the small orders involved.

Group GAP ID 2nd part Hall-Senior number Hall-Senior symbol Nilpotency class Occurrence of cyclic group:Z4 Number of subgroups = Number of normal subgroups = Number of conjugacy classes of subgroups (=1 iff characteristic subgroup) Size of orbit under automorphisms in the subgroup generated by 2-automorphisms (=1 iff the subgroup is a 2-automorphism-invariant subgroup) Size of orbit under automorphisms in the 2-core of the automorphism group (=1 iff the subgroup is a 2-core-automorphism-invariant subgroup) Size of orbit under automorphisms in the subgroup generated by 2'-automorphisms (=1 iff the subgroup is a coprime automorphism-invariant subgroup)
cyclic group:Z8 1 3 $(3)$ 1 Z4 in Z8 1 1 1 1
direct product of Z4 and Z2 2 2 $(21)$ 1 Z4 in direct product of Z4 and Z2 2 2 2 1
dihedral group:D8 3 4 $8\Gamma_2a_1$ 2 cyclic maximal subgroup of dihedral group:D8 1 1 1 1
quaternion group 4 5 $8\Gamma_2a_2$ 2 cyclic maximal subgroups of quaternion group 3 3 1 3
elementary abelian group:E8 5 1 $(1^3)$ 1 -- -- -- -- --

Note the following significance for the last column: any subgroup of order 4 in a group of order 8 is already an isomorph-normal subgroup, and we know that isomorph-normal coprime automorphism-invariant implies Sylow-relatively weakly closed (more generally isomorph-normal coprime automorphism-invariant implies fusion system-relatively weakly closed). Thus, if the last column value is 1, then the subgroup is a weakly closed subgroup of the whole group relative to its occurrence as a Sylow subgroup in any bigger group.

## Klein four-subgroups

Group GAP ID 2nd part Hall-Senior number Hall-Senior symbol Nilpotency class Occurrence of cyclic group:Z4 Number of subgroups = Number of normal subgroups = Number of conjugacy classes of subgroups (=1 iff characteristic subgroup) Size of orbit under automorphisms in the subgroup generated by 2-automorphisms (=1 iff the subgroup is a 2-automorphism-invariant subgroup) Size of orbit under automorphisms in the 2-core of the automorphism group (=1 iff the subgroup is a 2-core-automorphism-invariant subgroup)
cyclic group:Z8 1 3 $(3)$ 1 -- -- -- --
direct product of Z4 and Z2 2 2 $(21)$ 1 first omega subgroup of direct product of Z4 and Z2 1 1 1
dihedral group:D8 3 4 $8\Gamma_2a_1$ 2 Klein four-subgroups of dihedral group:D8 2 2 2
quaternion group 4 5 $8\Gamma_2a_2$ 2 -- -- -- --
elementary abelian group:E8 5 1 $(1^3)$ 1 V4 in E8 7 7 1

## Numerical information on counts of subgroups

The table below provides information on the counts of subgroups of order 4 in groups of order 8. Note the following:

General assertion Implication for the counts in this case
congruence condition on number of subgroups of given prime power order The number of subgroups of order 4 is odd.
The number of normal subgroups of order 4 is odd.
The number of p-core-automorphism-invariant subgroups (which in this case means the number of subgroups invariant under automorphisms in the 2-core of the automorphism group) of order 4 is odd.
index two implies normal (or more generally, any maximal subgroup of a group of prime power order is normal and has prime index) The number of subgroups of order 4 equals the number of normal subgroups of order 4.
In a group of prime power order $p^n$, the maximal subgroups are precisely the subgroups of index $p$. They all contain the Frattini subgroup, and via the fourth isomorphism theorem, they correspond precisely to maximal subgroups of the Frattini quotient, which is elementary abelian of order $p^r$ where $r$ is the minimum size of generating set for the group. The number of maximal subgroups is $(p^r - 1)/(p - 1)$. The number of subgroups of order 4 in a group of order 8 is 1,3, or 7, these values occurring when the minimum size of generating set is respectively 1, 2, and 3.
Group Second part of GAP ID Hall-Senior number Hall-Senior symbol Nilpotency class Minimum size of generating set Number of subgroups of order 4
(must be odd)
Number of normal subgroups of order 4
(must be odd)
Number of 2-core-automorphism-invariant subgroups of order 4
(must be odd)
Number of 2-automorphism-invariant subgroups of order 4 Number of characteristic subgroups of order 4
cyclic group:Z8 1 3 $(3)$ 1 1 1 1 1 1 1
direct product of Z4 and Z2 2 2 $(21)$ 1 2 3 3 1 1 1
dihedral group:D8 3 4 $8\Gamma_2a_1$ 2 2 3 3 1 1 1
quaternion group 4 5 $8\Gamma_2a_2$ 2 2 3 3 3 0 0
elementary abelian group:E8 5 1 $(1^3)$ 1 3 7 7 7 0 0

### Failure of congruence condition and replacement on elementary abelian groups

By the Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime, it is true that for an odd prime $p$, the singleton collection of the elementary abelian group of order $p^2$ is a collection of groups satisfying a universal congruence condition for the prime $p$, i.e., if there exists an elementary abelian subgroup of order $p^2$, the number of such subgroups is congruent to 1 mod $p$. In particular, the number of elementary abelian normal subgroups of order $p^2$ is congruent to 1 mod $p$. Further, the number of elementary abelian $p$-core-automorphism-invariant subgroups of order $p^2$ is also congruent to 1 mod $p$. Thus, in particular, the existence of an elementary abelian subgroup of order $p^2$ guarantees the existence of an elementary abelian normal subgroup of order $p^2$, and even the existence of an elementary abelian $p$-core-automorphism-invariant subgroup of order $p^2$.

However, this fails for the prime $p = 2$. Specifically, note that:

• The congruence condition fails for dihedral group:D8, which has exactly two Klein four-subgroups, so the number of Klein four-subgroups is nonzero and not congruent to 1 mod 2.
• Replacement of a Klein four-subgroup by a 2-core-automorphism-invariant Klein four-subgroup also fails for dihedral group:D8, which has a Klein four-subgroup but no 2-core-automorphism-invariant Klein four-subgroup.
• Replacement by a normal subgroup holds in groups of order 8 for the silly reason that any subgroup of order 4 is already normal. However, we can use the failure of the stronger version of replacement to construct an example of order 16, namely dihedral group:D16, where replacement by a normal subgroup also fails.