Supergroups of Klein four-group

This article discusses possible supergroups of the Klein four-group.

Note that unlike the discussion of the subgroup structure of Klein four-group, thisdiscussion is necessarily not comprehensive, because there are infinitely many groups containing the Klein four-group as a subgroup.

Direct products
Each of the groups listed below arises as the external direct product of Klein four-group and some nontrivial group. In particular, each of these groups contains the Klein four-group as a direct factor -- and hence as both a normal subgroup and a quotient group.

Note that since order of direct product is product of orders, if the other group has order $$a$$, the direct product has order $$4a$$.

The general procedure
Suppose $$G$$ is a group of order eight containing a subgroup $$N$$ of order four isomorphic to the Klein four-group. Note that since $$N$$ is of index two, it is a normal subgroup. Further, the quotient group is isomorphic to $$Q$$, the cyclic group of order two.

The classification has two steps:


 * Find all homomorphisms from $$Q$$ to $$\operatorname{Aut}(N)$$.
 * For each such homomorphism, find all possible extensions. These are classified by the elements of $$H^2(Q,N)$$ for the action.

The classification
$$\operatorname{Aut}(N)$$ is isomorphic to the symmetric group of degree three. This has three subgroups of order two, all of which are conjugate. Thus, up to equivalence under automorphisms of $$N$$, there is only one subgroup of order two.

Thus, there are (up to equivalence under automorphisms) two homomorphisms from $$Q$$ to $$\operatorname{Aut}(N)$$: the trivial homomorphism and an isomorphism to one of the subgroups of order two.

For the trivial map, the cohomology group $$H^2(Q,N)$$ has order two, and the two extensions are:


 * elementary abelian group of order eight: This corresponds to the identity element of $$H^2(Q,N)$$.
 * direct product of Z4 and Z2: This corresponds to the non-identity element of $$H^2(Q,N)$$.

For the nontrivial map, the cohomology group $$H^2(Q,N)$$ is trivial, and the unique extension corresponding to it is the dihedral group of order eight.