# Supergroups of cyclic group:Z4

From Groupprops

This article gives specific information, namely, supergroups, about a particular group, namely: cyclic group:Z4.

View supergroups of particular groups | View other specific information about cyclic group:Z4

This article discusses some of the supergroups of the cyclic group of order four.

## Classification of supergroups of order eight

`Further information: Classification of finite p-groups with cyclic maximal subgroup`

### The general procedure

We classify groups of order eight containing the cyclic group of order four as a normal subgroup . The quotient group must therefore be isomorphic to the cyclic group of order two.

The general procedure is as follows:

- Determine all homomorphisms from to .
- For each of these automorphisms, find all possible extensions, classified by the second cohomology group for that action.

### The classification

is cyclic of order two, so there are two possible homomorphisms from to : the trivial homomorphism and the unique isomorphism.

The groups corresponding to the trivial homomorphism are:

- Direct product of Z4 and Z2: This corresponds to the identity element in .
- Cyclic group:Z8: This corresponds to the non-identity element in .

The groups corresponding to the unique isomorphism are:

- Dihedral group:D8: This corresponds to the identity element in .
- Quaternion group: This corresponds to the non-identity element in .