Supergroups of dihedral group:D8

This article discusses some of the groups that admit the dihedral group of order eight as a subgroup, quotient group, or subquotient.

Note that unlike the discussion of the subgroup structure of dihedral group:D8, this discussion is necessarily not comprehensive, because there are infinitely many groups containing the dihedral group of order eight. However, we provide a comprehensive discussion of all the groups of order sixteen containing this as a subgroup or quotient group.

Subgroups: making all the automorphisms inner
The outer automorphism group of dihedral group:D8 is cyclic group:Z2. There are two possibilities for a group admitting dihedral group:D8 as a NSCFN-subgroup (a normal fully normalized subgroup that is also a self-centralizing subgroup). The significance of these is that all automorphisms of dihedral group:D8 extend to inner automorphisms in these bigger groups.

For both of these, the quotient by the normal subgroup dihedral group:D8 is its outer automorphism group cyclic group:Z2, and hence they both have order 16. The two groups are given below:

Quotients: Schur covering groups
The Schur multiplier of dihedral group:D8 is cyclic group:Z2. The corresponding Schur covering groups (i.e., stem extensions with normal subgroup the Schur multiplier and quotient group the dihedral group itself) are all of order 16. Each of these has a normal subgroup isomorphic to cyclic group:Z2 with corresponding quotient group dihedral group:D8. In fact, the normal subgroup is precisely the center in all these cases. The list is below:

Direct products
Each of the groups listed below arises as the external direct product of dihedral group:D8 and some nontrivial group. In particular, each of these contains dihedral group:D8 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 order of the direct product is $$8a$$.

Central products with common subgroup of order two identified
We consider the central product of dihedral group:D8 with some other group with a common central subgroup of order two. The dihedral group arises as a normal subgroup (specifically, a central factor) but not a quotient group of the whole group.

We restrict attention to central products with groups that have a cyclic central subgroup of order two that is not a direct factor -- because if the cyclic central subgroup of order two is a direct factor, the central product can be realized as a direct product.

Note that by the product formula, if the order of the other group is $$a$$, the order of the central product is $$8a/2 = 4a$$.

Groups containing this as a Sylow subgroup
Note that the $$2$$-Sylow subgroup is a normal Sylow subgroup if and only if the $$2$$-Sylow number equals $$1$$.

If the 2-Sylow subgroup has a normal complement, i.e., the group is a 2-nilpotent group, then the fusion system for the prime 2 is the inner fusion system. If, however, the group is not 2-nilpotent, then the fusion system for the prime 2 must be the unique non-inner fusion system on the dihedral group.

The general procedure
Note first that any subgroup of index two is normal, so the groups we are interested in classifying have the dihedral group of order eight as a normal subgroup of index two.

We can use cohomology theory to begin this analysis. Specifically, we are interested in groups of order sixteen where the dihedral group of order eight is the normal subgroup and the quotient group is the cyclic group of order two. We denote by $$N$$ the dihedral group of order eight and $$Q$$ the quotient group, which is cyclic of order two.

The classification proceeds in three steps:


 * Determine the set of possible homomorphisms $$Q \to \operatorname{Out}(N)$$. In this case, both $$Q$$ and $$\operatorname{Aut}(N)$$ (the outer automorphism group of $$N$$) are cyclic of order two.
 * For each such homomorphism, determine whether an extension exists.
 * If an extension exists, classify the extensions using the second cohomology group $$H^2(Q,Z(N))$$ corresponding to the induced action on $$Z(N)$$ from the homomorphism to $$\operatorname{Out}(N)$$.

It turns out that $$\operatorname{Hom}(Q,\operatorname{Out}(N))$$ is isomorphic to cyclic group:Z2. Further, for each choice of homomorphism, the induced action on $$Z(N)$$ is trivial, so in both cases, we get a copy of $$H^2(Q,Z(N)) \cong H^2(\mathbb{Z}_2,\mathbb{Z}_2)$$, which is isomorphic to $$\mathbb{Z}_2$$. For more information, see second cohomology group for trivial group action of Z2 on Z2.

Details on extension sets

 * Extensions for trivial outer action of Z2 on D8: This discusses the case where the homomorphism from $$Q$$ to $$\operatorname{Out}(N)$$ is the trivial map. There are two sub-cases for the extension.
 * Extensions for nontrivial outer action of Z2 on D8: This discusses the case where the homomorphism from $$Q$$ to $$\operatorname{Out}(N)$$ is a nontrivial map. There are two sub-cases for the extension.

Groups containing a normal subgroup of order two with this as quotient
The list of groups with a normal subgroup isomorphic to cyclic group:Z2 and the quotient isomorphic to dihedral group:D8 is completely classified by second cohomology group for trivial group action of D8 on Z2. The cohomology group is isomorphic to elementary abelian group:E8, with some repetitions (i.e., multiple extensions give isomorphic extension groups). The elements are given below:

Groups containing it as a normal subgroup with quotient cyclic of order four
These are groups containing a normal subgroup $$N$$ isomorphic to dihedral group:D8 and quotient group $$Q$$ isomorphic to cyclic group:Z4. All these extensions are classified as follows: for each element of $$\operatorname{Hom}(Q,\operatorname{Out}(N))$$, there is a $$H^2(Q,Z(N))$$ worth of extensions for the induced action on $$Z(N)$$, which is necessarily trivial.

Here is information on the extension sets:


 * Extensions for trivial outer action of Z4 on D8: There are two extensions.
 * Extensions for nontrivial outer action of Z4 on D8: There are two extensions.

Here is a list of all four extensions:

Groups containing it as a normal subgroup with quotient a Klein four-group
These are groups containing a normal subgroup $$N$$ isomorphic to dihedral group:D8 and quotient group $$Q$$ isomorphic to Klein four-group. All these extensions are classified as follows: for each element of $$\operatorname{Hom}(Q,\operatorname{Out}(N))$$, there is a $$H^2(Q,Z(N))$$ worth of extensions for the induced action on $$Z(N)$$, which is necessarily trivial.

The group $$H^2(Q,Z(N))$$ for the trivial action is isomorphic to elementary abelian group:E8. More information is available at second cohomology group for trivial group action of V4 on Z2.