# Supergroups of dihedral group:D8

View supergroups of particular groups | View other specific information about 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 and quotients: essential minimalist examples

### Subgroups: making all the automorphisms inner

Further information: extensions for nontrivial outer action of Z2 on D8

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:

Group Second part of GAP ID (order is 16) Embedding of $D_8$ as a subgroup
dihedral group:D16 7 D8 in D16
semidihedral group:SD16 8 D8 in SD16

### 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:

Schur covering group Second part of GAP ID (order is 16) Embedding of center (with quotient $D_8$)
dihedral group:D16 7 center of dihedral group:D16
semidihedral group:SD16 8 center of semidihedral group:SD16
generalized quaternion group:Q16 9 center of generalized quaternion group:Q16

## General lists

### 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 $8a$.

Other factor of direct product Order Second part of GAP ID Value of direct product Order Second part of GAP ID Hall-Senior symbol (if applicable)
cyclic group:Z2 2 1 direct product of D8 and Z2 16 11 $16\Gamma_2a_1$
cyclic group:Z3 3 1 direct product of D8 and Z3 24 10 --
cyclic group:Z4 4 1 direct product of D8 and Z4 32 25 $32\Gamma_2e_1$
Klein four-group 4 2 direct product of D8 and V4 32 46 $32\Gamma_2a_1$
cyclic group:Z5 5 1 direct product of D8 and Z5 40 10 --
symmetric group:S3 6 1 direct product of D8 and S3 48 38 --
cyclic group:Z6 6 2 direct product of D8 and Z6 48 45 --
cyclic group:Z7 7 1 direct product of D8 and Z7 56 9 --
cyclic group:Z8 8 1 direct product of Z8 and D8 64 115  ?
direct product of Z4 and Z2 8 2 direct product of D8 and Z4 and Z2 64 196  ?
dihedral group:D8 8 3 direct product of D8 and D8 64 226  ?
quaternion group 8 4 direct product of D8 and Q8 64 230  ?
elementary abelian group:E8 8 5 direct product of D8 and E8 64 261 $64\Gamma_2a_1$

### 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 $8a/2 = 4a$.

Other component of central product Order Second part of GAP ID Choice of central subgroup of order two Value of central product Order Second part of GAP ID Comments
cyclic group:Z4 4 1 unique choice central product of D8 and Z4 16 13 Same as the central product of quaternion group and cyclic group:Z4.
cyclic group:Z8 8 1 unique choice central product of D8 and Z8 32 38 Same as the central product of quaternion group and cyclic group:Z8. Also, coincides with the central product of M16 and cyclic group:Z8 with an identified cyclic group:Z4.
direct product of Z4 and Z2 8 2 squares direct product of SmallGroup(16,13) and Z2 32 48 Same as the corresponding central product where the dihedral group is replaced by the quaternion group.
dihedral group:D8 8 3 unique choice inner holomorph of D8 32 49 The extraspecial group of "+" type for $2^5$.
quaternion group 8 4 unique choice central product of D8 and Q8 32 50 The extraspecial group of "-" type for $2^5$.

### Semidirect products with it as normal piece

Other piece Way it's acting Value of semidirect product Comments
cyclic group:Z2 Outer automorphism that fixes pointwise the cyclic maximal subgroup dihedral group:D16
cyclic group:Z2 Outer automorphism that moves cyclic maximal subgroup central product of D8 and Z4

### Wreath products with it as base

Wreathing group Way it's acting Value of wreath product Comments
cyclic group:Z2 Regular group action wreath product of D8 and Z2 Also the $2$-Sylow subgroup of symmetric group of degree eight

## Groups containing this as a Sylow subgroup

Further information: fusion systems for dihedral group:D8

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.

Group Order GAP ID second part $2'$-part of order $2$-Sylow number Embedding information (if available) Complement, i.e., $p'$-Hall subgroup (if it exists). Note this need not be a normal complement Choice of fusion system (see fusion systems for dihedral group:D8)
dihedral group:D24 24 6 3 3 cyclic group:Z3 inner (because it has a normal complement)
SmallGroup(24,8) 24 8 3 3 cyclic group:Z3 inner (because it has a normal complement)
direct product of D8 and Z3 24 10 3 1 cyclic group:Z3 inner (because it has a normal complement)
symmetric group:S4 24 12 3 3 D8 in S4 cyclic group:Z3 non-inner non-simple fusion system for dihedral group:D8
dihedral group:D40 40 6 5 5 cyclic group:Z5 inner (because it has a normal complement)
SmallGroup(40,8) 40 8 5 5 cyclic group:Z5 inner (because it has a normal complement)
direct product of D8 and Z5 40 10 5 5 cyclic group:Z5 inner (because it has a normal complement)
symmetric group:S5 120 34 15 15 D8 in S5 -- non-inner non-simple fusion system for dihedral group:D8
projective special linear group:PSL(3,2) 168 42 21 21 D8 in PSL(3,2) (subgroup of order 21, link not available) simple fusion system for dihedral group:D8

## Groups containing this as a subgroup of index two

### 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.

### Details

Choice of homomorphism from $Q$ to $\operatorname{Out}(N)$ Choice of cohomology class for that homomorphism Extension group GAP ID second part (order is 16) Is the $D_8$ complemented, i.e., is it a semidirect product? Is the $D_8$ characteristic?
trivial map trivial class direct product of D8 and Z2 11 Yes No
trivial map nontrivial class central product of D8 and Z4 13 Yes No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map dihedral group:D16 7 Yes No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map semidihedral group:SD16 8 No Yes

## 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:

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is 16) Stem extension? Base characteristic in whole group? Hall-Senior family (equivalence class up to being isoclinic) Nilpotency class of whole group (at least 2, at most 3) Derived length of whole group (always exactly 2) Minimum size of generating set of whole group (at least 2, at most 3) Subgroup information on base in whole group
trivial 1 direct product of D8 and Z2 11 No No $\Gamma_2$ 2 2 3
nontrivial and symmetric 1 nontrivial semidirect product of Z4 and Z4 4 No Yes $\Gamma_2$ 2 2 2 Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4
nontrivial and symmetric 2 SmallGroup(16,3) 3 No No $\Gamma_2$ 2 2 2
non-symmetric 1 dihedral group:D16 7 Yes Yes $\Gamma_3$ 3 2 2 center of dihedral group:D16
non-symmetric 1 generalized quaternion group:Q16 9 Yes Yes $\Gamma_3$ 3 2 2 center of generalized quaternion group:Q16
non-symmetric 2 semidihedral group:SD16 8 Yes Yes $\Gamma_3$ 3 2 2 center of semidihedral group:SD16
Total (6 rows) 8 (equals order of the cohomology group) -- -- -- -- -- -- -- -- --

## Groups containing this as a subgroup of index four

### Groups containing it as a normal subgroup with quotient cyclic of order four

Further information: second cohomology group for trivial group action of Z4 on Z2

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:

Here is a list of all four extensions:

Choice of homomorphism from $Q$ to $\operatorname{Out}(N)$ Choice of cohomology class for that homomorphism Extension group GAP ID second part (order is 32) Is the $D_8$ complemented, i.e., is it a semidirect product? Is the $D_8$ characteristic?
trivial map trivial class direct product of D8 and Z4 25 Yes No
trivial map nontrivial class central product of D8 and Z8 38 No No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map SmallGroup(32,9) 9 Yes No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map wreath product of Z4 and Z2 11 Yes Yes

### Groups containing it as a normal subgroup with quotient a Klein four-group

Further information: second cohomology group for trivial group action of V4 on Z2

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.

Choice of homomorphism from $Q$ to $\operatorname{Out}(N)$ Choice of cohomology class for that homomorphism Extension group GAP ID second part (order is 32) Is the $D_8$ complemented, i.e., is it a semidirect product? Is the $D_8$ characteristic?
trivial map trivial class direct product of D8 and V4 46 Yes No
trivial map symmetric nontrivial class direct product of SmallGroup(16,13) and Z2 48 Yes No
trivial map one of the non-symmetric classes inner holomorph of D8 49 Yes No
trivial map one of the non-symmetric classes central product of D8 and Q8 50 No No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map direct product of D16 and Z2 39 Yes No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map direct product of SD16 and Z2 40 No No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map central product of D16 and Z4 42 No No
nontrivial map doesn't make sense since no natural origin in case of nontrivial map holomorph of Z8 43 Yes Yes
nontrivial map doesn't make sense since no natural origin in case of nontrivial map SmallGroup(32,44) 44 No Yes