Supergroups of dihedral group:D8

This article gives specific information, namely, supergroups, about a particular group, namely: dihedral group:D8.
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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

Further information: group cohomology of dihedral group:D8#Schur multiplier, second cohomology group for trivial group action of D8 on Z2

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:

• 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:

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