Abelian subgroup structure of groups of order 16

This article gives specific information, namely, abelian subgroup structure, about a family of groups, namely: groups of order 16.
View abelian subgroup structure of group families | View abelian subgroup structure of groups of a particular order |View other specific information about groups of order 16

Summary on existence, congruence, and replacement

Order of subgroup	Existence of abelian subgroup of that order guaranteed?	Existence of abelian normal subgroup of that order guaranteed?	Abelian-to-normal replacement for that order guaranteed?	Number of abelian subgroups of that order always odd if nonzero?
1	Yes	Yes	Yes	Yes
2	Yes	Yes	Yes	Yes
4	Yes	Yes	Yes	Yes
8	Yes	Yes	Yes	Yes
16	No	No	Yes	Yes

Abelian normal subgroups

FACTS TO CHECK AGAINST for abelian normal subgroup of group of prime power order:
EXISTENCE: existence of abelian normal subgroups of small prime power order
CONGRUENCE, IMPLIES ABELIAN-TO-NORMAL REPLACEMENT: congruence condition on number of subgroups of given prime power order (covers cases of $p$ and $p^{2}$) | congruence condition on number of abelian subgroups of prime-cube order | congruence condition on number of abelian subgroups of prime-fourth order | Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order | congruence condition on number of abelian subgroups of prime index | congruence condition on number of abelian subgroups of prime-square index for odd prime |abelian-to-normal replacement theorem for prime-square index | abelian-to-normal replacement theorem for prime-cube index for odd prime
CONGRUENCE CONDITIONS BASED ON EXPONENT: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

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RELATION WITH REPRESENTATION THEORY: degree of irreducible representation divides index of abelian normal subgroup

Abelian normal subgroups of order 8

• Existence: By the existence of abelian normal subgroups of small prime power order, there always exists an abelian normal subgroup of order 8 in any group of order 16. To see this, note that if $n>k(k-1)/2$, then any group of order $p^{n}$ contains an abelian normal subgroup of order $p^{k}$. Set $p=2,k=3,n=4$ and get the desired conclusion.
• Count: First, note that since index two implies normal, all abelian subgroups of order 8 are normal. Using the congruence condition on number of abelian subgroups of prime index, we get that the number of abelian subgroups is odd, i.e., it is one of the numbers 1,3,5,7,... In fact, we can say more: for a non-abelian group of order 16, the number of abelian normal subgroups of order 8 is either 1 or 3, and for an abelian group of rank $r$, the number is $2^{r}-1$ (so one of the numbers 1,3,7,15).

We can be even more specific regarding the count: for a non-abelian group of order 16, the number of abelian subgroups of order 8 is 1 iff the group has class exactly three (and this subgroup is the centralizer of derived subgroup, where the derived subgroup has order 4), and is 3 iff the group has class exactly two (and these 3 are precisely the subgroups of order 8 containing the center, which is a subgroup of order 4).

Below is the information on abelian normal subgroups of order 8:

Group	GAP ID 2nd part	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Minimum size of generating set	Number of subgroups isomorphic to elementary abelian group:E8	Number of subgroups isomorphic to direct product of Z4 and Z2	Number of subgroups isomorphic to cyclic group:Z8	Total number of abelian subgroups of order 8 = number of abelian normal subgroups of order 8	Orbit sizes	Subgroup information
cyclic group:Z16	1	5	$(4)$	1	1	0	0	1	1	1	Z8 in Z16
direct product of Z4 and Z4	2	3	$(2^{2})$	1	2	0	3	0	3	3	direct product of Z4 and Z2 in direct product of Z4 and Z4
SmallGroup(16,3)	3	9	$16\Gamma _{2}c_{1}$	2	2	1	2	0	3	1,2
nontrivial semidirect product of Z4 and Z4	4	10	$16\Gamma _{2}c_{2}$	2	2	0	3	0	3	1,2
direct product of Z8 and Z2	5	4	$(31)$	1	2	0	1	2	3	1,2
M16	6	11	$16\Gamma _{2}d$	2	2	0	1	2	3	1,2	direct product of Z4 and Z2 in M16, Z8 in M16
dihedral group:D16	7	12	$16\Gamma _{3}a_{1}$	3	2	0	0	1	1	1	Z8 in D16
semidihedral group:SD16	8	13	$16\Gamma _{3}a_{2}$	3	2	0	0	1	1	1	Z8 in SD16
generalized quaternion group:Q16	9	14	$16\Gamma _{3}a_{3}$	3	2	0	0	1	1	1	Z8 in Q16
direct product of Z4 and V4	10	2	$(21^{2})$	1	3	1	6	0	7	1,6
direct product of D8 and Z2	11	6	$16\Gamma _{2}a_{1}$	2	3	2	1	0	3	1,2
direct product of Q8 and Z2	12	7	$16\Gamma _{2}a_{2}$	2	3	0	3	0	3	3
central product of D8 and Z4	13	8	$16\Gamma _{2}b$	2	3	0	3	0	3	3
elementary abelian group:E16	14	1	$(1^{4})$	1	4	15	0	0	15	15

We now construct a table derived from the above, that lists the total number of abelian normal subgroups of order eight and exponent bounded by some specific number (2, 4, or 8).

Note that the number of abelian normal subgroups of order eight and exponent dividing two need not be odd. However, the number of abelian normal subgroups of order eight and exponent dividing four must be odd, and so must the number of abelian normal subgroups of order eight and exponent dividing eight. See congruence condition on number of abelian subgroups of order eight and exponent dividing four and congruence condition on number of abelian subgroups of prime-cube order.

Group	GAP ID 2nd part	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Minimum size of generating set	Number of abelian (normal) subgroups of order 8, exponent dividing 2	Number of abelian (normal) subgroups of order 8, exponent dividing 4 must be odd	Number of abelian (normal) subgroups of order 8, exponent dividing 8 must be odd
cyclic group:Z16	1	5	$(4)$	1	1	0	0	1
direct product of Z4 and Z4	2	3	$(2^{2})$	1	2	0	3	3
SmallGroup(16,3)	3	9	$16\Gamma _{2}c_{1}$	2	2	1	3	3
nontrivial semidirect product of Z4 and Z4	4	10	$16\Gamma _{2}c_{2}$	2	2	0	3	3
direct product of Z8 and Z2	5	4	$(31)$	1	2	0	1	3
M16	6	11	$16\Gamma _{2}d$	2	2	0	1	3
dihedral group:D16	7	12	$16\Gamma _{3}a_{1}$	3	2	0	0	1
semidihedral group:SD16	8	13	$16\Gamma _{3}a_{2}$	3	2	0	0	1
generalized quaternion group:Q16	9	14	$16\Gamma _{3}a_{3}$	3	2	0	0	1
direct product of Z4 and V4	10	2	$(21^{2})$	1	3	1	7	7
direct product of D8 and Z2	11	6	$16\Gamma _{2}a_{1}$	2	3	2	3	3
direct product of Q8 and Z2	12	7	$16\Gamma _{2}a_{2}$	2	3	0	3	3
central product of D8 and Z4	13	8	$16\Gamma _{2}b$	2	3	0	3	3
elementary abelian group:E16	14	1	$(1^{4})$	1	4	15	15	15

Abelian normal subgroups of order 4

We note that all groups of order 4 are abelian, so congruence condition on number of subgroups of given prime power order yields that the number of abelian normal subgroups of order 4 is congruent to 1 mod 2, i.e., it is odd.

Below is the information on abelian normal subgroups of order 4:

Group	GAP ID 2nd part	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Minimum size of generating set	Number of normal subgroups isomorphic to Klein four-group	Number of normal subgroups isomorphic to cyclic group:Z4	Total number of abelian normal subgroups of order 4	Orbit sizes	Subgroup information
cyclic group:Z16	1	5	$(4)$	1	1	0	1	1	1	Z4 in Z16
direct product of Z4 and Z4	2	3	$(2^{2})$	1	2	1	6	7	1,6
SmallGroup(16,3)	3	9	$16\Gamma _{2}c_{1}$	2	2	3	0	3	3
nontrivial semidirect product of Z4 and Z4	4	10	$16\Gamma _{2}c_{2}$	2	2	1	2	3
direct product of Z8 and Z2	5	4	$(31)$	1	2	1	2	3	1,1,1
M16	6	11	$16\Gamma _{2}d$	2	2	1	2	3	1,1,1
dihedral group:D16	7	12	$16\Gamma _{3}a_{1}$	3	2	0	1	1	1	derived subgroup of dihedral group:D16
semidihedral group:SD16	8	13	$16\Gamma _{3}a_{2}$	3	2	0	1	1	1	derived subgroup of semidihedral group:SD16
generalized quaternion group:Q16	9	14	$16\Gamma _{3}a_{3}$	3	2	0	1	1	1
direct product of Z4 and V4	10	2	$(21^{2})$	1	3	7	4	11	3,4,4
direct product of D8 and Z2	11	6	$16\Gamma _{2}a_{1}$	2	3	5	2	7
direct product of Q8 and Z2	12	7	$16\Gamma _{2}a_{2}$	2	3	1	6	7
central product of D8 and Z4	13	8	$16\Gamma _{2}b$	2	3	3	4	7
elementary abelian group:E16	14	1	$(1^{4})$	1	4	35	0	35	35

Abelian normal subgroups of order 2

The abelian normal subgroups of order 2 are precisely the subgroups of order 2 contained inside the socle, which is the first omega subgroup of the center and is an elementary abelian 2-group. If the socle has rank $s$, the number of abelian normal subgroups of order 2 inside it is $2^{s}-1$. Thus, all the counts here are among the numbers 1,3,7,15.

Group	GAP ID 2nd part	Hall-Senior number	Hall-Senior symbol	Nilpotency class	Minimum size of generating set	Isomorphism class of center	Rank of center, call it $s$	Number of normal subgroups isomorphic to cyclic group:Z2, equals $2^{s}-1$	Orbit sizes	Subgroup information
cyclic group:Z16	1	5	$(4)$	1	1	cyclic group:Z16	1	1	1	Z2 in Z16
direct product of Z4 and Z4	2	3	$(2^{2})$	1	2	direct product of Z4 and Z4	2	3	3
SmallGroup(16,3)	3	9	$16\Gamma _{2}c_{1}$	2	2	Klein four-group	2	3	1,2	derived subgroup of SmallGroup(16,3), PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
nontrivial semidirect product of Z4 and Z4	4	10	$16\Gamma _{2}c_{2}$	2	2	Klein four-group	2	3	1,1,1	[SHOW MORE] derived subgroup of nontrivial semidirect product of Z4 and Z4, central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4, subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4
direct product of Z8 and Z2	5	4	$(31)$	1	2	direct product of Z8 and Z2	2	3	1,2
M16	6	11	$16\Gamma _{2}d$	2	2	cyclic group:Z4	1	1	1	derived subgroup of M16
dihedral group:D16	7	12	$16\Gamma _{3}a_{1}$	3	2	cyclic group:Z2	1	1	1	center of dihedral group:D16
semidihedral group:SD16	8	13	$16\Gamma _{3}a_{2}$	3	2	cyclic group:Z2	1	1	1	center of semidihedral group:SD16
generalized quaternion group:Q16	9	14	$16\Gamma _{3}a_{3}$	3	2	cyclic group:Z2	1	1	1	center of semidihedral group:SD16
direct product of Z4 and V4	10	2	$(21^{2})$	1	3	direct product of Z4 and V4	3	7	1,6
direct product of D8 and Z2	11	6	$16\Gamma _{2}a_{1}$	2	3	Klein four-group	2	3	1,2	derived subgroup of direct product of D8 and Z2, PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
direct product of Q8 and Z2	12	7	$16\Gamma _{2}a_{2}$	2	3	Klein four-group	2	3	1,2	derived subgroup of direct product of Q8 and Z2, PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
central product of D8 and Z4	13	8	$16\Gamma _{2}b$	2	3	cyclic group:Z4	1	1	1	derived subgroup of central product of D8 and Z4
elementary abelian group:E16	14	1	$(1^{4})$	1	4	elementary abelian group:E16	4	15	15

Abelian characteristic subgroups

For groups of order 16, there may or may not exist abelian characteristic subgroups. The situation is discussed below based on the nilpotency class:

• For an abelian group, the whole group is an abelian characteristic subgroup, and is the unique subgroup that is maximal among abelian characteristic subgroups. There may also exist other nontrivial abelian characteristic subgroups.
• For a non-abelian group of nilpotency class two, the center is an abelian characteristic subgroup of order four. There are two possibilities: either the group is a group in which every abelian characteristic subgroup is central (in which case the center is the unique maximum among abelian characteristic subgroups) or it is not, in which case there exists an abelian characteristic subgroup of order eight, which is a self-centralizing subgroup and hence a critical subgroup. The former case arises for direct product of Q8 and Z2 (ID: (16,12)) and central product of D8 and Z4 (ID: (16,13)) and the latter case (i.e., the existence of an abelian characteristic subgroup of order 8) occurs for all the remaining groups of nilpotency class two: SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4, M16, and direct product of D8 and Z2.
• For a group of nilpotency class exactly three, there is a unique subgroup maximal among abelian characteristic subgroups, which is also the unique abelian normal subgroup of order eight. This is the centralizer of derived subgroup and in all cases, it is isomorphic to cyclic group:Z8. It is a self-centralizing subgroup and hence is a critical subgroup.