Subgroup structure of groups of order 24

This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 24.
View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 24

To understand these in a broader context, see subgroup structure of groups of order 3.2^n | subgroup structure of groups of order 2^3.3^n

Numerical information on counts of subgroups by order

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Note that, by Lagrange's theorem, the order of any subgroup must divide the order of the group. Thus, the order of any proper nontrivial subgroup is one of the numbers 2,4,8,3,6,12.

Here are some observations on the number of subgroups of each order:

1. Congruence condition on number of subgroups of given prime power order: In particular:
   • The number of subgroups of order 2 is congruent to 1 mod 2 (i.e., it is odd), and the number of conjugacy classes is positive.
   • The number of subgroups of order 4 is congruent to 1 mod 2 (i.e., it is odd), and the number of conjugacy classes is positive.
   • The number of subgroups of order 8 is congruent to 1 mod 2 (i.e., it is odd), and the number of conjugacy classes is positive.
   • The number of subgroups of order 3 is congruent to 1 mod 3, and the number of conjugacy classes is positive.
2. By the fact that Sylow implies order-conjugate, we obtain that Sylow number equals index of Sylow normalizer, and in particular, divides the index of the Sylow subgroup. Combined with (1), we get the following:
   • The number of 2-Sylow subgroups (subgroups of order 8) is either 1 or 3, and there is a unique conjugacy class of such subgroups.
   • The number of 3-Sylow subgroups (subgroups of order 3) is either 1 or 4, and there is a unique conjugacy class of such subgroups.
3. In the case of a finite nilpotent group, the number of subgroups of a given order is the product of the number of subgroups of order equal to each of its maximal prime power divisors, in the corresponding Sylow subgroup. In particular, we get (number of subgroups of order 3) = 1, (number of subgroups of order 6) = (number of subgroups of order 2), (number of subgroups of order 12) = (number of subgroups of order 4), and (number of subgroups of order 8) = 1.
4. In the special case of a finite abelian group, we have (number of subgroups of order 3) = (number of subgroups of order 8) = 1, and (number of subgroups of order 2) = (number of subgroups of order 4) = (number of subgroups of order 6) = (number of subgroups of order 12). This is because subgroup lattice and quotient lattice of finite abelian group are isomorphic.
5. Finite supersolvable implies subgroups of all orders dividing the group order: For any finite supersolvable group, there are subgroups of every possible order, i.e., there are proper nontrivial subgroups of orders 2,3,4,6,8,12. All finite nilpotent groups are supersolvable.
6. In particular, by (2), the normalizer of the 3-Sylow subgroup is nontrivial (order either 6 or 24). Therefore, there exists an element of order 2 normalizing a 3-Sylow subgroup, and so we obtain that there must exist a subgroup of order 6.

Table of number of subgroups

Group	Second part of GAP ID (ID is (24,second part))	Nilpotency class	Supersolvable?	Derived length	Number of subgroups of order 2 (must be odd by (1))	Number of subgroups of order 4 (must be odd by (1))	Number of subgroups of order 8 (must be 1 or 3 by (2))	Number of subgroups of order 3 (must be 1 or 4 by (2))	Number of subgroups of order 6 (must be positive by (5))	Number of subgroups of order 12	Total number of subgroups (includes trivial subgroup and whole group)
nontrivial semidirect product of Z3 and Z8	1	not nilpotent	Yes	2	1	1	3	1	1	1	10
cyclic group:Z24	2	1	Yes	1	1	1	1	1	1	1	8
special linear group:SL(2,3)	3	not nilpotent	No	3	1	3	1	4	4	0	15
dicyclic group:Dic24	4	not nilpotent	Yes	2	1	7	3	1	1	3	18
direct product of S3 and Z4	5	not nilpotent	Yes	2	7	7	3	1	3	3	26
dihedral group:D24	6	not nilpotent	Yes	2	13	7	3	1	3	3	32
direct product of Dic12 and Z2	7	not nilpotent	Yes	2	3	7	3	1	3	3	22
semidirect product of Z3 and D8 with action kernel V4	8	not nilpotent	Yes	2	9	7	3	1	5	3	30
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	Yes	1	3	3	1	1	3	3	16
direct product of D8 and Z3	10	2	Yes	2	5	3	1	1	5	3	20
direct product of Q8 and Z3	11	2	Yes	2	1	3	1	1	1	3	12
symmetric group:S4	12	not nilpotent	No	3	9	7	3	4	4	1	30
direct product of A4 and Z2	13	not nilpotent	No	2	7	7	1	4	4	1	26
direct product of D12 and Z2 (also direct product of S3 and V4)	14	not nilpotent	Yes	2	15	19	3	1	7	7	54
direct product of E8 and Z3	15	1	Yes	1	7	7	1	1	7	7	32
Possibility set	--	1, 2 if nilpotent	Yes, No	1, 2, 3	1, 3, 5, 7, 9, 13, 15	1, 3, 7, 19	1, 3	1, 4	1, 3, 4, 5, 7	0, 1, 3, 7

Table of number of conjugacy classes of subgroups

Group	Second part of GAP ID (ID is (24,second part))	Nilpotency class	Supersolvable?	Derived length	Number of conjugacy classes of subgroups of order 2 (must be positive)	Number of conjugacy classes of subgroups of order 4 (must be positive)	Number of conjugacy classes of subgroups of order 8 (must be 1)	Number of conjugacy classes of subgroups of order 3 (must be 1)	Number of conjugacy classes of subgroups of order 6 (must be positive by (5))	Number of conjugacy classes of subgroups of order 12	Total number of conjugacy classes of subgroups (includes trivial subgroup and whole group
nontrivial semidirect product of Z3 and Z8	1	not nilpotent	Yes	2	1	1	1	1	1	1	8
cyclic group:Z24	2	1	Yes	1	1	1	1	1	1	1	8
special linear group:SL(2,3)	3	not nilpotent	No	3	1	1	1	1	1	0	7
dicyclic group:Dic24	4	not nilpotent	Yes	2	1	3	1	1	1	3	12
direct product of S3 and Z4	5	not nilpotent	Yes	2	3	3	1	1	3	3	16
dihedral group:D24	6	not nilpotent	Yes	2	3	3	1	1	3	3	16
direct product of Dic12 and Z2	7	not nilpotent	Yes	2	3	3	1	1	3	3	16
semidirect product of Z3 and D8 with action kernel V4	8	not nilpotent	Yes	2	3	3	1	1	3	3	16
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	Yes	1	3	3	1	1	3	3	16
direct product of D8 and Z3	10	2	Yes	2	3	3	1	1	3	3	16
direct product of Q8 and Z3	11	2	Yes	2	1	3	1	1	1	3	12
symmetric group:S4	12	not nilpotent	No	3	2	3	1	1	1	1	11
direct product of A4 and Z2	13	not nilpotent	No	2	3	2	1	1	2	1	12
direct product of D12 and Z2 (also direct product of S3 and V4)	14	not nilpotent	Yes	2	7	7	1	1	7	7	32
direct product of E8 and Z3	15	1	Yes	1	7	7	1	1	7	7	32
Possibility set	--	1, 2 if nilpotent	Yes, No	1, 2, 3	1, 2, 3, 7	1, 2, 3, 7	1	1	1, 2, 3, 7	0, 1, 3, 7	7, 8, 11, 12, 16, 32

Table of number of isomorphism classes of subgroups

The number of isomorphism classes is bounded from above both by the number of conjugacy classes and the number of isomorphism classes of all groups of that order. Also, if subgroups of the order exist, the number of isomorphism classes is at least 1. This restricts the number of isomorphism classes fairly narrowly.

Group	Second part of GAP ID (ID is (24,second part))	Nilpotency class	Supersolvable?	Derived length	Number of isomorphism classes of subgroups of order 2 (must be 1)	Number of isomorphism classes of subgroups of order 4 (must be 1 or 2)	Number of isomorphism classes of subgroups of order 8 (must be 1)	Number of isomorphism classes of subgroups of order 3 (must be 1)	Number of isomorphism classes of subgroups of order 6 (must be 1 or 2)	Number of isomorphism classes of subgroups of order 12	Total number of isomorphism classes of subgroups
nontrivial semidirect product of Z3 and Z8	1	not nilpotent	Yes	2	1	1	1	1	1	1	8
cyclic group:Z24	2	1	Yes	1	1	1	1	1	1	1	8
special linear group:SL(2,3)	3	not nilpotent	No	3	1	1	1	1	1	0	7
dicyclic group:Dic24	4	not nilpotent	Yes	2	1	1	1	1	1	1	8
direct product of S3 and Z4	5	not nilpotent	Yes	2	1	2	1	1	2	2	11
dihedral group:D24	6	not nilpotent	Yes	2	1	2	1	1	2	2	11
direct product of Dic12 and Z2	7	not nilpotent	Yes	2	1	2	1	1	2	2	13
semidirect product of Z3 and D8 with action kernel V4	8	not nilpotent	Yes	2	1	2	1	1	2	3	12
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	Yes	1	1	2	1	1	1	2	10
direct product of D8 and Z3	10	2	Yes	2	1	2	1	1	1	2	10
direct product of Q8 and Z3	11	2	Yes	2	1	1	1	1	1	1	8
symmetric group:S4	12	not nilpotent	No	3	1	2	1	1	1	1	9
direct product of A4 and Z2	13	not nilpotent	No	2	1	1	1	1	1	1	8
direct product of D12 and Z2 (also direct product of S3 and V4)	14	not nilpotent	Yes	2	1	1	1	1	2	2	10
direct product of E8 and Z3	15	1	Yes	1	1	1	1	1	1	1	8
Possibility set	--	1, 2 if nilpotent	Yes, No	1, 2, 3	1	1, 2	1	1	1, 2	0, 1, 2, 3	7, 8, 9, 10, 11, 12, 13

Sylow subgroups

2-Sylow subgroups

Here is the occurrence summary:

Group of order 8	GAP ID (second part)	Information on fusion systems	Number of groups of order 24 in which it is a 2-Sylow subgroup with a normal complement (i.e., uses inner fusion system) -- equivalently, the whole group is 2-nilpotent	List of these groups	Second part of GAP IDs of these groups	Number of groups of order 24 in which it is a 2-Sylow subgroup without a normal complement (i.e., uses one of the outer fusion systems)	List of these groups	Second part of GAP IDs of these groups
cyclic group:Z8	1	--	2	nontrivial semidirect product of Z3 and Z8, cyclic group:Z24	1, 2	0	--	--
direct product of Z4 and Z2	2	--	3	direct product of S3 and Z4, direct product of Dic12 and Z2, direct product of Z6 and Z4	5, 7, 9	0	--	--
dihedral group:D8	3	fusion systems for dihedral group:D8	3	dihedral group:D24, semidirect product of Z3 and D8 with action kernel V4, direct product of D8 and Z3	6, 8, 10	1	symmetric group:S4	12
quaternion group	4	fusion systems for quaternion group	2	dicyclic group:Dic24, direct product of Q8 and Z3	4, 11	1	special linear group:SL(2,3)	3
elementary abelian group:E8	5	fusion systems for elementary abelian group:E8	2	direct product of D12 and Z2, direct product of E8 and Z3	14, 15	1	direct product of A4 and Z2	13

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

Group	Second part of GAP ID (ID is (24,second part))	2-Sylow subgroup	Second part of GAP ID	Number of 2-Sylow subgroups	Number of 3-Sylow subgroups (=1 iff the group is 2-nilpotent, i.e., the 2-Sylow subgroup is a retract, i.e., it has a normal complement and the whole group is a semidirect product)
nontrivial semidirect product of Z3 and Z8	1	cyclic group:Z8	1	3	1
cyclic group:Z24	2	cyclic group:Z8	1	1	1
special linear group:SL(2,3)	3	quaternion group	4	1	4
dicyclic group:Dic24	4	quaternion group	4	3	1
direct product of S3 and Z4	5	direct product of Z4 and Z2	2	3	1
dihedral group:D24	6	dihedral group:D8	3	3	1
direct product of Dic12 and Z2	7	direct product of Z4 and Z2	2	3	1
semidirect product of Z3 and D8 with action kernel V4	8	dihedral group:D8	3	3	1
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	direct product of Z4 and Z2	2	1	1
direct product of D8 and Z3	10	dihedral group:D8	3	1	1
direct product of Q8 and Z3	11	quaternion group	4	1	1
symmetric group:S4	12	dihedral group:D8	3	3	4
direct product of A4 and Z2	13	elementary abelian group:E8	5	1	4
direct product of D12 and Z2 (also direct product of S3 and V4)	14	elementary abelian group:E8	5	3	1
direct product of E8 and Z3	15	elementary abelian group:E8	5	1	1

3-Sylow subgroups

Note that the 3-Sylow subgroup is isomorphic to cyclic group:Z3 in all cases. By the congruence condition on Sylow numbers as well as the divisibility condition on Sylow numbers, the only possibilities for the number of 3-Sylow subgroups is 1 or 4. In the former case, we have a normal Sylow subgroup. In the latter case, the normalizer of the Sylow subgroup has order 6, and is thus either cyclic group:Z6 or symmetric group:S3.

Group	Second part of GAP ID (ID is (24,second part))	Number of 3-Sylow subgroups	Normalizer of Sylow subgroup
nontrivial semidirect product of Z3 and Z8	1	1	whole group
cyclic group:Z24	2	1	whole group
special linear group:SL(2,3)	3	4	cyclic group:Z6
dicyclic group:Dic24	4	1	whole group
direct product of S3 and Z4	5	1	whole group
dihedral group:D24	6	1	whole group
direct product of Dic12 and Z2	7	1	whole group
semidirect product of Z3 and D8 with action kernel V4	8	1	whole group
direct product of Z6 and Z4 (also, direct product of Z12 and Z2)	9	1	whole group
direct product of D8 and Z3	10	1	whole group
direct product of Q8 and Z3	11	1	whole group
symmetric group:S4	12	4	symmetric group:S3
direct product of A4 and Z2	13	4	cyclic group:Z6
direct product of D12 and Z2 (also direct product of S3 and V4)	14	1	whole group
direct product of E8 and Z3	15	1	whole group