Subgroup structure of groups of order 12

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This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 12.
View subgroup structure of group families | View subgroup structure of groups of a particular order |View other specific information about groups of order 12
To understand these in a broader context, see subgroup structure of groups of order 3.2^n | subgroup structure of groups of order 2^2.3^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.

In fact, we can say something even stronger. Either the 2-Sylow subgroup of the 3-Sylow subgroup is a normal Sylow subgroup, so the group is an internal semidirect product of one of its Sylow subgroups by the order. Thus, it is a metabelian group. This fact generalizes to the observation order is product of Mersenne prime and one more implies normal Sylow subgroup, with the Mersenne prime here being M_2 = 2^2 - 1 = 3 and the order being 3 \cdot 4 = 12.

We have the following constraints on the counts of subgroups. Note that (1) and (2) give constraints on the counts of groups of particular orders, whereas (3) and (4) give constraints on the relationships between the counts.

  1. Congruence condition on number of subgroups of given prime power order: If p is a prime and p^r divides the order of the group, the number of subgroups of order p^r is congruent to 1 mod p.
    • Case p = 2, r = 1: The number of subgroups of order 2 is congruent to 1 mod 2, i.e., it is odd, and the number of conjugacy classes of subgroups is positive.
    • Case p = 2, r = 2: The number of subgroups of order 4 is congruent to 1 mod 2, i.e., it is odd, and the number of conjugacy classes of subgroups is positive.
    • Case p = 3: The number of subgroups of order 3 is congruent to 1 mod 3.
  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 the congruence condition, we get the following:
    • The number of 2-Sylow subgroups subgroups of order 4) is either 1 or 3, and the number of conjugacy classes of subgroups is 1.
    • The number of 3-Sylow subgroups (subgroups of order 3)is either 1 or 4, and the number of conjugacy classes of subgroups is 1.
  3. By size considerations, we also get that at least one of the Sylow numbers must be 1, i.e., we have either a normal 2-Sylow subgroup or a normal 3-Sylow subgroup. For more, see order is product of Mersenne prime and one more implies normal Sylow subgroup.

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 1 or 3 by (2)) Number of subgroups of order 3 (must be 1 or 4 by (2)) Number of subgroups of order 6 Total number of subgroups (includes trivial subgroup and whole group)
dicyclic group:Dic12 1 not nilpotent Yes 2 1 3 1 1 8
cyclic group:Z12 2 1 Yes 1 1 1 1 1 6
alternating group:A4 3 not nilpotent No 2 3 1 4 0 10
dihedral group:D12 4 not nilpotent Yes 2 7 3 1 3 16
direct product of Z6 and Z2 5 1 Yes 1 3 1 1 3 10
Possibility set -- 1 if nilpotent Yes, No 1, 2 1, 3, 7 1, 3 1, 4 0, 1, 3 6, 8, 10, 16

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 subgroups of order 2 (must be positive) Number of subgroups of order 4 (must be 1) Number of conjugacy classes of subgroups of order 3 (must be 1) Number of isomorphism classes of subgroups of order 6 (at most 3 by size considerations) Total number of isomorphism classes subgroups (includes trivial subgroup and whole group)
dicyclic group:Dic12 1 not nilpotent Yes 2 1 1 1 1 6
cyclic group:Z12 2 1 Yes 1 1 1 1 1 6
alternating group:A4 3 not nilpotent No 2 1 1 1 0 5
dihedral group:D12 4 not nilpotent Yes 2 3 1 1 3 10
direct product of Z6 and Z2 5 1 Yes 1 3 1 1 3 10
Possibility set -- 1 if nilpotent Yes, No 1, 2 1, 3 1 1 0, 1, 3 5, 6, 10

Table of number of isomorphism classes of subgroups

The number of isomorphism classes of subgroups of a given order is bounded from above by both the number of conjugacy classes of subgroups of that order and the number of isomorphism classes of all groups of that order. Moreover, if the number of conjugacy classes is positive, so is the number of isomorphism classes.

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) Number of isomorphism classes of subgroups of order 3 (must be 1) Number of isomorphism classes of subgroups of order 6 (at most 2) Total number of isomorphism classes of subgroups (includes trivial subgroup and whole group)
dicyclic group:Dic12 1 not nilpotent Yes 2 1 1 1 1 6
cyclic group:Z12 2 1 Yes 1 1 1 1 1 6
alternating group:A4 3 not nilpotent No 2 1 1 1 0 5
dihedral group:D12 4 not nilpotent Yes 2 1 1 1 2 7
direct product of Z6 and Z2 5 1 Yes 1 1 1 1 1 6
Possibility set -- 1 if nilpotent Yes, No 1, 2 1 1 1 0, 1, 2 5, 6, 7

Sylow subgroups

2-Sylow subgroups

Here is the occurrence summary:


Group of order 4 GAP ID (second part) Information on fusion systems Number of groups of order 12 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 12 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:Z4 1 2 dicyclic group:Dic12, cyclic group:Z12 1, 2 -- -- --
Klein four-group 2 fusion systems for Klein four-group 2 dihedral group:D12, direct product of Z6 and Z2 4, 5 1 alternating group:A4 3


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 (12,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)
dicyclic group:Dic12 1 cyclic group:Z4 1 3 1
cyclic group:Z12 2 cyclic group:Z4 1 3 1
alternating group:A4 3 Klein four-group 2 1 4
dihedral group:D12 4 Klein four-group 2 3 1
direct product of Z6 and Z2 5 Klein four-group 2 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, we have a self-normalizing Sylow subgroup. You can see from the table above the cases where the number of 3-Sylow subgroups is respectively 1 and 4.