Subgroup structure of groups of order 24

Numerical information on counts of subgroups by 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:


 * Congruence condition on number of subgroups of given prime power order: The number of subgroups of order 2 is congruent to 1 mod 2 (i.e., it is odd). The same is true for the number of subgroups of order 4, as well as the number of subgroups of order 8. The number of subgroups of order 3 is congruent to 1 mod 3.
 * 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 is either 1 or 3, and the number of 3-Sylow subgroups is either 1 or 4.
 * 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.
 * 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.
 * 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.

2-Sylow subgroups
Here is the occurrence summary:

  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.

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.