Subgroup structure of groups of order 12

Numerical information on counts of subgroups by 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:

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.
 * 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 $$p = 2$$: The number of subgroups of order 2 is congruent to 1 mod 2, i.e., it is odd. Also, the number of subgroups of order 4 is odd.
 * Case $$p = 3$$: 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.