TITLE: If a group has exactly 24 elements of order 6, how many subgroups of order 6 does it have?
QUESTION [6 upvotes]: I have no idea how the two are related. I can figure out that the group should have at least 30 elements because the number has to be more than 24 and divisible by 6.

REPLY [6 votes]: Edit: The problem is not well-determined. The group $C_6 \times C_6$ has 24 elements of order 6, and 12 subgroups of order 6. The group $S_3 \times C_3 \times C_3$ also has 24 elements of order 6, but beyond the 12 cyclic subgroups this necessarily gives, it also has the subgroup $S_3 \times \{0\} \times \{0\}$.
The following two hints lead to an answer only under additional assumptions: e.g., $G$ is abelian, or we consider only cyclic subgroups.
Hint 1: How many elements of order 6 are there in a subgroup of order 6?
Hint 2: Can an element of order 6 belong to more than one subgroup of order 6?