The general question is this: given a positive integer $n$, are there any non-trivial lower bounds on the number of subgroups of a group of order $n$?
Some more specific thinking: we know that in the case of cyclic groups, the number of subgroups is equal to the number of divisors of the number of elements in the group. Among all groups of the same order, does the cyclic group have the smallest number of subgroups? To put it another way, is it true that any group of order $n$ has at least $d(n)$ subgroups, where $d(n)$ is the number of divisors of $n$?
I'd appreciate any help with regard to the above questions.