# General lower bound on the number of subgroups of a finite group

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.

• @YCor Thanks for your advice. I think I've made the question clearer now. Sep 29 at 6:27
• There's an answer to this on Math Stack Exchange, by Derek Holt. (The answer is yes, any group of order $n$ has at least $d(n)$ subgroups.) Sep 29 at 7:07
• I still do not know whether there are any published proofs of this result anywhere. It would seem very surprising if there were not! Sep 29 at 8:17