[Math] Proper subgroups of non-cyclic p-group cannot be all cyclic

Question

Prove or disprove?

I'm leaning towards it being true but not sure. Any hint would be greatly appreciated.

In case of it being false, i.e a non-cyclic p-group can have all cyclic proper subgroups, is there any way to count them?

Answer 1

If you take the smallest non-cyclic $p$-group, then its proper subgroups are smaller $p$-groups and thus have to be cyclic. So, it can happen.