How to find all groups that have exactly 3 subgroups?
Any group must have identity and itself as subgroups, so we just need to find all the groups that only have one proper subgroup. I think that for a prime $p$ the group $\mathbb Z/p^2\mathbb Z$ has only one proper subgroup (for example, $\mathbb Z/4\mathbb Z$, $\mathbb Z/9\mathbb Z$). Are there any other possibilities?