In general, for what $n$ do there exist two groups of order $n$? How about three groups of order $n$?
I know that if $n$ is prime, there only exists one group of order $n$, by Lagrange's Theorem, but how do you classify all other such $n$ that have $2, 3, 4, ...$ groups?
This question came to me during a group theory class, when we were making a table of groups of order $n$. For instance, all groups of order $4$ are isomorphic to either $C_4$ or $C_2\times C_2$.