Number of groups of a given order 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 $C_2\times C_2$. 

Exactly 2 groups. There is a paper, which claims to classify "Orders for which there exist exactly two groups".
This link contains the text of the paper in text (!) format. I didn't find a pdf.
Disclamer: I didn't check if the proofs in the paper are correct. I also don't know, if the paper was published in any peer-reviewed journal (probably it wasn't).
Exactly 3 groups. Proposition 2 of the paper above says, that if $n$ is not cube-free, then there are at least 5 groups of the order $n$. So you are only interested in cube-free numbers. Then you at least know all numbers $n$ with exactly 3 groups for $n<50000$ from here. Based on this one will know the probable answer and then one can try to prove it.
