given any $n\in\mathbb{N}$ how many non isomorphic groups are there Just for curiosity: given any $n\in\mathbb{N}$  how many non isomorphic groups are there? Is that number finite?countably infinite?uncountable? I know that depending on the $n$ it is finite for some cases though.
Thank you for discussion. 
 A: Well, there is always finite since the number of maps $G\times G\to G$ is finite--so there are only finitely many groups of order $n$, let alone non-isomorphic groups of order $n$. 
A: The number of distinct abstract groups of given finite order $n$ is discussed in a very interesting paper by John H. Conway, Heiko Dietrich and E.A. O’Brien. Check the table at the end of the article (the number of groups for each order < 2048)! Observe that, as Tobias pointed out, the number of groups of prime power orders are substantially larger.
A: For most orders, it is not known how many groups there are of that order. It is expected (though nothing concrete has been proven that I know), that there tends to be many more groups of prime power orders than of other orders of the same approximate magnitude. For example, there are more than $50$ billion groups of order $1024$ and only about $100$ million groups of order at most $1023$.
http://www.math.ku.dk/~olsson/manus/three-group-numbers.pdf is a classification of those orders for which there are precisely $1$, $2$ or $3$ groups up to isomorphism. For larger numbers, the sort of arguments given there tend to become impractical.
