Finding all the groups of an order up to isomorphism In general, how does one best approach a question of the form, "Find all the groups of order n up to isomorphism?"
Abelian groups seem to be easier to find.
The first step is factoring out n, but beyond that? Could someone outline a good set of criteria to look at for n or direct me to a good reading? 
 A: For finite abelian groups, there is indeed a nice classification based on a decomposition of cyclic groups (this is sometimes called the fundamental theorem of abelian groups.
For non-abelian groups, usually the Sylow theorems come to mind.  These theorems allow you to calculate special subgroups of prime power order, based on the prime decomposition of your order $n$.  In special cases, the Sylow theorems give you a complete classification of groups of a particular order.  For example, the theorems can be used to show that any group of order $n = pq$, where $p,q$ are distinct primes, is a (semi-) direct product of cyclic groups.
And while not a "classification," some more advanced topics, such as character theory (representation theory), are used to prove seemingly simple results in finite group theory.  For example, Burnside's $p^aq^b$ lemma is used using characters and --- as far as I know --- the character theory used is the simplest way to prove the result.  
You can find the first two results, for sure, in any self-respecting algebra book, such as Artin's Algebra, Dummit & Foote's Abstract Algebra, etc.  The last result I mentioned is in Dummit & Foote, I think, if you're interested.
