>> So, in summary, G *definitely* has subgroups for every prime factor, 
>> and *might* have subgroups for some or all of the composite factors as 
>> well?
> 
> I believe that is correct.  In addition I think that G must have a 
> subgroup for every prime power which still divides the order of G. 
> Weather or not the composite factors have associated subgroups will 
> indeed depend on the particular group G represents.
> 
> As an example consider the group of rotations of a tetrahedron.  It has 
> 12 elements but no subgroup of order 6.

http://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)

It seems that for a *commutative* group, my original statements are 
correct. I found many other interesting properties, but failed to find 
any confirmation of your assertion about prime subgroup orders...

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