Orchid XP v8 wrote:
> 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.

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