i think an example of such a group would be Q8. Q8/Z(Q8) is of order 4, and is isomorphic to the klein 4-group, but any subgroup of Q8 of order 4 is cyclic. and in point of fact, Q8 cannot be realized as a semi-direct product of subgroups; even though Q8/<i> (for example) is isomorphic to {1,-1}, we have {1,-1} ∩ <i> = {1,-1}, so Q8 doesn't "split" over <i> (a similar logic holds for <j> and <k>). it also doesn't "split" over Z(Q8), for essentially the same reason.

(as this shows G/H can be isomorphic to a subgroup of G even if we don't have a semi-direct product).

of course, 2 is prime, so the result Amer wants to prove is just a simple example of the sylow theorems, or (even more pointedly) of cauchy's theorem for finite groups.