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Is it possible to classify those groups whose order is $n^2$ for some natural number $n$ but which do not have any subgroups of order $n$?

To be a bit more specific (in case a full classification is not known): Will a solvable group of order $n^2$ always have a subgroup of order $n$?

In fact, I have not been able to find any examples at all of such groups (obviously, the order would have to be fairly large).

My motivation for asking is that in my question Sudokus as composition tables of finite groups it turned out that having a subgroup of order $n$ is sufficient for being able to arrange the composition table to form a sudoku, but it seems unknown if this condition is necessary, and a natural starting point for determining if it is would be to look at examples.

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Tobias Kildetoft
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    Someone else had the same bright idea :) –  Jul 30 '14 at 18:28
  • @T.Bongers Wow, I can't believe I somehow missed that question. I noticed some upvotes on my old question and recalled that I had thought of this one back then but never posted it. I should have known it might have been due to a question linking it (I didn't even check the linked questions). – Tobias Kildetoft Jul 30 '14 at 19:57

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