TITLE: Groups of order $n^2$ that have no subgroup of order $n$
QUESTION [10 upvotes]: For which $n$ is there a group of order $n^2$ without a subgroup of order $n$.
Such groups can not be nilpotent. 
This question is related to Sudokus as composition tables of finite groups.

REPLY [8 votes]: Just to expand my comment, the smallest $n$ for which this occurs is $n=24$, and the single example of a group of order $24^2$ with no subgroup of order $24$ is ${\mathtt{SmallGroup}}(576,8661)$, a semidirect product $N \rtimes C_9$, where $N$ is elementary abelian of order $64$, and the $C_9$ acts irreducibly on $N$.
It is a Frobenius group of degree $64$ and can also be accessed as $\mathtt{PrimitiveGroup}(64,1)$.