I have a non-Lagrangian group $G$ of order $pq^3$, $Q$ a Sylow $q$-subgroup of G and a $H$ a subgroup of $Q$ with $|H|=q^2$. It is clear that $Q \subseteq N_G(H)$. I must prove that $G$ doesn't posses subgroups of order $pq^2$.
I supposed that $G$ posses a subgroup $P$ of order $pq^2$. $P$ will posses a normal Sylow $p$-subgroup or a normal Sylow $q$-subgroup. In the first case, $P$ will contain a subgroup of order $pq$ , making $G$ a Lagrangian group, a contradiction. But I don't find a way for the second case.