Let $G$ be a finite group and $n > 1$ a divisor of $|G|$. Let $P_n(G)$ be the subgroup of $G$ which is generated by all elements which satisfy $x^n = 1$. Since $n > 1$, by Cauchy's theorem we have $P_n(G) > 1$. It is also easy to see that $P_n(G)$ is a characteristic subgroup of $G$. Hence if $G$ is characteristically simple (ie. direct product of isomorphic simple groups), then $P_n(G) = G$ for all $n > 1$, $n$ divisor of $|G|$ . There are plenty of other examples in finite $p$-groups, dihedral group $D_8$ also has this property.
My question: Is there a finite group $G$ that is not characteristically simple or a $p$-group, but for which $P_n(G) = G$ for every divisor $n > 1$ of $|G|$?