Finite groups with elements of order a prime power

Question

Consider a finite group $G$ where the order of each element is a power of a certain prime number $p$, then $G$ is a $p$-group.

My question: are there groups that are not $p$-groups, but for which the order of each element is a power of a prime number?

Answer 1

Let $G=\mathfrak{S}_3$ be the group of permutations of $\{1,2,3\}$. The order of the elements is either $2$ or $3$. (Or $1$ for the trivial permutation).