A a matter of historical interest, R. Brauer classified finite groups of order divisible by a prime $p$ to the first power, but of order less than $p^{3}$ for some prime $p,$ using some block theory. This was later generalized by M. Herzog, using the Theory of blocks with cyclic defect group,  to classify finite groups with a cyclic Sylow $p$-subgroup of order $p^{a},$ but which have order less than $p^{3a}.$ Both of these results predate the classification of finite simple groups by some margin. Brauer's classification would suffice to deal with this question.

The only non-Abelian simple groups which appear in Brauer's list are ${\rm PSL}(2,p)$ and ${\rm SL}(2,p-1)$ (the latter only when $p$ is a Fermat prime).