About the order of finite simple groups there exists a very interesting result which stated as follows:

Let $G$ be an non-solvable simple group of order $g$. If $p\mid g$, where $p>g^{1\over 3}$ is a prime, then $p>3$ and either $G\cong L_2(p)$ or $p$ is a Fermat prime and $G\cong L_2(p-1)$.

This result proved by Brauer in 1958 and surely they proved it directly. Now using the classification of finite simple groups, we may prove this result too.

As a similar result we need all finite simple groups $S$ such that $S$ is of order $g$ and $p\mid g$, also $g$ is a divisor of ${p(p-1)^2(p+1)\over 4}<{p^4\over 4}$. I guess that if $p>5$, then $L_2(p)$ is the only answer. Any references or hints would be highly appreciated. Thanks in advance.