Checking the list of finite simple groups, it seemed to me that all groups have order a multiple of $3$. This clear for alternating groups and checked case by case for sporadic groups. For groups of Lie type it looked like the orders are always multiples of $q(q^2 - 1)$ for a prime power $q$, and this quantity is always a multiple of $3$.
Upon closer inspection there is an outlier, namely the Suzuki groups. Are these the only exception? Is there a reason why this is the case, or is it just a corollary of the classification?
I have seen that there are many constructions of the Suzuki groups. Could you recommend a reference to read about them?
First of all : We must assume that the group is non-abelian, otherwise the cyclic groups with prime order , except $\mathbb Z_3$ , would already be counterexamples.
The Suzuki groups have orders not divisible by $3$, $5$ is however always a prime factor of the order of those groups. All the other finite simple non-abelian groups have order divisible by $3$.