Is there a non-solvable number NOT divisible by $3\ $?

Question

Here

https://oeis.org/A056866

it is claimed that every non-solvable number is divisible by $4$ and either $3$ or $5$. However, I did not find a number in the list not divisible by $3$.

So, my question :

Is there a non-solvable number NOT divisble by $3$ ?

Answer 1

The encyclopedia of integer sequences has a list of possible criteria for a number being a non-solvable number, nicely laid out here on MathOverflow. One of these is $2^{2p}(2^{2p}+1)(2^p-1)$, p odd prime.

The Suzuki groups are the only finite non-abelian simple groups with order not divisible by 3, and they have order $$2^{2(2n+1)}(2^{2(2n+1)} + 1)(2^{(2n+1)} −1)$$ which matches this criterion.

As mentioned by Derek Holt, it's also worth noting that since $2^{2(2n+1)}+1$ is divisible by $5$, it follows that all non-solvable groups whose order is not divisible by $3$ must be divisible by $5$.