I find that there are either $1$ or $3$ Sylow-$2$ subgroups, and $1, 4$ or $16$ Sylow-$3$ subgroups. I need one of them to be $1$, so that it is normal, so I can mod out by it and have $2$ $p$-groups $N$, $G/N$, which are thus both solvable which implies that $G$ is solvable.
Hint: If $G$ has $3$ Sylow $2$-subgroups, then $G$ acts non-trivially on them by conjugation, therefore there is a non-trivial mapping $f:G\to S_3$. Then $|G/\ker f|=2,3,6$. This action is transitive, so $|G/\ker f|=3,6$ and $|\ker f|=16,8$ and both these groups are solvable.