i know $\displaystyle SL_2(F_3)$ is solvable.

it's sylow 2-subgroup is isomorphic to $\displaystyle Q_8$, which pretty much wraps it up (this group contains the center, to finish off the composition series).

is there an easier way to demonstrate this group is solvable, without using the above isomorphism? can we show, for example, the sylow 2-subgroup is normal, without ever calculating any matrix products, and can we prove the center is contained in this group?