Characterization of $L_3(2^n)$ by Sylow 2-subgroups

The following theorem is proved: If the Sylow 2-subgroups of a finite simple group $G$ are isomorphic to the Sylow 2-subgroups of $L_3(2^n)$, $n\geqslant2$, then $G$ is isomorphic with $L_3(2^n)$.