The general linear group $GL(2,3) $ of $2\times 2$ matrices over modulo $3$ does not have normal Sylow $2$-subgroup as it has three Sylow $2$-subgroups. I have to find the maximal normal $2$- subgroup of $GL(2,3)$. According to me it has normal $2$- subgroup of order $8$, one can see https://groupprops.subwiki.org/wiki/2-core_of_general_linear_group:GL(2,3). Can I say it is maximal normal $2$- subgroup. Please suggest me . Thanks .