Maximal normal $2$-subgroup of $GL(2,3)$.

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 .

• If it has a normal subgroup $N$ of order $8$, and its Sylow $2$-subgroup of order $16$ is not normal, then $N$ must be its largest normal $2$-subgroup. BTW, it has three Sylow $2$-subgroups. – Derek Holt Jan 28 at 8:13
• @DerekHolt I am also saying that it’s Sylow subgroups are not normal – neelkanth Jan 28 at 8:27
• Yes it’s has three Sylow subgroups ... I will edit ... – neelkanth Jan 28 at 8:29
• @DerekHolt so my argument is correct ? – neelkanth Jan 28 at 8:31
• If it had two normal subgroups of order eight, $N_1$ and $N_2$. Then $N_1N_2$ would be a subgroup. Furthermore, it would be normal. And its order would be $8^2/|N_1\cap N_2|$. Implying that $N_1N_2$ would have to be a Sylow 2-subgroup. But that wasn't normal, so... – Jyrki Lahtonen Jan 28 at 8:57