A follow-up question to Alternating subgroups of $\mathrm{SU}_n $.
$\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PSU_m $ be the projective unitary group, a compact simple adjoint Lie group corresponding to the root system $ A_{m-1} $.
Let $ \PSL_n(q) $ be the finite simple group of Lie type $ A_{n-1}(q) $ given by taking the special linear group with entries from the field with $ q $ elements and modding out by the center.
Let $ \PSU_n(q^2) $ be the finite simple group of Lie type $ ^2 A_{n-1}(q^2) $ given by taking the special unitary group with entries from the field with $ q^2 $ elements and modding out by the center.
$ \PSU_2 $ contains a $ 60 $ element subgroup isomorphic to $ \PSL_2(4) \cong \PSL_2(5) $. It is the largest of the exceptional finite subgroups of $ \PSU_2 $ and it is maximal (the only closed subgroup containing it is the whole group).
The references in The finite subgroups of SU(n) show that $ \PSU_3 $ contains a subgroup of order $ 360 $ isomorphic to $ \PSL_2(9) $ and that again this subgroup is the largest of the exceptional finite subgroups. Again it is maximal. Also $ \PSU_3 $ contains a maximal group of order $ 168 $ isomorphic to $ \PSL_2(7)\cong \PSL_3(2) $. There is also a 60 element $ \PSL_2(4) \cong \PSL_2(5) $ subgroup of $ \PSU_3 $ but it is unclear if it is maximal.
The reference Hanany and He - A Monograph on the Classification of the Discrete Subgroups of SU(4) from the same MO question shows that $ \PSU_4 $ contains a subgroup of order $ 25{,}920 $ isomorphic to $ \PSU_4(4) $ and that again this subgroup is the largest of the exceptional finite subgroups. Again it is maximal. Also $ \PSU_4 $ contains a group of order $ 360 $ isomorphic to $ \PSL_2(9) $ which seems maximal, and a group of order $ 168 $ isomorphic to $ \PSL_2(7)\cong \PSL_3(2) $ and a group of order 60 isomorphic to $ \PSL_2(4) \cong \PSL_2(5) $. It is unclear whether or not the $ 168 $ and $ 60 $ elements subgroups of $ \PSU_4 $ are maximal.
That leads me to ask: Does $ \PSU_m $ always have a maximal $ \PSL_n(q) $ or $ \PSU_n(q^2) $?
