I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying to determine which maximal subgroups of $GL(n^2, \mathbb{C})$ have invariant vectors when the following representation $\rho$ of $GL(n^2, \mathbb{C})$: $$ \rho = \mathrm{Sym}: GL(n^2) \to GL(V) \qquad V \cong \mathrm{Sym}^n \mathbb{C}^n $$ is restricted to the maximal subgroup. An example that has been found to stabilize the one-dimensional subspace $\wedge^n \mathbb C^n \times \wedge^n \mathbb C^n$ is the subgroup $GL(n) \times GL(n)$.
So first I am considering small rank groups, and I would like to start out with the maximal subgroups of $GL(9)$. A literature reference would be most appreciated.