So I know any group of order $48$ has a normal subgroup of order $8$ or a normal subgroup of order $16$ (or possibly both).
I am trying to expand upon this topic, figuring which of these conditions hold for $G = GL_2(\Bbb Z_3)$, as well as the group’s composition factors.
Indeed, every group of order $48$ has a normal subgroup of order $8$ or $16$. This was already shown here. The group ${\rm GL}(2,3)$ of order $48$ has a normal subgroup of order $8$, namely the quaternion group $Q_8$. Of course the special linear subgroup $SL(2,3)$ of order $24$ is normal, too. The only subgroup of order $16$ (up to isomorphism) is the semi-dihedral group $D_{16}$, which is not normal.
Composition factors of ${\rm GL}(2,3)$: here is a hint. It helps to consider the action of the group on the set of $1$-dimensional subspaces of $\mathbb{F}_2^3$.