Maximal subgroups of GL3 over F2 ($24,24,21$)

The maximal subgroups are point stabilizers and plane stabilizers of order 24, and Singer cycle normalizers of order 21. A proper subgroup containing a 7-element has a unique Sylow 7-subgroup, since the other count gives an impossible index-three subgroup of the simple group. A subgroup of order dividing 24 has a fixed point or a three-point orbit on the seven nonzero vectors. A collinear orbit gives an invariant plane; an independent triple gives its fixed vector sum.

Every involution of $G{L}_3(2)$ is a transvection $I+vf$, because its nilpotent part has square zero and rank one. Two distinct commuting transvections either have common image line or common fixed plane. This gives two types of elementary abelian groups of order four, displayed above. No larger elementary abelian $2$-subgroup is possible: a transvection commuting with all three elements of a common-image group must have that same image, and the dual argument applies to a common-fixed-plane group. Their normalizers are respectively point and plane stabilizers of order $24$. The common fixed-space dimensions one and two distinguish their conjugacy classes.