 Maximal subgroups of GL3 over F2

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.