Abstract: The maximal subgroups of the finite classical groups are
divided by a theorem of Aschbacher into nine classes. In this
paper, the authors show how to construct those maximal subgroups
of the finite classical groups of linear, symplectic or unitary
type that lie in the first eight of these classes. The ninth class
consists roughly of absolutely irreducible groups that are almost
simple modulo scalars, other than classical groups over the same
field in their natural representation. All of these constructions
can be carried out in low-degree polynomial time.
