John Bray, Derek Holt and Colva Roney-Dougal are writing a book of which the aim is to classify maximal subgroups of all almost simple extensions of the classical groups G over finite fields in dimension up to 12. (The geometric type subgroups in dimensions greater that 12 have been classified in an earlier book by Kleidman and Liebeck.)

In this talk, we shall discuss a few specific aspects of this project. We concentrate on almost simple subgroups (i.e. non-geometric type) in which the natural characteristic of the subgroup H is unequal to the characteristic in the classical group G. The simple candidates for Socle(H) have been listed in a paper of Hiss and Malle. But it becomes technically complicated when we try to ascertain which almost simple extensions of H are subgroups of which almost simple extensions of G.

Techniques used involve use of representations of H over number fields. Some nontrivial problems in number theory arise in a few cases.