In this paper I shall try to summarise the current situation regarding the problem of finding the maximal subgroups of the sporadic simple groups, and to give some idea of the techniques that have been used to attack this problem. The fundamental lemma underlying the method is:

Lemma 1. If Gis a simple group, and M is a maximal subgroup of G, and K is a minimal normal subgroup of M, then

(i)I ii a characteristically simple group (i.e.a direct prdduot of isomorphic simple groups),

(ii)M=NG(K).

Proof. Elementary.

The general strategy is therefore to classify the characteristically simple subgroups into conjugacy classes, and find their normalizers. We then have a list which contains all the maximal subgroups, and it is usually a straightforward matter to eliminate the non-maximal subgroups from this list.

There is a fundamental dichotomy between the cases when K is an elementary Abelian p-group (in this case M is called a p-local subgroup), and the cases when K is non-Abelian. The most difficult problems are concerned with proving uniqueness rather than existence, and with non-local rather than local subgroups.

Before I go into details of the methods perhaps I should give a survey of the results that have been obtained to date. The actual lists of maximal subgroups are of course much too long to include here, but can all be found in the ATLAS [3].