TITLE: Thompson's Conjecture
QUESTION [5 upvotes]: I have heard that the following is a conjecture due to Thompson:
The number of maximal subgroups of a (finite) group $G$ does not exceed the order $|G|$ of the group.
My question is: did Thompson really conjecture this? If so, is there any literature on the subject?

REPLY [8 votes]: I've seen this conjecture attributed to Wall (1961), for example: On a conjecture of G.E. Wall. This is a recent article (journal version appeared in 2007), and it gives a bunch of references. The conjecture remains open. Here is a very recent article which does not attack the conjecture itself, but uses it as an inspiration for a different conjecture.