Dear Alexander,
Indeed as Derek Holt mentioned there exists an infinite
two-parametric series of maximal subgroups of desired nature.
(Examples mentioned by Derek, correspond to the case when
for one of parameters, say m, we have m =2, 3.)
The proof of the existence of such series was given many years ago,
by Kalu\vznin and myself, see first reference.
(In fact we rediscovered and investigated for the purpose
of the proof what is now called Johnson association scheme.)
Later on in works by M.K., E.Halberstadt and V.A.Ustimenko
all values were found when we indeed have maximality.
All details are summarised in the survey paper
(second reference), pp. 99-102.
The resuts are obtained without the use of CFSG.
I will send you copies of these pages by air mail
(I doubt that this Kluwer volume is available in Ukraine.)
Surely question by Sushchanskii in Kourovka was
influenced by this result
(he also is a disciple of Kalu\vznin.)

References
1. L.~A.~Kalu\v znin and M.~H.~Klin,
On some maximal subgroups of the symmetric and alternating groups,
\textit{Mat.~Sb.,} 87 (1972), 91--121 (in Russian).

2. I.~A.~Farad\v zev, M.~H.~Klin, M.~E.~Muzichuk,
Cellular rings and automorphism groups of graphs,
in \textit{Investigation in Algebraic Theory of Combinatorial
Objects} (eds. I.~A. Farad\v zev, A.~A.Ivanov, M.~H.~Klin,
A.~J.Woldar {\it et al.}), Kluwer, Dordrecht
(1994), pp.~1--152.

Best regards,
Mikhail

Please, find here my present address:
Dr. Mikhail KLin
Department of Mathematics and Computer Science
Ben-Gurion University of the Negev
P.O.Box 653, Beer-Sheva 84105, Israel
Tel: (0)7/6477-802 (office: note new phone from 03.99)
(0)7/641-37-15 (home: note new phone from 28.10.98)
e-mail: klin@indigo.cs.bgu.ac.il
klin@cs.bgu.ac.il