A New Maximal Subgroup of $E_8$ in Characteristic $3$

Abstract: We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type

E_{8}

in characteristic

3

. This has type

F_{4}

, and was missing from previous lists of maximal subgroups produced by Seitz and Liebeck--Seitz. We also prove a result about the finite group

H =^{3}! D_{4} (2)

, that if

H

embeds in

E_{8}

(in any characteristic

p

) and has two composition factors on the adjoint module then

p = 3

and

H

lies in this new maximal

F_{4}

subgroup.

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