A new maximal subgroup of 𝐸₈ in characteristic 3

Abstract

We prove the existence and uniqueness up to conjugacy of a new maximal subgroup of the algebraic group of type E 8 E_8 in characteristic 3 3 . This has type F 4 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 ) H={}^3\!D_4(2) , namely that if H H embeds in E 8 E_8 (in any characteristic p p ) and has two composition factors on the adjoint module then p = 3 p=3 and H H lies in a conjugate of this new maximal F 4 F_4 subgroup.