The maximal subgroups of the exceptional groups $F_4(q)$, $E_6(q)$ and ${}^2E_6(q)$ and related almost simple groups
David A. Craven
TL;DR
This work provides a complete enumeration of the maximal subgroups of finite simple groups of type $F_4(q)$, $E_6(q)$, and ${}^2E_6(q)$ across all finite fields, and determines the Lie primitive almost simple subgroups as part of the larger program classifying almost simple maximal subgroups in exceptional groups of Lie type. It synthesizes techniques from algebraic group theory, representation theory, and computation (including Magma), notably employing the trilinear form on the 27-dimensional minimal $E_6$-module to constrain embeddings. The paper corrects prior omissions (e.g., a maximal subgroup in ${}^2F_4(8)$) and provides a comprehensive set of tables detailing which subgroups occur, their numbers of conjugacy classes, and how outer automorphisms act on them; this enables the full determination of maximal subgroups of all almost simple groups with these socles. The results mark the first new comprehensive listing for exceptional groups in three decades and supply a resource of explicit constructions and accompanying Magma files proving the claims. Overall, the work advances understanding of the subgroup structure of exceptional groups, with implications for applications in permutation groups, geometry, and related areas of finite group theory. The combination of deep theoretical methods with extensive computational verification yields a robust, publishable atlas of maximal subgroups and their automorphism-stabilizers in these important families.
Abstract
This article produces a complete list of all maximal subgroups of the finite simple groups of type $F_4$, $E_6$, and twisted $E_6$ over all finite fields. Along the way, we determine the collection of Lie primitive almost simple subgroups of the corresponding algebraic groups. We give the stabilizers under the actions of outer automorphisms, from which one can obtain complete information about the maximal subgroups of all almost simple groups with socle one of these groups. We also provide a new maximal subgroup of ${}^2\!F_4(8)$, correcting the maximal subgroups for that group from the list of Malle. This provides the first new exceptional groups of Lie type to have their maximal subgroups enumerated for three decades. The techniques are a mixture of algebraic groups, representation theory, computational algebra, and use of the trilinear form on the 27-dimensional minimal module for $E_6$. We provide a collection of auxiliary Magma files that prove the author's computational claims, yielding existence and the number of conjugacy classes of all maximal subgroups mentioned in the text.