  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. We
also 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 ${}^2F_4(8)$, correcting the list of maximal subgroups for that
group from the list of Gunter 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 files that prove
the author's claims, yielding existence and uniqueness of all maximal subgroups
mentioned in the text.
