  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.
