Classification of some 3-subgroups of the finite groups of Lie type E₆

We consider the finite exceptional group of Lie type G=E₆ ^ε(q) (universal version) with 3|q−ε, where E₆ ⁺¹(q)=E₆(q) and E₆ ⁻¹(q)=²E₆(q). We classify, up to conjugacy, all maximal-proper 3-local subgroups of G, that is, all 3-local M<G which are maximal with respect to inclusion among all proper subgroups of G which are 3-local. To this end, we also determine, up to conjugacy, all elementary-abelian 3-subgroups containing Z(G), all extraspecial subgroups containing Z(G), and all cyclic groups of order 9 containing Z(G). These classifications are an important first step towards a classification of the 3-radical subgroups of G, which play a crucial role in many open conjectures in modular representation theory.