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

We consider the finite exceptional group of Lie type $G=E_6^{\epsilon }(q)$ (universal version) with $3|q?\epsilon$, where $E_6^{+1}(q)=E_6(q)$ and $E_6^{?1}(q){=}^2{E}_6(q)$. 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.