Radical subgroups of the finite exceptional groups of Lie type E6

We consider the finite exceptional groups of Lie type E6+1(q)=E6(q) and E6−1(q)=E62(q), both the universal versions. We classify, up to conjugacy, the maximal p-local subgroups and radical p-subgroups of G=E6ε(q) for p⩾5 with p∤q and q≡εmodp, and for p=3 with 3∤q and q≡−εmod3. As an application, the essential p-rank of the Frobenius category FD(G) is determined, where D is a Sylow p-subgroup of G. Moreover, if p=3, then we show that there is a subgroup H=F4(q) of G containing D such that FD(G)=FD(H), that is, H controls 3-fusion in G.