Subgroups of Chevalley groups over rings

In the present paper, we study the subgroup lattice of a Chevalley group $\operatorname{G}(\Phi,R)$ over a commutative ring $R$, containing the subgroup $D(R)$, where $D$ is a subfunctor of $\operatorname{G}(\Phi,\_)$. Assuming that over any field $F$ the normalizer of the group $D(F)$ is “closed to be maximal”, we formulate some technical conditions, which imply that the lattice is standard. We also study the conditions concerning the normalizer of $D(R)$ in the case, where $D(R)$ is the elementary subgroup of another Chevalley group $\operatorname{G}(\Psi,R)$ embedded into $\operatorname{G}(\Phi,R)$.