In 2012 the second author obtained a description of the lattice of subgroups of a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice is a disjoint union of ``sandwiches'' parameterized by subrings $R$ such that $K\subseteq R\subseteq A$. In the current article a similar result is proved for $\Phi=C_n$, $n\ge3$, and $2=0$ in $K$. In this setting one has to introduce more sandwiches, namely, sandwiches which are parameterized by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. The result we get generalizes Ya.\,N.\,Nuzhin's theorem of 2013 concerning the root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained, but under the condition that $A$ is an algebraic extension of a field~$K$.