It is a well established fact (by Greenberg) that if $p$ is a prime of good ordinary reduction of an elliptic curve $E/\mathbb{Q}$, then the dual of the Selmer group, denoted by $X(E/\mathbb{Q}_{cyc})$, has no finite non-trivial $\Lambda$-submodules.

What about the fine Selmer groups? When does the dual of the fine Selmer group *have* non-trivial finite $\Lambda$-submodules, if possible?