  Let $\H_g$ be a genus $g$ handlebody and $\MCG_{2n}(\T_g)$ be the
$2n$-punctured mapping class group of $\T_g=\partial\H_g$. In this paper we
study two particular subgroups of $\MCG_{2n}(\T_g)$ which generalize Hilden
groups. As well as Hilden groups are related to plate closures of braids, these
generalizations are related to Heegaard splittings of manifolds and to bridge
decompositions of links. Connections between these subgroups and motion groups
of links in closed 3-manifolds are also provided.
