We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this talk, we discuss these groups from group (co)homological point of view. The results include the determination of their abelianization, lower bounds of their second homology and remarks on (co)homology of higher degrees relating to Miller-Morita-Mumford classes.