  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 paper, we discuss these groups
from a group (co)homological point of view. The results include the
determination of their abelianizations, lower bounds of the second homology and
remarks on the (co)homology of higher degrees. As a by-product of this
investigation, we determine the second homology of the mapping class group of a
surface of genus 3.
