  Previously known three-dimensional Dirac semimetals (DSs) occur in two types
-- topological DSs and nonsymmorphic DSs. Here we present a novel
three-dimensional DS that exhibits both features of the topological and
nonsymmorphic DSs. We introduce a minimal tight-binding model for the space
group 100 that describes a layered crystal made of two-dimensional planes in
the $p4g$ wallpaper group. Using this model, we demonstrate that double
glide-mirrors allow a noncentrosymmetric three-dimensional DS that hosts both
symmetry-enforced Dirac points at time-reversal invariant momenta and
twofold-degenerate Weyl nodal lines on a glide-mirror-invariant plane in
momentum space. The proposed DS allows for rich topological physics manifested
in both topological surface states and topological phase diagrams, which we
discuss in detail. We also perform first-principles calculations to predict
that the proposed DS is realized in a set of existing materials BaLa$X$B$Y_5$,
where $X$ = Cu or Au, and $Y$ = O, S, or Se.
