  We propose two novel two-dimensional (2D) topological materials,
(PtN$_4$)$_3$C$_{32}$ and (MnN$_4$)$_3$C$_{32}$, with a special geometry that
we named as kagome-in-honeycomb (KIH) lattice structure, to illustrate the
coexistence of the paradigmatic states of kagome physics, Dirac fermions and
flat bands, that are difficult to be simultaneously observed in
three-dimensional realistic systems. In such system, MN$_4$(M=Pt or Mn)
moieties are embedded in honeycomb graphene sheet according to kagome lattice
structure, thereby resulting in a KIH lattice. Using the first-principles
calculations, we have systemically studied the structural, electronic, and
topological properties of these two materials. In the absence of spin-orbit
coupling (SOC), they both exhibit the coexistence of Dirac/quadratic-crossing
cone and flat band near the Fermi level. When SOC is included, a sizable
topological gap is opened at the Dirac/quadratic-crossing nodal point. For
nonmagnetic (PtN$_4$)$_3$C$_{32}$, the system is converted into a
$\mathbb{Z}_2$ topological quantum spin Hall insulator defined on a curved
Fermi level, while for ferromagnetic (MnN$_4$)$_3$C$_{32}$, the material is
changed from a half-semi-metal to a quantum anomalous Hall insulator with
nonzero Chern number and nontrivial chiral edge states. Our findings not only
predict a new family of 2D quantum materials, but also provide an
experimentally feasible platform to explore the emergent kagome physics,
topological quantum Hall physics, strongly correlated phenomena, and theirs
fascinating applications.
