  We study characteristic band structures of the fermions on a square kagome
lattice, one of the two-dimensional lattices hosting a corner-sharing network
of triangles. We show that the band structures of the nearest-neighbor
tight-binding model exhibit many characteristic features, including a flat band
which is ubiquitous among frustrated lattices. On the flat band, we elucidate
its origin by using the molecular-orbital representation, and also find
localized exact eigenstates called compact localized states. In addition to the
flat band, we also find two spin-1 Dirac cones with different energies. These
spin-1 Dirac cones are not described by the simplest effective Dirac
Hamiltonian, but by incorporating some modifications that cause the bending of
the middle band as well as the particle-hole asymmetry. We also investigated
the Hofstadter problem on a square kagome lattice in the presence of an
external field, and find that the profile of the Chern numbers around the
modified spin-1 Dirac cones coincides with the conventional one.
