  The Heisenberg model on a face center cubic (fcc) lattice is a typical
three-dimensional frustrated spin system expected to have magnetization
plateaus and supersolid phases. There are model compounds $A_2$CoTeO$_6$ ($A$ =
Ca, Sr, Pb) for the fcc lattice but with lattice distortions. Motivated by the
presence of the compounds, we investigate the ground state of the spin-1/2
antiferromagnetic Heisenberg model on a tetragonally distorted fcc lattice in
the magnetic field using a large-size cluster mean-field method for the sake of
finding new supersolid phases. We find five supersolid phases in the model,
indicating possibility to observe supersolid phases in these compounds. We also
find that one of the supersolid phases is similar to the nonclassical coplanar
phase obtained in the XXZ model on the triangular lattice.
