  We have investigated orbital ordering in the half-doped manganite
La$_{0.5}$Sr$_{1.5}$MnO$_4$, which displays spin, charge and orbital ordering,
by means of unrestricted Hartree-Fock calculations on the multiband $p$-$d$
model. From recent experiment, it has become clear that
La$_{0.5}$Sr$_{1.5}$MnO$_4$ exhibits a cross-type $(z^2-x^2/y^2-z^2)$ orbital
ordering rather than the widely believed rod-type $(3x^2-r^2/3y^2-r^2)$ orbital
ordering. The calculation reveals that cross-type $(z^2-x^2/y^2-z^2)$ orbital
ordering results from an effect of in-plane distortion as well as from the
relatively long out-of-plane Mn-O distance. For the "Mn$^{4+}$" site, it is
shown that the elongation along the c-axis of the MnO$_6$ octahedra leads to an
anisotropic charge distribution rather than the isotropic one.
