$\gamma$-subdifferential is a concept which can be used for global optimization. If $x_*$ is a global minimizer of an arbitrary function f$$ then $0\in \partial_{\gamma}f(x_*)$, where $\partial_{\gamma}f(x_*)$ is the $\gamma$-subdifferential of $f$ at $x_*$. In particular, $\partial_{\gamma}f(x_*)\not = \emptyset$ at a global minimizer $x_*$. In this paper we investigate the nonemptiness and the monotonicity of $\gamma$-subdifferentials of $\gamma$-convex functions. Some sufficient conditions are stated for the nonemptiness of the $\gamma$-subdifferential of a symmetrically $\gamma$-convex function at a point. It is proved that for a symmetrically $\gamma$-convex function, the G\^{a}teaux derivative (when it exists) at a point belongs to the $\gamma$-subdifferential at that point. A relation between the $\gamma$-subdifferential and the Clarke generalized gradient of a symmetrically $\gamma$-convex function is also presented.