In this paper we consider the reachability problem for quantized control systems, i.e. systems that take inputs from a finite set of symbols. Previous work addressed this problem for linear systems and for some specific classes of nonlinear driftless systems. In this paper we attack the study of more general nonlinear systems. To do so we find it useful to pose the problem in more abstract terms, and make use of the wealth of tools available in group theory, which enables us to proceed in our agenda of better understanding effects of quantization of inputs on dynamic systems.