This paper considers the 'reduction' problem for distributed control systems. In particular, we consider controllability of systems containing multiple instances of diffeomorphic components where the overall system dynamics is invariant with respect to a discrete group action. A subclass of such systems are systems with a set of identical components where the overall system dynamics are invariant with respect to physically interchanging these components. The main result is a proposition which shows that for an equivalence class of symmetric systems of this type, controllability of the entire class of systems can be determined by analysing the member of the equivalence class with the smallest state space. The reduction methods developed are illustrated by considering the controllability of a team of mobile robots and a platoon of underwater vehicles.