In this article, we examine the controllability of Laplacian dynamic networks on cographs. Cographs appear in modeling a wide range of networks and include as special instances, the threshold graphs. In this article, we present necessary and sufficient conditions for the controllability of cographs, and provide an efficient method for selecting a minimal set of input nodes from which the network is controllable. In particular, we define a sibling partition in a cograph and show that the network is controllable if all nodes of any cell of this partition except one are chosen as control nodes. The key ingredient for such characterizations is the intricate connection between the modularity of cographs and their modal properties. Finally, we use these results to characterize the controllability conditions for certain subclasses of cographs.