Studying the control properties of complex networks provides insight into how designers and engineers can influence these systems to achieve a desired behavior. Topology of a network has been shown to strongly correlate with certain control properties; here we uncover the fundamental structures that explain the basis of this correlation. We develop the control profile, a statistic that quantifies the different proportions of control-inducing structures present in a network. We find that standard random network models do not reproduce the kinds of control profiles that are observed in real-world networks. The profiles of real networks form three well-defined clusters that provide insight into the high-level organization and function of complex systems.