This article investigates the nonlinear normal modes (NNMs) of a three degrees-of-freedom cyclically symmetric system with bilinear contact springs. Further, two different cases of zero and nonzero clearance are assumed for the contact model. These systems are found to exhibit qualitatively different characteristics. Interestingly, the former system admits positive-scaling of solutions, which is reminiscent of linear systems. Four classes of NNMs are found for both zero and nonzero clearance. The NNM solutions are calculated using analytical or numerical techniques. The analytical solution involves solving the equations of motion using appropriate ansatzes, while the numerical solution uses a shooting-based continuation scheme. The NNMs' features and their variations with contact spring stiffness and system energy are also studied. This study highlights certain simplifications applicable to cyclically symmetric structures with contact. These simplifications will be of practical interest for designing real-life structures.