The relative equilibria of a plane three-link pendulum with a suspension rotating about a vertical axis in a gravitational force field are investigated. The pendulum is modelled as a system of three point masses, joined in tandem by massless non-elastic rods using cylindrical hinges. All the trivial equilibrium positions, their stability and branching are investigated. Attention is mainly paid to the non-trivial equilibrium positions, that is, to those equilibrium configurations of the pendulum for which not all the links are stretched out along the vertical axis. Questions of the existence, stability and the branching of these non-trivial equilibrium positions are investigated at fairly high angular velocities of rotation. The plane of the geometric parameters of the pendulum is subdivided into domains with a different number of non-trivial relative equilibria. Conditions are presented for each non-trivial equilibrium position which is found, and these are imposed on the system parameters for which an equilibrium position exists, their stability is investigated and a geometric description of the configuration of the pendulum is given.