Quasinormality is a classical constraint qualification originally introduced by Hestenes in 1975 and subsequently extensively studied in nonlinear programming and in problems with abstract constraints. In this paper, we extend this concept to the setting of nonlinear semidefinite programming (NSDP). We show that the proposed condition is strictly weaker than Robinson’s constraint qualification, while still guaranteeing the existence of exact penalty functions, local error bounds, and boundedness of dual sequences generated by augmented Lagrangian methods. As a consequence, convergence to Karush–Kuhn–Tucker points can be established for a broader class of NSDPs under mild regularity assumptions. In addition, a pseudonormality condition is introduced and explored.