There is a dual program linked with every nonlinear program. The dual objective function is called the Lagrangian; it is defined in terms of the original problem. This note presents a characterization of the Lagrangian subgradients under general conditions. The theorem follows from a result of Danskin [1] that can be used (see [2]) to characterize the Lagrangian directional derivatives. These characterizations are theoretically interesting and may be useful in computing optimal dual solutions. Rockafellar's outstanding book, [3], is used as a basic reference; it contains a wealth of background and historical information.