The topic of variational integrators for mechanical systems whose dynamics evolve on nonlinear spaces has seen strong growth recently. Within this class of variational integrators is the subclass of Lie group variational integrators that can be used for mechanical systems whose dynamics evolve on Lie groups. This class of mechanical systems includes all systems that can be modeled as rigid bodies or connections of rigid bodies. In this paper, we present a Lie group variational integrator for the full (translation and orientation) motion of a rigid body under the possible influence of nonconservative forces and torques. We use a discretization scheme for such systems which is based on the discrete Lagrange-d'Alembert principle to obtain the Lie group variational integrator. We apply the composition of the Lie group variational integrator with its adjoint and a Crouch-Grossman method to the example of a conservative underwater system. We show numerically that with respect to energy these manifold methods, as expected, behave as a symplectic integrator and a nonsymplectic integrator, respectively.