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We present a discontinuous Galerkin surface integral equation method, herein referred to as IEDG, for time harmonic electromagnetic wave scattering from nonpenetrable targets. The proposed IEDG algorithm allows the implementation of the combined field integral equation (CFIE) using square-integrable, , trial and test functions without any considerations of continuity requirements across element boundaries. Due to the local characteristics of basis functions, it is possible to employ nonconformal surface discretizations of the targets. Furthermore, it enables the possibility to mix different types of elements and employ different order of basis functions within the same discretization. Therefore, the proposed IEDG method is highly flexible to apply adaptation techniques. Numerical results are included to validate the accuracy and demonstrate the versatility of the proposed IEDG method. In addition, a complex large-scale simulation is conducted to illustrate the potential benefits offered by the proposed method for modeling multiscale electrically large targets.