In this paper, we propose and analyze a mixed H1-conforming finite element method for solving Maxwell's equations in terms of electric field and Lagrange multiplier, where the multiplier is introduced accounting for the divergence constraint. We mainly focus on the case that the physical domain is nonconvex and its boundary includes reentrant corners or edges, which may lead the solution of Maxwell's equations to be a non-H1 very weak function and thus cause many numerical difficulties. The proposed method is formulated in the stabilized form by adding an additional mesh- dependent stabilization term to the mixed variational formulation. A general framework of stability and error analysis is established. Specifically, a pair of H1-conforming finite elements, namely, the CP2-P1 elements, for electric field and multiplier is studied, and its stability and error bounds are also derived. Numerical experiments for source problems as well as eigenvalue problems on the L-shaped and cracked domains are presented to illustrate the high performance of the proposed mixed H1-conforming finite element method.