We consider a class of linear-quadratic-Gaussian mean-field games having a major agent and numerous heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets Gamma(k) of R-m. The decentralized strategies of individual agents and the consistency condition system are represented in a unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on Gamma(k) The well-posedness of the consistency system is established in both the local and global cases through the contraction mapping and discounting methods, respectively. A related epsilon-Nash-equilibrium property is also verified.