This paper investigates a novel transmissive reconfigurable intelligent surface (RIS) transceiver architectureenabled multigroup multicast downlink communication system. Under this setup, an optimization problem is formulated to maximize the minimum rate of users across all groups, subject to the maximum available power of each RIS unit. Due to the nondifferentiable nature of the objective function, the max-min rate problem is challenging to solve. To tackle this difficult problem, we develop an iterative solution by leveraging the successive convex approximation (SCA) and the penalty function method. However, the above approach has high computational complexity and may lead to compromised performance. To overcome these drawbacks, we design an efficient second-order cone programming (SOCP)-based method using the weighted minimum mean squared error (WMMSE) framework to reduce computational complexity. Furthermore, to further reduce the computational complexity, we also propose a low-complexity and solver-free algorithm that analytically updates all variables by combining the smooth approximation theory and the majorization-minimization (MM) method. Numerical results are provided to verify the convergence and effectiveness of our proposed three algorithms. It is also demonstrated that the SOCP-based method outperforms the penalty-based algorithm in terms of both the achieved min rate and the computational complexity. In contrast, the lowcomplexity design achieves significantly lower complexity with only slightly degraded performance.