In this paper, we investigate the maximum flow routing strategy with the service function chain (SFC) constraints in the space information networks (SINs), where a SFC consists of a specific ordered sequence of service functions, and the mission flow must go through these functions in a predefined order. The time-varying SIN is modeled by the time-expanded graph (TEG). We formulate the maximum flow routing strategy problem with the SFC constraints as a linear programming (LP) problem. Furthermore, for a large-scale SIN, as the complexity of solving the LP problem is still very high, we propose a novel low-complexity SFC-constrained graph theory based (SFC-GT) algorithm. Specifically, we formulate this problem as one special single commodity maximum flow problem, where this flow must satisfy the SFC constraints. We first define the SFC-constrained residual network and the SFC-constrained augmenting path. Afterwards, we iteratively search the SFC-constrained augmenting path and update the SFC-constrained residual network. Simulation results demonstrate our proposed SFC-GT algorithm can achieve near-optimal performance with much less complexity.