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This paper analyzes the input-to-state stability (ISS) properties of subhomogeneous cooperative systems for the first time. A new definition of finite-time ISS (FISS) for nonlinear positive systems is given. For stability analysis, the max-separable Lyapunov function is used, and the min-separable Lyapunov-like function is presented for the first time in this paper. Using these functions, we obtain our main results on solution estimates for subhomogeneous positive systems. According to the proposed method of analysis, the ISS properties of these systems can be easily judged only on the basis of their differential and algebraic characteristics. Simulation examples verify the validity of our results.