Particle flow Gaussian particle flow (PFGPF) uses an invertible particle flow to generate a proposal density. It approximates the predictive and posterior distributions as Gaussian densities. In this paper, we use bank of PFGPF filters to construct a Particle flow Gaussian sum particle filter (PFGSPF), which approximates the predictive and posterior as Gaussian mixture model. This approximation is useful in complex estimation problems where a single Gaussian approximation is not sufficient. We compare the performance of this proposed filter with PFGPF and others in challenging numerical simulations.