In this paper we consider problem of the existence and stability of invariant manifolds in a network of diffusively coupled identical systems. It is shown that the existence of a symmetry in the network implies the existence of linear invariant manifolds. This correspond to so called partial synchronization, or clusterization, phenomenon occurring when some subsystems from the network operate in a synchronous manner. Conditions guaranteeing global asymptotic stability of the partial synchronization manifolds are presented.