A subgroup \(H\) of a finite group \(G\) is ‘normally embedded’ in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some normal subgroup of \(G\). This paper provides characterizations of the normally embedded subgroups of a solvable direct product \(G=A\times B\) in terms of Hall systems of \(G\) (Theorem 5.1) and describes the Fitting set associated to a normally embedded subgroup of an arbitrary direct product of two finite groups (Prop. 6.4). A large number of useful technical results about the subgroups of a finite direct product \(G=A\times B\) is employed to establish the main results.