A ring extension $R\subseteq S$ is said to be FO if it has only finitely many intermediate rings. $R\subseteq S$ is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension $R\subseteq S$ to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair...