In this paper we investigate a property of the algebras of complexes (or power algebras or globals) which is a natural generalization of the notion of having all subgroups to be quasinormal in group theory. We say that an algebra $\mathcal A$ has the \emph{complex algebras of subalgebras} if the set of all non-empty subuniverses of this algebra forms a subuniverse of the algebra of complexes of $\mathcal A$. For example, all conservative and all entropic algebras have this property. Among other things, we prove that the class of finite algebras which have the complex algebra of subalgebras is not closed under finite direct products and it is not globally determined.