This paper considers the lattice of subquasivarieties of a regular variety. In particular we show that if V is a strongly irregular variety that is minimal as a quasivariety, then the smallest quasivariety containing both V and SI (the variety of semilattices) is never equal to the regularization V of V.

We use this result to describe the lattice of subquasivarieties of V in several special but quite common, cases and give a number of applications and examples.