Lattices of subclasses. III

We prove that for certain $Q$-universal quasivarieties $\mathbf{K}$, the lattice of $\mathbf{K}$-quasivarieties contains continuum many subquasivarieties with the undecidable quasi-equational theory and for which the finite membership problem is also undecidable. Moreover, we prove that certain $Q$-universal quasivarieties have continuum many subquasivarieties with no independent quasi-equational basis.