Steiner triple systems of order $21$ with a transversal subdesign $\mathrm{TD}(3, 6)$

A Steiner triple system (STS) contains a transversal subdesign $\mathrm{TD}(3, w)$ if its point set has three pairwise disjoint subsets $A$, $B$, $C$ of size $w$ and $w^2$ blocks of the STS intersect with each of $A$, $B$, $C$ (those $w^2$ blocks form a $\mathrm{TD}(3, w)$). We prove several structural properties of Steiner triple systems of order $3w + 3$ that contain one or more transversal subdesigns $\mathrm{TD}(3, w)$. Using exhaustive search, we find that there are $2\ 004\ 720$ isomorphism classes of STS(21) containing a subdesign $\mathrm{TD}(3, 6)$ (or, equivalently, a $6 \times 6$ Latin square).