  We study clones modulo minor homomorphisms, which are mappings from one clone
to another preserving arities of operations and respecting permutation and
identification of variables. Minor-equivalent clones satisfy the same sets of
identities of the form $f(x_1,\dots,x_n)\approx g(y_1,\dots,y_m)$, also known
as minor identities, and therefore share many algebraic properties. Moreover,
it was proved that the complexity of the $\operatorname{CSP}$ of a finite
structure $\mathbb{A}$ only depends on the set of minor identities satisfied by
the polymorphism clone of $\mathbb{A}$. In this article we consider the poset
that arises by considering all clones over a three-element set with the
following order: we write $\mathcal{C} \preceq_{\mathrm{m}} \mathcal{D}$ if
there exists a minor homomorphism from $\mathcal{C}$ to $\mathcal{D}$. We show
that the aforementioned poset has only three submaximal elements.
