  A non-degenerate toric variety $X$ is called $S$-homogeneous if the subgroup
of the automorphism group $\text{Aut}(X)$ generated by root subgroups acts on
$X$ transitively. We prove that maximal $S$-homogeneous toric varieties are in
bijection with pairs $(P,\mathcal{A})$, where $P$ is an abelian group and
$\mathcal{A}$ is a finite collection of elements in $P$ such that $\mathcal{A}$
generates the group $P$ and for every $a\in\mathcal{A}$ the element $a$ is
contained in the semigroup generated by $\mathcal{A}\setminus\{a\}$. We show
that any non-degenerate homogeneous toric variety is a big open toric subset of
a maximal $S$-homogeneous toric variety. In particular, every homogeneous toric
variety is quasiprojective. We conjecture that any non-degenerate homogeneous
toric variety is $S$-homogeneous.
