  In this paper a conformal classification of three dimensional left-invariant
sub-Riemannian contact structures is carried out; in particular we will prove
the following dichotomy: either a structure is locally conformal to the
Heisenberg group $\mathbb H_3$, or its conformal classification coincides with
the metric one. If a structure is locally conformally flat, then its conformal
group is locally isomorphic to $SU(2,1)$.
