  We analyse the Higgs sector of an $S_3$ model with three Higgs doublets and
no CP violation. After electroweak breaking there are nine physical Higgs
bosons, one of which corresponds to the Standard Model one. We study the scalar
and gauge sectors of this model, taking into account the conditions set by the
minimisation and stability of the potential. We calculate the masses, trilinear
and quartic Higgs-Higgs, and Higgs-gauge couplings. We consider two possible
alignment scenarios, where only one of the three neutral scalars has couplings
to the gauge bosons and corresponds to the SM Higgs, and whose trilinear and
quartic couplings reduce exactly to the SM ones. We also obtain numerically the
allowed parameter space for the scalar masses in each of the alignment
scenarios. We use the calculated trilinear and quartic couplings to find the
analytical structure of the one-loop neutral scalar mass matrix, without
fermionic contributions. We show that it is possible to have a compact mass
spectrum where the contribution to the oblique parameters might be small. We
explore some scenarios for the loop contributions to the neutral scalar masses.
