  Higgs-flavon fields appear as a part of the Froggatt-Nielsen (FN) mechanism,
which attempts to explain the hierarchy of Yukawa couplings. We explore the
possibility that the 750 GeV diphoton resonance recently reported at the LHC13,
could be identified with a low-scale Higgs-flavon field $H_F$ and find the
region of the parameter space consistent with CMS and ATLAS data. It is found
that the extra vector-like fermions of the ultraviolet completion of the FN
mechanism are necessary in order to reproduce the observed signal. We consider
a standard model (SM) extension that contains two Higgs doublets (a standard
one and an inert one) and one complex FN singlet. The inert doublet includes a
stable neutral boson, which provides a viable dark matter candidate, while the
mixing of the standard doublet and the FN singlet induces flavor violation in
the Higgs sector at the tree-level. Constraints on the parameters of the model
are derived from the LHC Higgs data, which include the search for the lepton
flavor violating decay of the SM Higgs boson $h\to \bar{\mu}\tau $. It is also
found that in some region of the parameter space the model may give rise to a
large branching ratio for the $H_F \to hh$ decay, of the order of 0.1, which
could be searched for at the LHC.
