  The subjet multiplicity has been measured in neutral current e+p interactions
at Q**2 > 125 GeV**2 with the ZEUS detector at HERA using an integrated
luminosity of 38.6 pb-1. Jets were identified in the laboratory frame using the
longitudinally invariant K_T cluster algorithm. The number of jet-like
substructures within jets, known as the subjet multiplicity, is defined as the
number of clusters resolved in a jet by reapplying the jet algorithm at a
smaller resolution scale y_cut. Measurements of the mean subjet multiplicity, <
n_sbj >, for jets with transverse energies E_T,jet >15 GeV are presented.
Next-to-leading-order perturbative QCD calculations describe the measurements
well. The value of alpha_s(M_Z), determined from < n_sbj > at y_cut=10**-2 for
jets with 25 < E_T,jet < 71 GeV, is alpha_s (M_Z) = 0.1187 +/- 0.0017 (stat.)
+0.0024 / -0.0009 (syst.) +0.0093 / -0.0076 (th.).
