One possible and natural derivation from the collisionless cold dark matter (CDM) standard cosmological framework is the assumption of the existence of interactions between dark matter (DM) and photons or neutrinos. Such possible interacting dark matter (IDM) model would imply a suppression of small-scale structures due to a large collisional damping effect, even though the weakly interacting massive particle (WIMP) can still be the DM candidate. Because of this, IDM models can help alleviate alleged tensions between standard CDM predictions and observations at small mass scales. In this work, we investigate the properties of DM halo substructure or subhalos formed in a high-resolution cosmological N-body simulation specifically run within these alternative models. We also run its CDM counterpart, which allowed us to compare subhalo properties in both cosmologies. We show that, in the lower mass range covered by our simulation runs, both subhalo concentrations and abundances are systematically lower in IDM compared to the CDM scenario. Yet, as in CDM, we find that median IDM subhalo concentration values increase towards the innermost regions of their hosts for same mass subhalos. Also similarly to CDM, we find IDM subhalos to be more concentrated than field halos of the same mass. Our work has a direct application on studies aimed at the indirect detection of DM where subhalos are expected to boost the DM signal of their host halos significantly. From our results, we conclude that the role of halo substructure in DM searches will be less important in interacting scenarios than in CDM but far from being negligible.