In the present work, we investigate the effects of dark matter (DM) on hybrid star properties. We assume that dark matter is mixed with both hadronic and quark matter and interacts with them through the exchange of a Higgs boson. The hybrid star properties are obtained from equations of state calculated with a Maxwell prescription. For the hadronic matter, we use the NL3* parameter set, and for the quark matter, the MIT bag model with a vector interaction. We see that dark matter does not influence the phase transition points (pressure and chemical potential) but shifts the discontinuity on the energy density, which ultimately reduces the minimum mass star that contains a quark core. Moreover, it changes considerably the star family mass-radius diagrams and moves the merger polarizability curves inside the confidence lines. Another interesting feature is the influence of DM in the quark core of the hybrid stars constructed. Our results show an increase of the core radius for higher values of the dark particle Fermi momentum.