The low-lying states in ¹⁰⁶Zr and ¹⁰⁸Zr have been investigated by means of β-γ and isomer spectroscopy at the radioactive isotope beam factory (RIBF), respectively. A new isomer with a half-life of 620 ± 150 ns has been identified in ¹⁰⁸Zr. For the sequence of even-even Zr isotopes, the excitation energies of the first 2⁺ states reach a minimum at N = 64 and gradually increase as the neutron number increases up to N = 68, suggesting a deformed subshell closure at N = 64. The deformed ground state of ¹⁰⁸Zr indicates that a spherical subshell gap predicted at N = 70 is not large enough to change the ground state of ¹⁰⁸Zr to the spherical shape. The possibility of a tetrahedral shape isomer in ¹⁰⁸Zr is also discussed.