Crystalline symmetries can generate exotic band-crossing features, which can lead to unconventional fermionic excitations with interesting physical properties. We show how a cubic Dirac point---a four-fold-degenerate band-crossing point with cubic dispersion in a plane and a linear dispersion in the third direction---can be stabilized through the presence of a nonsymmorphic glide mirror symmetry in the space group of the crystal. Notably, the cubic Dirac point in our case appears on a threefold axis, even though it has been believed previously that such a point can only appear on a sixfold axis. We show that a cubic Dirac point involving a threefold axis can be realized close to the Fermi level in the non-ferroelectric phase of LiOsO$_3$. Upon lowering temperature, LiOsO$_3$ has been shown experimentally to undergo a structural phase transition from the non-ferroelectric phase to the ferroelectric phase with spontaneously broken inversion symmetry. Remarkably, we find that the broken symmetry transforms the cubic Dirac point into three mutually-crossed nodal rings. There also exist several linear Dirac points in the low-energy band structure of LiOsO$_3$, each of which is transformed into a single nodal ring across the phase transition.
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