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Due to the lack of full rotational symmetry in condensed matter physics, solids exhibit new excitations beyond Dirac and Weyl fermions, of which the six-fold excitations have attracted considerable interest owing to the presence of the maximum degeneracy in bosonic systems. Here, we propose that a single linear dispersive six-fold excitation can be found in the electride Li$_{12}$Mg$_3$Si$_4$ and its derivatives. The six-fold excitation is formed by the floating bands of elementary band representation -- $A@12a$ -- originating from the excess electrons centered at the vacancies (${\it i.e.}$, the $12a$ Wyckoff sites). There exists a unique topological bulk-surface-edge correspondence for the spinless six-fold excitation, resulting in trivial surface 'Fermi arcs' but nontrivial hinge arcs. All energetically-gapped $k_z$-slices belong to a two-dimensional (2D) higher-order topological insulating phase, which is protected by a combined symmetry ${\mathcal T}{\widetilde S_{4z}}$ and characterized by a quantized fractional corner charge $Q_{corner}=\frac{3|e|}{4}$. Consequently, the hinge arcs are obtained in the hinge spectra of the $\widetilde S_{4z}$-symmetric rod structure. The state with a single six-fold excitation, stabilized by both nonsymmorphic crystalline symmetries and time-reversal symmetry, is located at the phase boundary and can be driven into various topologically distinct phases by explicit breaking of symmetries, making these electrides promising platforms for the systematic studies of different topological phases.