For finite m and q we study the lattice $$\mathbf {L}(\mathbf {V})=(L(\mathbf {V}),+,\cap ,\{\vec {0}\},V)$$ L ( V ) = ( L ( V ) , + , ∩ , { 0 → } , V ) of subspaces of an m-dimensional vector space $$\mathbf {V}$$ V over a field $$\mathbf {K}$$ K of cardinality q. We present formulas for the number of d-dimensional subspaces of $$\mathbf {V}$$ V , for the number of complements of a subspace and for the number of e-dimensional subspaces including a given d-dimensional subspace. It was shown in Eckmann and Zabey (Helv Phys Acta 42:420–424, 1969) that $$\mathbf {L}(\mathbf {V})$$ L ( V ) possesses an orthocomplementation only in case $$m=2$$ m = 2 and $${{\,\mathrm{char}\,}}\mathbf {K}\ne 2$$ char K ≠ 2 . Hence, only in this case $$\mathbf {L}(\mathbf {V})$$ L ( V ) can be considered as an orthomodular lattice. On the contrary, we show that a complementation $$'$$ ′ on $$\mathbf {L}(\mathbf {V})$$ L ( V ) can be chosen in such a way that $$(L(\mathbf {V}),+,\cap ,{}')$$ ( L ( V ) , + , ∩ , ′ ) is both weakly orthomodular and dually weakly orthomodular. Moreover, we show that $$(L(\mathbf {V}),+,\cap ,{}^\perp ,\{\vec {0}\},V)$$ ( L ( V ) , + , ∩ , ⊥ , { 0 → } , V ) is paraorthomodular in the sense of Giuntini et al. (Stud Log 104:1145–1177, 2016).