An algebraic variational multiscale-multigrid method is proposed for large eddy simulation of turbulent flow. Level-transfer operators from plain aggregation algebraic multigrid methods are employed for scale separation. In contrast to earlier approaches based on geometric multigrid methods, this purely algebraic strategy for scale separation obviates any coarse discretization besides the basic one. Operators based on plain aggregation algebraic multigrid provide a projective scale separation, enabling an efficient implementation of the proposed method. The application of the algebraic variational multiscale-multigrid method to turbulent flow in a channel produces results notably closer to reference (direct numerical simulation) results than other state-of-the-art methods both for mean streamwise and root-mean-square velocities. For predicting highly sensitive components of the Reynolds-stress tensor in the context of turbulent recirculating flow in a lid-driven cavity, the algebraic variational multiscale-multigrid method also shows a remarkably good performance in predicting reference results from experiment and direct numerical simulation compared to other methods.