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Virtual materials design requires not only the simulation of a huge number of systems, but also of systems with ever larger sizes and through increasingly accurate models of the electronic structure. These can be provided by density functional theory (DFT) using not only simple local approximations to the unknown exchange and correlation functional, but also more complex approaches such as hybrid functionals, which include some part of Hartree–Fock exact exchange. While hybrid functionals allow many properties such as lattice constants, bond lengths, magnetic moments and band gaps, to be calculated with improved accuracy, they require the calculation of a nonlocal potential, resulting in high computational costs, that scale rapidly with the system size. This limits their wide application. Here, we present a new highly-scalable implementation of the nonlocal Hartree-Fock-type potential into FLEUR—an all-electron electronic structure code that implements the full-potential linearized augmented plane-wave (FLAPW) method. This implementation enables the use of hybrid functionals for systems with several hundred atoms. By porting this algorithm to GPU accelerators, we can leverage future exascale supercomputers which we demonstrate by reporting scaling results for up to 64 GPUs and up to 12,000 CPU cores for a single k-point. As proof of principle, we apply the algorithm to large and complex iron garnet materials (YIG, GdIG, TmIG) that are used in several spintronic applications.