Because of their multiscale heterogeneous structure, the fine-scale simulation of archi- tected materials, such as lattices, is known to be computationally demanding. One of the efficient methods to finely model and compute these structures is to make use of the Iso- Geometric Analysis (IGA) based on spline composition [1]. An inexact FETI-DP solver based on reduction model [2] has been proposed to overcome some of the computational difficulties of the approach. The solver appears particularly attractive to perform full fine-scale simulations at minimal memory storage (therefore offering the opportunity to compute the solution on a single laptop). In this work, we develop a highly scalable HPC solver for the IGA of lattices built by spline composition. The solver, based on Krylov subspace methods, is an alternative to the inexact FETI-DP solver for representative problems with tremendous computational costs being out of reach for general purpose computers. To start with, our solver leverages a fast multiscale assembly procedure benefiting from the repetitiveness of unit cell [3]. This allows to drastically reduce the memory costs compared to standard solvers. Then, a highly scalable matrix-free preconditioner for the high-order IGA discretizations is constructed by exploiting an efficient preconditionner, based on a geometric multigrid method involving a coarse low-order spline problem. The latter is eventually solved using algebraic multigrid methods.