A theory for forming Ir-atomic nanoclusters in a dielectric matrix of Na4Ir3O8 structure and spinel-like structures is suggested. The atomic order in the Na4Ir3O8 structure is investigated by group-theoretical methods of phase transition theory. The critical irreducible representation τ , generating appearance of enantiomorphic P4132 (P4332)-phases from high symmetry spinel-like phase with space group Fd3m, is six dimensional irreducible representation k10(τ1) (in Kovalev designation). Ir and Na atoms form an intriguing atom ordering, giving rise to a network of corner shared Ir triangles, called a hyperkagome lattice. It is shown that inside the hyperkagome lattice, there are closed metal contours of chemical bonds formed by Ir-clusters – decagons. Unusual physical properties of solid solutions on the basis of Na4Ir3O8 are expected. The existence of hyperkagome lattices in six types in ordered spinel structures is theoretically predicted.