We examine the formation of groups of multiple supermassive black holes (SMBHs) in gas-poor galactic nuclei due to high merger rate of galaxies at high redshifts. We calculate the relative likelihood of binary, triple and quadruple SMBH systems, by considering the time-scales for relevant processes and by combining merger trees with N-body simulations for the dynamics of stars and SMBHs in galactic nuclei. Typical haloes today with mass M0≈ 1014 M⊙ have an average mass of Mz= 6= 5 × 1011 M⊙ at z∼ 6, while rare haloes with current mass M0≳ 1015 M⊙ have an average mass of Mz= 6= 5 × 1012 M⊙ at that redshift. These cluster-size haloes are expected to host single galaxies at z∼ 6. We expect about 30 per cent galaxies within haloes with present-day mass M0≈ 1014 M⊙ to contain more than two SMBHs at redshifts 2 ≲z≲ 6. For larger present-day haloes, with M0≳ 1015 M⊙, this fraction is almost 60 per cent. The existence of multiple SMBHs at high redshifts can potentially explain the mass deficiencies observed in cores of massive elliptical galaxies, which are up to five times the mass of their central BHs. Multiple SMBHs would also lead to an enhanced rate of tidal disruption of stars, modified gravitational wave signals compared to isolated BH binaries and slingshot ejection of SMBHs from galaxies at high speeds in excess of 2000 km s−1.