AlGaN growth by hydride vapor phase epitaxy on patterned sapphire substrates has been investigated. Growth on honeycomb-shaped holes is disturbed by parasitic growth of c-plane oriented AlGaN crystallites on n-plane sapphire facets. Triangular hole-like structures allow for complete suppression of parasitic c-plane oriented AlGaN crystallites and coalescence of c-plane AlGaN at very low layer thickness. Additionally, triangular columnar sapphire patterns allow for biaxial strain relaxation and higher crystalline perfection. A total pressure of 400 hPa is favored for lateral overgrowth of ternary AlGaN by hydride vapor phase epitaxy. Lower pressures lead to strong composition inhomogeneity and higher pressure to lower crystal perfection. Additionally, very low V/III ratios of less than 10 yield best results. Lower ammonia supply support the growth of beneficial {11-22}-AlGaN crystallites on sapphire m-sidewalls which supresses undesired AlGaN growth on sapphire n-facets.