The effects of growth conditions and composition on the crystal growth of La-2126-phase superconducting materials have been studied by the floating-zone method. The composition of La-2126-phase single crystals was (La 1− x Sr x ) 2 CaCu 2 O 6+ δ (x=0.075–0.25). The crystals were grown under oxygen pressures from 1bar to 11bar. A smooth planar solid–liquid growth interface of (La 1− x Sr x ) 2 CaCu 2 O 6+ δ (x=0.075) tended to break down into a cellular interface when the growth velocity was higher than 0.5mm/h, or the Sr concentration x>0.075. When the planar solid–liquid growth interface broke down into a cellular interface, the single-crystal size in an as-grown rod decreased abruptly and the as-grown rod was not of single phase. Various small sizes of CaO second phases were found inside the as-grown single crystals of (La 1− x Sr x ) 2 CaCu 2 O 6+ δ (x ⩾0.125). Based on the occurrence of these CaO second phases inside the as-grown crystals, the estimated solubility of as-grown single crystals is x=0.075 for (La 1− x Sr x ) 2 CaCu 2 O 6+ δ . The melting point of the La-2126 materials increases with increasing oxygen pressure. Cubic-centimeter-size single crystals were successfully grown for (La 1− x Sr x ) 2 CaCu 2 O 6+ δ (x=0.075). The superconducting transition temperature of the as-grown (La 1− x Sr x ) 2 CaCu 2 O 6+ δ single crystals increases from Tc=28K to 46K with increasing the Sr concentration x from 0.075 to 0.25 at 11bar oxygen pressure.