We report the discovery of superconductivity in a series of noncentrosymmetric compounds La 4 TX (T = Ru, Rh, Ir; X = Al, In), which have a cubic crystal structure with the space group F m 43 . La 4 RuAl, La 4 RhAl, La 4 IrAl, La 4 RuIn and La 4 IrIn exhibit bulk superconducting transitions with critical temperatures (T c ) of 0.61-3.35 K, while La 4 RhIn was not successfully synthesized. The specific heats of La 4 RuAl and La 4 RhAl are best accounted for by an s-wave model with a fully open superconducting gap, while a nodal gap cannot be excluded for La 4 IrAl, which has stronger antisymmetric spinorbit coupling (ASOC). In all cases, the upper critical fields are well described by the Werthamer-Helfand-Hohenberg model, and the values are well below the Pauli limit, indicating that orbital limiting is the dominant pair-breaking mechanism. Density functional theory (DFT) calculations reveal that the degree of band splitting by the ASOC shows considerable variation between the different compounds. This indicates that the strength of the ASOC is highly tunable across this series of superconductors, suggesting that they are good candidates for examining the relationship between ASOC and superconducting properties in noncentrosymmetric superconductors.