The level structure of I-124 (Z = 53, N = 71) has been studied via the Sn-122(Li-7, (5)n) I-124 reaction with a beam energy of 54 MeV. Through in-beam and out-of-beam gamma-ray spectroscopy, the sophisticated low-lying levels including isomeric states and numerous collective states have been established for the first time. A positive-parity collective band built on the 10(+) state at 1297 keV is interpreted as being associated with the combination of a proton and a neutron in the same intruder h(11/2) orbital, namely the pi h(11/2)nu h(11/2) configuration. This band shows a typical quadrupole vibrational character. In contrast, the negative parity bands based on the pi g(7/2)nu h(11/2) configuration show a soft triaxial rotation. An isomeric 8(-) state at 689 keV with a half-life of 14 ns can be explained as a K isomer due to a proton with an Omega(p) = 9/2 in the g(9/2) orbital coupled to a neutron with an Omega n = 7/2 in the h(11/2) orbital. The excited states based on this pi g(9/2)nu h(11/2) configuration show a coupled rotational structure. Another coupled rotational band built on the 6(+) state at 714 keV is thought to be based on the pg(9/2)nu d(3/2) configuration.