Context. Filamentary molecular clouds, such as many of the infrared dark clouds (IRDCs), can undergo hierarchical fragmentation into substructures (clumps and cores) that can eventually collapse to form stars. Aims. We aim to determine the occurrence of fragmentation into cores in the clumps of the filamentary IRDC G304.74+01.32 (hereafter, G304.74). We also aim to determine the basic physical characteristics (e.g. mass, density, and young stellar object (YSO) content) of the clumps and cores in G304.74. Methods. We mapped the G304.74 filament at 350 μm using the Submillimetre APEX Bolometer Camera (SABOCA) bolometer. The new SABOCA data have a factor of 2.2 times higher resolution than our previous Large APEX BOlometer CAmera (LABOCA) 870 μm map of the cloud (9″ vs. 19 .̋ 86). We also employed the Herschel far-infrared (IR) and submillimetre, and Wide-field Infrared Survey Explorer (WISE) IR imaging data available for G304.74. The WISE data allowed us to trace the IR emission of the YSOs associated with the cloud. Results. The SABOCA 350 μm data show that G304.74 is composed of a dense filamentary structure with a mean width of only 0.18 ± 0.05 pc. The percentage of LABOCA clumps that are found to be fragmented into SABOCA cores is 36% ± 16%, but the irregular morphology of some of the cores suggests that this multiplicity fraction could be higher. The WISE data suggest that 65% ± 18% of the SABOCA cores host YSOs. The mean dust temperature of the clumps, derived by comparing the Herschel 250, 350, and 500 μm flux densities, was found to be 15.0 ± 0.8 K. The mean mass, beam-averaged H2 column density, and H2 number density of the LABOCA clumps are estimated to be 55 ± 10M⊙, (2.0 ± 0.2) × 1022 cm-2, and (3.1 ± 0.2) × 104 cm-3. The corresponding values for the SABOCA cores are 29 ± 3M⊙, (2.9 ± 0.3) × 1022 cm-2, and (7.9 ± 1.2) × 104 cm-3. The G304.74 filament is estimated to be thermally supercritical by a factor of ≳ 3.5 on the scale probed by LABOCA, and by a factor of ≳ 1.5 for the SABOCA filament. Conclusions. Our data strongly suggest that the IRDC G304.74 has undergone hierarchical fragmentation. On the scale where the clumps have fragmented into cores, the process can be explained in terms of gravitational Jeans instability. Besides the filament being fragmented, the finding of embedded YSOs in G304.74 indicates its thermally supercritical state, although the potential non-thermal (turbulent) motions can render the cloud a virial equilibrium system on scale traced by LABOCA. The IRDC G304.74 has a seahorse-like morphology in the Herschel images, and the filament appears to be attached by elongated, perpendicular striations. This is potentially evidence that G304.74 is still accreting mass from the surrounding medium, and the accretion process can contribute to the dynamical evolution of the main filament. One of the clumps in G304.74, IRAS 13039-6108, is already known to be associated with high-mass star formation, but the remaining clumps and cores in this filament might preferentially form low and intermediate-mass stars owing to their mass reservoirs and sizes. Besides the presence of perpendicularly oriented, dusty striations and potential embedded intermediate-mass YSOs, G304.74 is a relatively nearby (d ~ 2.5 kpc) IRDC, which makes it a useful target for future star formation studies. Owing to its observed morphology, we propose that G304.74 could be nicknamed the Seahorse Nebula.