GaSb nanostructures integrated on Si substrates are of high interest for p-type transistors and mid-IR photodetectors. Here, we investigate the metalorganic chemical vapor deposition and properties of GaSb nanostructures monolithically integrated onto silicon-on-insulator wafers using template-assisted selective epitaxy. A high degree of morphological control allows for GaSb nanostructures with critical dimensions down to 20 nm. Detailed investigation of growth parameters reveals that the GaSb growth rate is governed by the desorption processes of an Sb surface layer and, in turn, is insensitive to changes in material transport efficiency. The GaSb crystal structure is typically zinc-blende with a low density of rotational twin defects, and even occasional twin-free structures are observed. Hall/van der Pauw measurements are conducted on 20 nm-thick GaSb nanostructures, revealing high hole mobility of 760 cm2/(V s), which matches literature values for high-quality bulk GaSb crystals. Finally, we demonstrate a process that enables cointegration of GaSb and InAs nanostructures in close vicinity on Si, a preferred material combination ideally suited for high-performance complementary III-V metal-oxide-semiconductor technology.