Apemosyne was killed by her brother.
An oracle told Althaemenes that he would murder his own father. Althaemenes took Apemosyne and left Crete for Rhodes. A despondent Catreus sailed the seas searching for his children. In the middle of the night, his ship stopped at Rhodes and was mistaken for a pirate ship. Althaemenes and others attacked the ship, and the prophecy came to pass; Catreus died at the hands of his son.
But Catreus, son of Minos, had three daughters, Aerope, Clymene, and Apemosyne, and a son, Althaemenes. When Catreus inquired of the oracle how his life should end, the god said that he would die by the hand of one of his children. Now Catreus hid the oracles, but Althaemenes heard of them, and fearing to be his father's murderer, he set out from Crete with his sister Apemosyne, and put in at a place in Rhodes, and having taken possession of it he called it Cretinia. And having ascended the mountain called Atabyrium, he beheld the islands round about; and descrying Crete also and calling to mind the gods of his fathers he founded an altar of Atabyrian Zeus.2 But not long afterwards he became the murderer of his sister. For Hermes loved her, and as she fled from him and he could not catch her, because she excelled him in speed of foot, he spread fresh hides on the path, on which, returning from the spring, she slipped and so was deflowered. She revealed to her brother what had happened, but he, deeming the god a mere pretext, kicked her to death.