What are the sources supporting gender segregation in Islam?

I am familiar with references such as: >Tell the believing men to lower their gaze and to be mindful of their chastity: this will be most conducive to their purity – (and,) verily, Allah is aware of all that they do. And tell the believing women to lower their gaze and to be mindful of their chastity, and not to display their charms beyond what may be apparent thereof; hence let them draw their veils over their bosoms and do not show their adornments except to their husbands or their fathers or their husbands' fathers or their sons or their husbands' sons or their brothers or their brothers' sons or their sisters' sons or their women or what their right hands possess or male servants free of sexual desires or those children who never knows the private things of women; and do not stamp their feet so that it may show their hidden adornments; and repent towards God collectively O believers so that you may succeed. > >— [Qur'an: Sura 24 (An-Nur): ayat 30-31] However there is no direct reference to gender segregation. In fact the passage suggests that men and women do meet from time to time and should lower their gaze and be modest.