In Brazil the subprefectures (Portuguese: subprefeituras) are administrative divisions of some big cities, such as São Paulo and Rio de Janeiro. The head of a subprefecture, the subprefeito, is indicated by the municipality's mayor (in Brazil called prefeito).
In São Paulo there are 32 subprefectures. The largest in total area, Parelheiros, covers 353.5km2, and the most populous, Capela do Socorro, has more than 600,000 inhabitants.
It was used in Qing dynasty. Called ting (廳 or 厅) in Chinese, it is also on the same level as a department (州) and a district (縣), and is below a prefecture (府).
Some Japanese prefectures have branch offices called 支庁 (shichō) in Japanese, which are translated in English as "subprefectures", "branch offices", or "branches of the prefectural government". See details in Subprefectures of Japan and an example of Kushiro Subprefecture.
Taiwan
The word "subprefecture" is used to translate the term tīng (廳) from Chinese. This was a type of subdivision under Qing dynasty rule.
Examples include Tamsui Subprefecture (淡水廳 dàn shuĭ tīng) and Kavalan (Ga'malan) Subprefecture (噶瑪蘭廳 gá mă lán tīng, of Yilan City) (both in Taiwan).
Under Japanese rule, the term 廳 (chō) is translated to prefecture, so 支廳 (shichō) is translated to subprefecture.
Notes
Look up subprefecture in Wiktionary, the free dictionary.
↑Aihun Ting map from 1911 Atlas of Heilongjiang (in Chinese)