A subsidiary is a company that is controlled by its parent company. Even though a subsidiary works on its own, and its purposes are different from that of its parent company, the parent company is in control of its subsidiary.
A company's subsidiary is different from a company's division because a division works together with its parent company. A division's goal is usually similar to that of its parent company, and the division has to use its parent company's name. A subsidiary can use a name different from that of its parent company.