A division simply means a section of a large company. A division of a particular company is being run to deliver the exact services as the original company, although a division can be established with a different name entirely. A division might be created to serve as the extension of a particular company for some economic reasons. For example, the parent company might be selling different construction and building materials, while a division can be created to be selling a particular type of building materials.
Most times, a division is created to improve the public relevance of a company. A subsidiary, on the other hand, is a company owned by a parent company which is created to deliver and offer different services that are entirely different from what the parent company is offering. For example, the parent company might be into automobiles production, while a subsidiary might be selling construction materials.