Subsidiary and division are business arms of a company. Subsidiary is a divisional company owned by the parent company. A division is on the other hand is a portion of the business that operates under a different name. This is the main difference between subsidiary and division.
Division is the equivalent of a corporation or limited liability company. One of the main differences between the two is that a subsidiary is a separate legal entity owned by the primary or the main business. A division on the contrary is a part of the main business.
It is interesting to note that a division is not quite separate from the main business. It is equally important to note that if the division runs into any debt then it is the primary business that has to take the responsibility legally.
A corporation or a limited liability company would have a division. A lone company for that matter cannot be a subsidiary of any organization. On the other hand only an entity can be a subsidiary.
You might wonder whether a parent company can be smaller than a subsidiary. As a matter of fact a parent company can very well be smaller than a subsidiary. It is also possible that a parent company can be larger than all of its subsidiaries or some of its subsidiaries. It is to be observed that a parent company and a subsidiary need not do the same business or need not operate at the same place for that matter.
A division on the other hand should necessarily be doing the same business as the parent company. This is so because a division is after all a part of the primary business but with a different name. It is a portion of that very same business that operates at the same place or at a different place too.