If you travel for business but use your own credit card, who gets the frequent flier or credit card points?

I travel a fair amount for work, buying plane tickets and staying at hotels. I used my own credit card and am reimbursed by the company when I turn in receipts.

My credit card earns me a certain amount of points per dollar spent.

Although I’m the person with the name on the credit card, ultimately I am not spending my own money – it’s the company’s.

Should they get the points as well?

What about the frequent flier miles I collect?