Garlic will last in the fridge for up to 3 weeks as long as the garlic has not been peeled.
If the garlic has been peeled then the peeled garlic will last for up to 1 week before it goes bad.
If the garlic is not peeled the garlic does not have to be refrigerated and will last for a few weeks out of the refrigerator.
But to keep the garlic fresher longer place it in the refrigerator.
If you do cut the garlic up or peel the garlic then you need to refrigerate it to prevent it from going bad.
Once you mince, slice, cut the garlic cloves then you should use them within a day or refrigerate them to keep the garlic from going bad.
I've kept sliced and minced garlic in the refrigerator for up to a week and it was okay.