Dividing is done differently than the original planting. When dividing, you need to know where they are before you start digging! Therefore, it is done in the late spring, when the foliage starts dying (and never before they're done blooming for the year--the energy will have gone to the stem and you'll be shocked to find NO BULB under there until the energy returns to its underground storage. JUMPING THE GUN WILL KILL THE PLANT!!! I cannot emphasize this enough!)
After the foliage turns yellow, take a large shovel and dig WAY down so you're sure to get under the bulb. Do not blame yourself if you miss on some of your first attempts at this and shovel through a bulb. We're all human! It's well worth it once you get the knack of it.

Once you get under the bulb, carefully dig it up. Gently remove the soil. If it won't come off, use a hose to wash it off. This can make quite a mess, so be sure to wear old clothes.
Then, SURPRISE! you will see at least 4 bulbs are there, each with its own stem. Carefully (this may take some firmness) separate the bulbs.

THEN REPLANT THEM, THE SAME DAY, in the spring! Since you will have just dug them up, they will still be in the current growth cycle. If you cannot plant them back the same day, it is safer to wait till fall to replant. But it is MUCH BETTER to do it all at once. You will know where the other bulbs are! It is very risky to dig in an established bed when you don't know where the others are. The next spring you may find your transplants are the only ones left unless you are SURE where the others are!

replanting them is the same procedure as an original planting, only you will need to clean off the outer layer of old bulb skin from the bulb. This is done by carefully rubbing it off. It will help keep the bulb healthy longer if you clean off the decomposing outer skin. Bulb mites like to hide in this outer skin which is usually in some state of decomposition. It's Grody! But once this is done, you will see a PRISTINE skin underneath! It usually looks better than the ones you buy in the store once cleaned. This is because YOURS haven't been drying out for months on some rack, and like fresh vegetables, there's nothing like 'em!