I would wait til fall & dig up your lily & divide off the bulblets. Asiatic lilies? They do this & will multiply very fast this way. These tiny bulbs will bloom in a year or 2 years.

Do not snap the top growth! The foliage feeds the bulblet during the growing season. If you want to plant this variety at some other place in your garden, you can separate the bulblets and plant them there. Fall is a good time for this as Ngraham said. Otherwise, you can divide the clump when it will become really crowded. Personally, I like to separate baby bulbs when they are at least 1" in diameter.

My Turk's-Cap-Lillies drop seed and produce the bulblets. They are pretty hearty. I pot mine for a season or two and then plant them elswhere around the property. I have some monsters, the hummingbirds love them.