I don't know the name of it, but I use a tool called a 'garden claw.' It does the job & no chemicals to worry about, and it's good exercise. I keep a huge pitcher of homemade lemonade in the fridge for refreshment. That's how I deal with these awful weeds - lemonade. :)

It looks like a type of creeping Cinquefoli. Does it have yellow flowers ?
I think it's difficult to get rid of. Maybe you could try a lawn friendly weedkiller if there's too much to dig up.

Use a grass friendly, broadleaf weedkiller. The maxim that nature abhors a vacuum applies here. If the grass is giving space for weeds you might need to improve the grass. Drainage & aeration improvements are probably your best bet after the weedkiller

Think its what we call silverweed and it sure is keen to spread itself.
Just remove as much as you can as often as you can. They send out long runners.

Chemical warfare is the only way you'll get rid of that as it's so creeping. Lawn weedkiller will do the job.

Don't think it's Silverweed Stera, think it's Creeping Potentilla (Potentilla retans). Ghastly plant, this garden is inundated with it, pretty flower, but awful to get rid of. I have poisoned the grass off and the only thing that comes back is this! Digging it up is almost impossible as it has a really deep woody root which grows little nodules - to make more plants!! Bit like some of you complaining about Celandines, and not being unable to get rid of of them.

Meant to add the link - more haste less speed!
https://en.wikipedia.org/wiki/Potentilla_reptans