The drain line is clogged. Snake the drain...

I agree and the clog is likely close. You should be able to rod down the stand pipe through the trap. If the traps is easily acessible remove it and rodding will be easier

I've tried a snake, there's no clog

How old is the house? Has any work been done recently? Is this a new problem or something that has been going on for a while? What floor of the house is the laundry located? How far did you run the snake (approx. feet)?
You may have a clog farther down the line than you think. If something has changed in the house it might be related if you can answer the above questions. A new appliance that discharges a larger volume? The pipes could be on a cross configuration where some water flows across to the laundry sink.
These are things that a plumber would ask and look for when visiting your house.