First thing’s first:
DO NOT SCRUB YOUR GROUT!
You’re doing more damage to your grout, than you are cleaning it.
The best way to get your grout clean is by using a mild bleach solution (1 tablespoon per gallon)- or making it even easier with a bleach pen. I use the Clorox Bleach Pen.
So next time, skip the elbow grease, and bleach the grout!