Is there any blood group named A1B blood group?

Yes!

There are two subtypes of the A-bloodgroup:

80% have group A1, and 20% have group A2.

This is relevant in the donations of organs.

Group A2 activates the immune defence in the recipient a lot less, which is good.

People with group A2 are therefore preffered as donors.

But I think that if you'll ever need a kidney yourself, you're better off with having group A1.