filtering – How to delete certain terms in a polynomial with variables in subscript?

I have a polynomial in the following style,

polynomialdata= Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0] + Subscript[xy, 1, 1][0]*Subscript[xy, 2, 1][0] +  Subscript[xy, 1, 2][0]*Subscript[xy, 2, 2][0] +  Subscript[xy, 2, 1][0]*Subscript[xy, 2, 2][0] + Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0]*Subscript[xy, 2, 1][0]*Subscript[xy, 2, 2][0] + Subscript[xy, 2, 1][0]*Subscript[xy, 3, 1][0] + Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0]*Subscript[xy, 2, 1][0]*Subscript[xy, 3, 1][0]

I wonder how to remove certain terms in the polynomialdata such as Subscript[xy, 2, 1][0] Subscript[xy, 3, 1][0] or keeping terms such as Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0]*Subscript[xy, 2, 1][0]*Subscript[xy, 3, 1][0].

Probably one stupid way is to do the following

Keep[expr_] := 
 expr /. {Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0]-> 0, Subscript[xy, 1, 1][0]*Subscript[xy, 2, 1][0]-> 0, Subscript[xy, 1, 2][0]*Subscript[xy, 2, 2][0]-> 0, Subscript[xy, 2, 1][0]*Subscript[xy, 2, 2][0] -> 0, Subscript[xy, 2, 1][0]*Subscript[xy, 3, 1][0]-> 0}

Keep[data]

the result gives 0; which is not I expected.

I want the outcome to be Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0]*Subscript[xy, 2, 1][0]*Subscript[xy, 2, 2][0] + Subscript[xy, 1, 1][0]*Subscript[xy, 1, 2][0]*Subscript[xy, 2, 1][0]*Subscript[xy, 3, 1][0]. Maybe I could use DeleteCases function.

However, if the data has many terms, it will be very inefficient to do the above removing. Is there any way to solve it? Thank you very much in advance!