Calculate $[(12^{12} \div 12^{11})^2 \cdot 4^2] \div 2^4$.
Remembering proper order of operations, first simplify the terms in the parentheses using the quotient of powers rule:

$12^{12} \div 12^{11} = 12^{12-11} = 12$ so that the expression becomes  \[(12^2 \cdot 4^2) \div 2^4 = 12^2 \cdot 4^2 \div 2^4.\] Since $4^2 = 4 \cdot 4 = 2 \cdot 2 \cdot 2 \cdot 2 = 2^4$, we have \[12^2 \cdot 4^2 \div 2^4 = 12^2 \cdot 1 = \boxed{144}.\]