Let A be an m x n, (0, 1)-matrix. A submatrix of A is odd if the sum of its entries is an odd integer and even otherwise. The maximum number of 2 x 2 odd submatrices in a (0, 1)-matrix is related to the existence of Hadamard matrices and bounds on Turán numbers. Pinelis [On the minimal number of even submatrices of 0-1 matrices, Designs, Codes and Cryptography, 9:85-93, 1994] exhibits an asymptotic formula for the minimum possible number of p x q even submatrices of an m x n (0, 1)-matrix. Assuming the Hadamard conjecture, specific techniques are provided on how to assign the 0's and 1's, in order to yield the maximum number of 2 x 2 odd submatrices in an m x n (0, 1)-matrix. Moreover, formulas are determined that yield the exact maximum counts with one exception, in which case upper and lower bounds are given. These results extend and refine those of Pinelis.