11. (10 points) Divide the natural numbers $2, 3, 4, \cdots, n$ into two groups, satisfying (1) the product of any two numbers in the same group is not in this group; (2) any number and its square are not in the same group. Then the maximum value of $n$ is $\qquad$
【Analysis】If 2 is in the first group, then 4 is in the second group, 16 is in the first group, 8 can only be in the second group, at this point 32 cannot be in either the first or the second group, so the maximum value of $n$ does not exceed 31, which leads to the conclusion.

【Solution】Solution: If 2 is in the first group, then 4 is in the second group, 16 is in the first group, 8 can only be in the second group, at this point 32 cannot be in either the first or the second group, so the maximum value of $n$ does not exceed 31.

When $n=31$, the construction is as follows: $(2,3,5,7,11,13,16,17,19,23,24,29,31)$ and $(4,6,8,9,10,12,14,15,18,20,21,22,25,26,27,28,30)$, so the maximum value of $n$ is 31. Therefore, the answer is 31.