1. Count the elements of $S$

   The set is $S=\{2,4,8,16,\ldots,512\}=\{2^1,2^2,2^3,\ldots,2^9\}.$ So $S$ has $9$ elements.

2. Condition on the partition

   We must partition $S$ into three pairwise disjoint sets $A,B,C$ such that $A\cup B\cup C=S,$ and each has equal number of elements.

   Since $|S|=9$, each of $A,B,C$ must have $\frac{9}{3}=3$ elements.

3. Interpretation of “maximum number of such possible partitions”

   A partition into sets $A,B,C$ means the three groups are labeled. So we count the number of ways to distribute the $9$ distinct elements into $A,B,C$ with $|A|=|B|=|C|=3.$

4. Count the number of ways

   Choose $3$ elements for $A$ from $9$: $\binom{9}{3}.$

   Then choose $3$ elements for $B$ from the remaining $6$: $\binom{6}{3}.$

   The remaining $3$ elements automatically form $C$: $\binom{3}{3}=1.$

   Therefore total number of partitions is $\binom{9}{3}\binom{6}{3}\binom{3}{3}=84\cdot 20\cdot 1=1680.$

5. Check options

   • A: $1640$ ❌
   • B: $1520$ ❌
   • C: $1710$ ❌
   • D: $1680$ ✅

Hence, the required number of partitions is $\boxed{1680}.$