smc

Since the prime factorization of $110$ is $2\cdot 5\cdot 11$, we have that the number is equal to $2\cdot 5\cdot 11\cdot n^3$. This has $2\cdot 2\cdot 2=8$ factors when $n=1$. This needs a multiple of 11 factors, which we can achieve by setting $n=2^3$, so we have $2^{10}\cdot 5\cdot 11$ has $44$ factors. To achieve the desired $110$ factors, we need the number of factors to also be divisible by $5$, so we can set $n=2^3\cdot 5$, so $2^{10}\cdot 5^4\cdot 11$ has $110$ factors. Therefore, $n=2^3\cdot 5$. In order to find the number of factors of $81n^4$, we raise this to the fourth power and multiply it by $81$, and find the factors of that number. We have $3^4\cdot 2^{12}\cdot 5^4$, and this has $5\cdot 13\cdot 5=\boxed{325}$ factors.