Let us write $p = abc$, $q = def$ and $r = ghi$.
We first show that it is possible to make the maximum of the three numbers $p, q, r$ to be no more than $72$. Indeed, if we consider the following, we see that this is possible:
$$1 \times 8 \times 9 = 72, \quad 2 \times 5 \times 7 = 70, \quad 3 \times 4 \times 6 = 72.$$
Next, we will show that the maximum of the three numbers $p, q, r$ must be at least $72$. Since all the numbers $a, b, c, d, e, f, g, h, i$ lie in between $1$ and $9$ and are distinct, we see that the product $pqr$ of the three numbers $p, q, r$ must be a constant, which from the example above must equal $72 \times 70 \times 72$. In particular, we have $70^3 < pqr$. Thus the maximum of $p, q, r$ must be greater than $70$. Since $71$ cannot be written as a product of integers lying in between $1$ through $9$, we see that it is impossible to have any of $p, q, r$ to be $71$. Therefore, the maximum of $p, q, r$ has to be at least $72$.