Voluminous rectangular prisms

There are several rectangular prisms with integer edge lengths that have an internal diagonal of length $2024.$ What is the greatest volume among these prisms?

Let's say we have a retangular prism with length $\ell,$ width $w,$ and height $h.$ The volume is $V = \ell w h$ while the length of the internal diagonal is $L = \sqrt{\ell^2 + w^2 + h^2}.$ In this case, we then need to looks for all Pythagorean quadruples $a^2 + b^2 + c^2 = d^2$ that have $d=2024.$ While we certainly could use primitive Pythagorean quadruples for the odd divisors of $2024$ (namely, $11$, $23,$ and $253$), we might as well just brute force the entire thing and do a dumb search: In a series of nested loops, we can loop through $a = 1, \dots, 2024,$ $b = a, a+1, \dots, 2024,$ and $c = b, b+1, \dots, 2024$ and print out the triple $(a, b, c)$ any time that $a^2 +b^2 + c^2 = 2024^2.$ This yields the following forty-one quadruples:

\begin{align*} 24^2 + 640^2 + 1920^2 &= 2024^2 \\ 24^2 + 1152^2 + 1664^2 &= 2024^2 \\ 64^2 + 168^2 + 2016^2 &= 2024^2 \\ 64^2 + 1344^2 + 1512^2 &= 2024^2 \\ 96^2 + 152^2 + 2016^2 &= 2024^2 \\ 96^2 + 872^2 + 1824^2 &= 2024^2 \\ 96^2 + 936^2 + 1792^2 &= 2024^2 \\ 96^2 + 1088^2 + 1704^2 &= 2024^2 \\ 152^2 + 864^2 + 1824^2 &= 2024^2 \\ 168^2 + 1344^2 + 1504^2 &= 2024^2 \\ 192^2 + 856^2 + 1824^2 &= 2024^2 \\ 192^2 + 1304^2 + 1536^2 &= 2024^2 \\ 224^2 + 600^2 + 1920^2 &= 2024^2 \\ 224^2 + 672^2 + 1896^2 &= 2024^2 \\ 224^2 + 1176^2 + 1632^2 &= 2024^2 \\ 264^2 + 528^2 + 1936^2 &= 2024^2 \\ 264^2 + 1232^2 + 1584^2 &= 2024^2 \\ 280^2 + 576^2 + 1920^2 &= 2024^2 \\ 360^2 + 800^2 + 1824^2 &= 2024^2 \\ 360^2 + 1376^2 + 1440^2 &= 2024^2 \\ 368^2 + 1104^2 + 1656^2 &= 2024^2 \\ 424^2 + 480^2 + 1920^2 &= 2024^2 \\ 424^2 + 768^2 + 1824^2 &= 2024^2 \\ 424^2 + 1248^2 + 1536^2 &= 2024^2 \\ 480^2 + 576^2 + 1880^2 &= 2024^2 \\ 528^2 + 1144^2 + 1584^2 &= 2024^2 \\ 576^2 + 744^2 + 1792^2 &= 2024^2 \\ 576^2 + 928^2 + 1704^2 &= 2024^2 \\ 576^2 + 1216^2 + 1512^2 &= 2024^2 \\ 576^2 + 1368^2 + 1376^2 &= 2024^2 \\ 600^2 + 640^2 + 1824^2 &= 2024^2 \\ 672^2 + 936^2 + 1664^2 &= 2024^2 \\ 672^2 + 1176^2 + 1504^2 &= 2024^2 \\ 744^2 + 1088^2 + 1536^2 &= 2024^2 \\ 768^2 + 1304^2 + 1344^2 &= 2024^2 \\ 768^2 + 1304^2 + 1344^2 &= 2024^2 \\ 856^2 + 1248^2 + 1344^2 &= 2024^2 \\ 864^2 + 1216^2 + 1368^2 &= 2024^2 \\ 928^2 + 936^2 + 1536^2 &= 2024^2 \\ 936^2 + 1152^2 + 1376^2 &= 2024^2 \\ 1104^2 + 1104^2 + 1288^2 &= 2024^2 \end{align*}

The rectangular prism with integral edge lengths internal diagonal length of $2024$ with the largest volume is $1104 \times 1014 \times 1288$ which has volume $1,569,835,008.$ Note that this is not necessarily that far from the overall, non-integral solution of a cube with side length $2024 / \sqrt{3}$ which has volume $1,595,694,111.621353...$, which is only about $1.6\%$ larger than the volume of the integral rectangular prism, so not too shabby.