Three-card straight from an eight-deck shoe

Eight standard 52-card decks are shuffled together into one shoe of $416$ cards, and three cards are dealt at random. What is the probability that they form a three-card straight (three consecutive ranks, with the ace allowed low as in A-2-3 or high as in Q-K-A) but not a straight flush?

$\tfrac{768}{24817}\approx0.031$. There are $12$ rank runs, each rank appearing $32$ times in the shoe, giving $12\cdot32^3$ straights. Subtract the $12\cdot4\cdot8^3$ single-suit ones to get $368{,}640$, then divide by $\binom{416}{3}=11{,}912{,}080$.