Although we've seen this before, rare indeed is the Best Director lineup where the majority of nominees are representing films not nominated for Best Picture. Indeed, I believe since the category went five-wide in 1936, three lone directors have only occurred five times: 1954, 1955, 1962, 1963, 1966. Curious that they're all within a 12-year span, coinciding with the winnowing-away of the Hays Code and the growing appreciation for international cinema.
The lone directing nominees here represent David and Lisa, a drama about youths in a residential home for mental health issues; Divorce Italian Style, an Italian comedy about a man who plots the death of his wife so he can seduce his teenage cousin; and The Miracle Worker, a surprise Best Picture snub, about Anne Sullivan teaching Helen Keller how to communicate. The directors "snubbed" are The Longest Day's triumvirate of Ken Annakin, Andrew Marton, and Bernhard Wicki (with significant though uncredited contributions from Gerd Oswald and Darryl F. Zanuck); The Music Man's Morton DaCosta; and Mutiny on the Bounty's Lewis Milestone (with uncredited filming from Carol Reed, who was fired mid-production). Here's who made it: