The production, an adaptation of Mark Haddon's best-selling novel about a teenage mathematician with behavioural difficulties, has been tipped for Best New Play, Best Director and Best Actor for its lead Luke Treadaway, among other categories.
Treadaway faces tough competition from Rupert Everett (The Judas Kiss), James McAvoy (Macbeth), Mark Rylance (Twelfth Night) and Rafe Spall (Constellations).
The production will go up against Constellations, The Audience and This House for Best New Play.
Dame Helen Mirren has been named in the Best Actress category for her turn as Queen Elizabeth II in The Audience, and will fight it out with Billie Piper (The Effect), Kristin Scott Thomas (Old Times), and Hattie Morahan (A Doll's House).
Stephen Daldry is in the running for Best Director for the royal drama, and he goes up against Marianne Elliott (The Curious Incident of the Dog in the Night-Time), Jeremy Herrin (This House) and Simon McBurney (The Master and Margarita).
New musical Top Hat is also a big contender with seven nominations, while Sweeney Todd: The Demon Barber Of Fleet Street has six.
The stage adaptation of Whitney Houston's film The Bodyguard is nominated for Best New Musical - against Loserville, Soul Sister and Top Hat - while its lead, R&B star Heather Headley, is tipped for Best Actress In A Musical.
Singer Will Young has also been recognised for his West End debut in Cabaret - he's been named in the Best Actor In A Musical category, alongside Michael Ball (Sweeney Todd), Alex Bourne (Kiss Me, Kate) and Tom Chambers (Top Hat).
The winners will be revealed during a prizegiving in London on 28 April.