Crown Books, the fast-growing discount subsidiary of Dart Drug Corp., reported yesterday that it earned a profit of $2.9 million, or 31.9 percent of Dart's $9.1 million in earnings last year, and accounted for 20.8 percent of the parent company's $291.8 million total sales.

It was the first time Crown had reported sales and profits separately from those of Dart. The figures were listed in a prospectus filed with the Securities and Exchange Commission for the first public offering of Crown stock.

Crown said that 2.5 million of its shares, or about 32 percent of the total, will be offered next month at an opening price of $19 to $23 per share. At the lower figure, the sale will raise $47.5 million in new capital for Crown, of which $20 million will be used to more than double the size of the book chain, from 123 stores to about 275, the prospectus said. The rest will be used to pay off an interest-free $5 million loan from Dart and for "general corporate purposes."

By spinning off Crown and at the same time retaining majority ownership, Dart and its partner, California-based Thrifty Drug Stores, are staging a repeat of Dart's earlier coup in selling shares of its Trak Auto subsidiary. The earnings potential of the Trak and Crown stocks, which is reflected in Dart earnings, has driven the price of Dart shares on the over-the-counter market to $116 bid yesterday, up from $12 a year ago.

According to the figures in the Crown prospectus, Crown and Trak together accounted for $6.91 million of Dart's $9.1 million profits in the year ended Jan. 31. This appears to confirm market analysts' theories that earnings growth in the drug stores has lagged behind that of the book and auto-parts subsidiaries.

Robert M. Haft, 30, who founded Crown Books in 1977 with an assist from his father, Dart Chairman Herbert H. Haft, is listed in the stock prospectus as director, president and chief executive officer of the new Crown Books Corp. His salary will rise from $75,000 a year at present to "at least $200,000" as of July 1. The younger Haft is also president of Trak Auto, but Trak's initial prospectus, issued in March, did not report his Trak salary.