The directors of Garfinckel, Brooks Brothers, Miller & Rhoads Inc., in the largest business takeover in Washington history, have agreed to sell the company to Allied Stores Corp. of New York for $228 million.

Garfinckel's board, which spurned Allied's first offer three weeks ago, reluctantly agreed to a merger Monday night after Allied upped its offer by approximately $21 million and agreed to keep on Garfinckel's top executives.

The Garfinckel corporation will become an operating division of Allied, which owns 178 department stores and 48 specialty shops in 28 states.

Garfinckel's was one of three companies in the Washington region that were drawn into complex merger negotiations yesterday:

* Gray Drug Stores Inc. of Cleveland, which bought the Drug Fair chain only a few weeks ago, agreed to sell its entire business, including Drug Fair, to Sherwin Williams Inc. Gray, which had not planned to sell, quickly made a deal with Sherwin Williams after a Texas trucking company started buying its stock.

* The Norfolk & Western Railroad -- already planning to merge with Southern Railway of Washington -- disclosed it has purchased more than 8 percent of Piedmont Aviation and plans to "acquire a significant interest" in the rapidly growing regional airline. Details on Page D8

The three multimillion-dollar mergers reflect a rampant trend among U.S. corporations. For companies with plenty of cash or good credit, it is easier and cheaper to expand their operations by buying another going concern than to try to build new lines of business from scratch.

Gray, Garfinckel's and Piedmont all were drawn into the game because they are profitable companies with large numbers of public stockholders, none of whom owns enough of the company's stock to prevent an outsider from buying in.

Under the tentative merger agreement between Allied and Garfinckel, Allied will make a public offer to buy all of Garfinckel's common stock for $53 a share, $5 more than Allied originally offered. The relatively small number of shares of Garfinckel's preferred stock will be redeemed by Garfinckel before the merger.

The $53 price will produce windfall profits for longtime Garfinckel stockholders, because the stock sold for $16 a little more than a year ago.

Garfinckel's board accepted the Allied offer late Monday night after a four-hour meeting in New York City. Several board members refused to comment on the action, referring all inquiries to Garfinckel's Chairman David R. Waters, who along with other company officials could not be reached for comment.

The board had unanimously rejected Allied's initial offer, contending the Washington-based retail chain was worth far more than the $48 a share Allied bid. Waters denounced Allied as "lying, deceitful and treacherous" and charged Allied officials "violated the standards by which decent men do business" by making the surprise bid to buy his company.

Allied became interested in Garfinckel's after Waters tried to sell the New York chain the Miller & Rhoads Department Stores, which are generally considered the weakest link in the Garfinckel chain. Besides the Garfinckel stores in the Washington area, the company owns some 20 Brooks Brothers stores, the Ann Taylor women's wear stores, the Catherine's Stout Shoppes, a chain specializing in large size clothing, three department store groups, Millers in Tennessee, Harzfeld's in Kansas City and Miller & Rhoads in Richmond.

Garfinckel's was trying to sell Miller & Rhoads to raise cash to protect itself from an unfriendly takeover. The company had been vulnerable for more than two years since members of the Joseph R. Harris family of Washington had a falling out with Garfinckel's management and decided to sell their stock, the biggest single block of shares.

Gamble Skogmo Inc. of Minneapolis bought the Harris family stock and enough additional shares to control 20 percent of Garfinckel's, but a series of legal maneuvers kept Gamble from taking over. Then Gamble itself was bought out by Wickes Corp., a California conglomerate.

Wickes decided it wanted either to control Garfinckel or sell its stock and offered to sell the stock back to the company. To get the money to buy the stock, Waters decided to sell Miller & Rhoads.

But as Allied executives were negotiating for Miller & Rhodes, they were also making a deal to buy the shares owned by Wickes. Allied and Wickes reached an agreement last month, and Allied simultaneously announced a public offer for the remainder of the Garfinckel shares.