In 2001, a fledgling Internet company named Google opened its first overseas office in Japan, eager to tap a huge technology market.

But after eight years, Japan is one of a few major countries Google has yet to conquer. The Web giant still trails far behind Yahoo Japan, the front-runner here, operated by the Japanese telecommunications giant Softbank.

In a reversal of the rivalry in the United States, Yahoo Japan dominates Japan’s Web search market with 56.5 percent of all queries, according to the Internet research company, GA-Pro. Google, at 33.7 percent, is a distant second.