SAN FRANCISCO — Google’s parent company, Alphabet, said Tuesday that Eric Schmidt, its former chief executive, planned to relinquish his position on the board of directors in June.

His departure will end an era for the internet giant, in a shake-up of one of the coziest and most stable corporate boards in Silicon Valley. Another member, Diane Greene, who gained her seat in 2012 and had been running Google’s cloud computing business until this year, will also not seek re-election.

Mr. Schmidt, who was Google’s chief executive for a decade until 2011 and then its executive chairman for seven years, oversaw the meteoric rise of Google from a useful search engine into an internet powerhouse.

He was brought into the company in 2001 to provide oversight for its young founders, Larry Page and Sergey Brin. He helped take Google public and oversaw major acquisitions like YouTube and DoubleClick, which cemented Google as an industry giant.