Cho Yang-ho's wife and daughters were already being investigated over alleged assaults, smuggling and illegal hiring of foreign housekeepers.
The two younger women, who held management positions at Korean Air, became viral sensations for temper tantrums dubbed the "nut rage" and "water rage" scandals online.
And the flag carrier's 69-year-old patriarch was himself grilled by prosecutors late last month over alleged tax evasion and other crimes.
It also asserted that the school's admission and graduation of Cho's son Cho Won-tae, now the Korean Air president, were flawed and cancelled his bachelor's degree.
The son was allegedly short of the required academic standards when he transferred to the university from a US college in 1998, and was given a degree in business administration in 2003 despite not having the necessary points, the ministry said.
A month-long investigation also found the school awarded contracts worth millions of dollars to companies associated with the Cho family, disadvantaging potential competitors.
"We will take stern measures ... so that irregularities that have been found can be corrected at an early date," education minister Kim Sang-gon said in a statement, adding the ministry would ask prosecutors to start a criminal inquiry into Cho and his wife Lee Myung-hee.
Inha University rejected what it called "excessive" measures and said it would to dispute them in court.
Yesterday, police wrapped up their assault investigation into Lee Myung-hee and sent the case to prosecution authorities, urging her indictment. Lee has denied most of the charges.
The super-wealthy owners of chaebols -- the sprawling conglomerates that dominate the world's 11th-largest economy -- often attract controversy, but a series of scandals have made the Cho family one of the most notorious in South Korea.
Cho's elder daughter Cho Hyun-ah made global headlines in 2014 for kicking a cabin crew chief off a Korean Air plane for being served macadamia nuts in a bag rather than a bowl. She later served a short prison sentence.
Earlier this year, her younger sister Cho Hyun-min was accused of throwing a drink at an advertising agency manager's face in a fit of rage during a business meeting.
Their father issued a public apology over the "immature" behaviour of his offspring and removed his two daughters from their management roles.
Cho himself has already had brushes with the law, receiving a suspended jail sentence for tax evasion in 2000 and awaiting trial for diverting 30 billion won of company funds for renovating his own house in 2013 and 2014.
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