The elder Mr. Lee is in a coma, leaving Lee Jae-yong as the de facto leader of Samsung and its valuable electronics unit. While Samsung has at times cast the younger Mr. Lee as a strategic thinker who hammers out deals with big foreign partners like Apple and Google, he testified in court this month that he had little knowledge of the details of running the business.

“At least people recognized his grandfather and father for making Samsung what it is today,” said Chung Sun-sup, editor of chaebul.com, a website that specializes in monitoring chaebols. “But people cast a cold eye on Lee Jae-yong because he has little to show for his attempt to inherit Samsung except that he shares a Lee blood line.”

Being in prison does not necessarily mean that Mr. Lee will give up control, however. He has two sisters, but neither has been heavily involved in the electronics business and they generally have not been considered rivals to Mr. Lee.

Other big South Korean family businesses have been run from behind bars. The chairman of SK Group, another family-run South Korean business empire, was convicted in 2013 for misappropriating company funds. A South Korean lawmaker, citing official data, said the next year that the chairman, Chey Tae-won, had nearly 1,800 visitors during more than 500 days in jail, far exceeding official guidelines. SK officials and lawyers opened an office near his prison, local news media said.

Still, if Samsung operates independently of the Lee family, it may run fine on its own thanks to its deep ranks of professional managers. Samsung Electronics has turned in strong results in recent months and has bounced back from an embarrassing spate of fires involving its Galaxy Note 7 smartphones. In the second quarter the company reported record earnings as it overtook Intel to become the world’s biggest chip maker by sales.

Perhaps the most daunting challenge to Samsung is one that the company has not been able to solve even with Mr. Lee free and in charge. While the company dominates hardware, from memory and chips to the finished smartphone and television, it has long struggled in the realm of software and services.

Another challenge could arise with the death of the elder Mr. Lee, which could set off the type of infighting that split up other chaebol families and empires in recent years, said Park Sang-in, a professor of public administration at Seoul National University and a chaebol critic.