Lawrence and Bennett were arrested and charged with drink-driving after they were involved in a crash after a team-building dinner on 24 September.
The duo is set to appear at court on 15 October, while club captain Richard Keogh has been ruled out for the rest of the season because of an injury sustained on the night in question.
Derby released a club statement on Saturday (AEST) stating it had completed its own disciplinary hearing and fined Lawrence and Bennett the equivalent of six weeks' wages, the maximum amount within their contractual limit.
The two players will also be required to serve an additional 80 hours of community and rehabilitation, which will include a drink aware course.
Bennett released his own statement along with Derby's announcement, describing his shame at the incident.
"On the night of Tuesday 24 September, I made huge mistakes after consuming alcohol," it read. "What I did was wrong.
“Being a footballer and someone who young fans look up to, I know the example I set is key. I am ashamed. At 23, I am old enough, and I was brought up, to know better.
"I understand that such actions have serious consequences. I take full responsibility for my actions and accept the punishment that the club has issued.
"I have to and will learn from these events and will strive to be a better person.
"There's no excuse for what I've done. I'm sorry to my family, my friends, the club, all Derby supporters and to the wider football community, I let you all down."