I have recent experience of setting up a club type organisation. Nothing to do with cycling, but I believe there are some parallels. I've been in other clubs, cycling and other pursuits, sometimes in committee roles. I'm not a lawyer by the way.
As has been alluded to by others, there is a potential risk of litigation if somebody is harmed on a club ride. The injured party may seek somebody to sue, and if there is somebody in a leadership position, they are an obvious target. I believe it boils down to having to prove negligence. If you are out with a group of experienced cycling mates then I believe the same common law applies, but the risk is very low as your mates are experienced (less chance of an accident) and they are mates (less likely to want to sue). On the other hand, if you set up an organisation - whatever you decide to call it - there is the potential for employees of your company that are newcomers to cycling to want to join your rides. These people are neither experienced, or your mates, so in my opinion, the risk of harm and of litigation goes up.
That's not to say that setting up a cycling group at work is in the too-hard-to-do category, but it needs to be done right in order to protect yourself from litigation and riders - particularly inexperiences ones - from harm. In my opinion its more than a question of obtaining insurance. I would expect that to obtain insurance, the insurer would need some documentation that describes what is being insured - a constitution. And I'd expect your company to want you to do risk assessments. Just google for Cycling club risk assessments.
If you are a member of Cycling UK or British Cycling I would expect they could give advice.