LLC on H1B
This question is more legal in nature and related to immigration law. Generally speaking, you can do anything you like as long as you don't violate the terms of your visa. An immigration law specialist would be in a better position to discuss the terms of your H1B visa with you and explain what would violate them and what wouldn't.
As far as I know you are not restricted from owning shares or member interest in US companies, however your involvement in managing the company or working for it would probably be restricted. You should confirm the details with an attorney, however if that is correct then you should have a partner or a manager who is eligible to work for the company and run it on your behalf.