I'm pretty sure you need to setup sub domains like that with your web host, which then treats each subdomain as an individual webapp. So, yes, you would have to install and maintain Flarum in multiple, redundant instances, which is nuts.
My suggestion would be to use primary categories (aka "tags") as your different group sections, which will give you landing page URLs like domain.tld/t/group-name. Then use the Pages and Links extensions to create an in-site navigation for those different groups.
You can then create user groups in Admin > Permissions to match the different discussion groups.
As I understand it, the user accounts and groups functionality will be getting an overhaul in future releases so it will be possible to make categories (groups) private, or whatever.
Basically, things are pretty early still, so you should not get your hopes up for any immediate solution to a complex need like that.
Maybe others have a better perspective than I.
