Community group appears as an active group after editing the group and selecting the "Inactive" button.
The "inactive" check box for any group only keeps the group from being offered in future school years. Even inactivating an academic group still keeps the current section active.
Since most Community Groups are continuous and don't have to be offered, the inactive button is moot. You will still need to go in and either delete the section of the community group or set the group page access to not allow any other personas besides the Community Group Manager.