Group object is quite unique as it stores multiple types of data, from Public Group, Queue, Role, etc., you can see all the Type values in this article Group.
Note: The Type for Public Group is Regular
Sample query: SELECT Type, COUNT(Id) FROM Group GROUP BY Type
For user members in Group and Queue can be a query to the GroupMember object. Sample query:
All queue with queue members
SELECT Id, GroupId, Group.Name, UserOrGroupId FROM GroupMember WHERE Group.Type = 'Queue'
All public group with public group members
SELECT Id, GroupId, Group.Name, UserOrGroupId FROM GroupMember WHERE Group.Type = 'Regular'
All inactive users in a queue or public group
SELECT Id, GroupId, Group.Name, Group.Type, UserOrGroupId FROM GroupMember WHERE UserOrGroupId IN (SELECT Id FROM User WHERE IsActive = false) ORDER BY Group.Name
Reference: