The part of the url after GitLab server domain name, also known as 'slug' of the project is nothing but combination of namespace and the project name.
Namespace is the path of group or subgroups.
Let us say, the group is 'new-group' and project is called 'my-project'.
Slug will be something like /new-group/my-project.
Then in order to move this existing project from the group into the new subgroups we need to create the new subgroup (down arrow next to the 'New project' button). Then change the project namespace to /new-group/sub-group/my-project.
To do this, go to the project page, click the settings button (cog bottom left). Go to the Advanced settings section. And it's just below the rename project option.
Just select the new subgroup and your done!