Moving an existing GitLab project into a new subgroup
Turns out the "slug" for a project... the part of the URL after the GitLab server domain name is made up of the "namespace" and the project name. The name space is the group/subgroup path, so I was looking to transfer project to new namespace.
So for example if the group is "important-group" and project is called "project". Then the slug will be something like /important-group/project. To then move that to /important-group/sub-group/project, we need to create the new subgroup (down arrow next to the "New project" button). Then change the project namespace.
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!
Here is the GitLab docs link with more info on managing projects in GitLab, in case that is useful to anyone.