Posts usually show on their single post view all tags that belong to that post. Since they are displayed by your theme, these tags are not aware of tag groups.
If you want to display your tags sorted into groups, you can use the “Post Tags” feature. When activated, we insert a table with groups (optionally with parent groups) and tags at the end of the post content. We use of course only tags belonging to that particular post. It is also possible to hide the default tags in most themes so that they won’t appear twice.
Since your theme is hard-coded, we cannot insert the tags at any position. It is only possible to append them to the post content and change the order compared to anything inserted by other plugins.
If you need to show the tags of a post in another position, you can try one of the shortcodes in a widget and add the parameter
You can activate and configure this feature in the Front End settings.