We recently had a client ask us - "what are those sublinks under my main homepage link on Google after I search for my company's name?"
The context of this question, which we've been asked frequently over the years is always around wanting to "control" or "edit" them.
Well, those are called Sitelinks, and Google does not allow you to edit them.
According to Google, they are: "are meant to help users navigate your site. Our systems analyze the link structure of your site to find shortcuts that will save users time and allow them to quickly find the information they're looking for."
Google only "shows sitelinks for results when (they) think they'll be useful to the user. If the structure of your site doesn't allow (their) algorithms to find good sitelinks, or (Google doesn't) think that the sitelinks for your site are relevant for the user's query, (they) won't show them.
At the moment, sitelinks are automated. (Google is) always working to improve (their) sitelinks algorithms, and may incorporate webmaster input in the future. There are best practices you can follow, however, to improve the quality of your sitelinks. For example, for your site's internal links, make sure you use anchor text and
alt text that's informative, compact, and avoids repetition.
One thing you can do now, however, if you're seeing links that you just don't want to be viewed at all on Google, is block them via Google Webmaster Tools (Search Console). This article tells you how.