Is it possible to set order of links that appear in google search in the part where it lists 6 links?
It is not possible to change the order of sitelinks , it adjusts itself according to the user's needs, usually directing the user's most searched or previously searched for links, which can vary greatly according to the Google search engine changes, as I replied in Error creating "Google Sitelinks"
According to Google :
We display sitelinks of results only when we think they might be useful to the user. If your site structure does not allow our algorithms to find good sitelinks, or if you find that the links to your site are not relevant to the user's query, we will not display them.
Sitelinks are currently automated. We are constantly working to improve sitelink algorithms. It is possible that in the future we will incorporate suggestions and information provided by webmasters. However, there are best practices that can be followed to improve the quality of your sitelinks. For example, for your site's internal links, use anchor text and alt-rich, informative, non-repetitive alt attribute text.
In other words, the part that speaks:
It is possible that in the future we will incorporate suggestions and information provided by webmasters.
It states that in the future the webmaster may do something to control the links / suggestions, but so far sitelinks are automated.
It is also worth noting that sitelinks are only displayed as the algorithm finds it necessary / useful to the user. In other words, it may even show, but sitelinks may vary for each user, showing what the algorithm believes to be useful for each specific user.