Gigantimax Pokemon are pretty cool, and can be useful in battles. But the problem is, sometimes I catch them and send them to boxes, where I may forget them until I search for that specific Gigantimax Pokemon again later. After that, I may forget which pokemon are Gigantimax, and which aren't. Is there a way to have a filter that shows me all the Gigantimax Pokemon that I have?
There is unfortunately (as far as I can remember), no filter option to search your boxes for Pokémon that can Gigantimax. You can, however, determine if a Pokemon can Gigantimax by viewing the summary details. If it can, there will be a large X-like symbol next to it's name.
As Wondercricket said in his answer there is no built-in way to recognize a Gigantimax Pokemon from a regular one without checking its summary.
That said, what is worth mentioning is that marks that you apply on a Pokemon are visible while scrolling the box list.
This means that you can use marks to create your very own categorization system. While in the Sun/Moon era this was used mostly to mark perfect IV in a specific stat, now they are more often used to mark other things (since the judge feature kinda eliminates the need for specific IV markers).
For example, take in consideration this (obviously unofficial) table created by a Youtube user (link below) with some suggestions:
(source here, )
Obviously this is not a full solution but could help you to avoid the need to check the summary every time.