Distinct List Items - Sort of

Given this dataset in a List<CustomItem>, where CustomItem = string Name, int Index, int FID, and a few other random fields:

Name	Index	FID	Misc
user1	100	3	random
user1	200	3	random
user1	100	4	random
user1	200	4	random
user1	300	4	random
user20	100	2	random
user20	200	2	random
user20	300	2	random
user20	100	4	random
user20	200	4	random
user20	300	4	random

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I want to extract objects that have the highest Index value, given where Name and FID are equal.

From the above set, I would want these rows:

Name	Index	FID	Misc
user1	200	3	random
user1	300	4	random
user20	300	2	random
user20	300	4	random

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I'm using a non-LINQ method now whereby I start with the first record and iterate through the list doing a manual comparison of line by line to see if I should add it to my resultset.

The LINQ .Distinct doesn't seem to be what I want.  I think it's going to be some combinations of group by's and max and let's, but I'm kinda swimming in unfamiliar territory with this one.