Optimizing GROUP BY Query
Your indexes won't be used because they won't narrow down the table scan. 
As a matter of fact, you want the max time for each type, so even if type index were used as a first intention, you'd still have to fully scan the segment at this index to get the max time for a given type. As a result, MySQL would have scanned the whole table, as a sum of type-indexed segment scans.

Using (type, time) index would in theory give a direct access to the last (max) time, for each type sub-tree. But as far as I know, some older MySQL versions have trouble regarding index selection for this kind of queries.