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We have diminutive suffixes in English. However, many of the diminutives have become common words and are no longer recognized as diminutives. Many of the suffixes are borrowed from foreign languages, so a full list of them would be rather large. Usage is common, but informal. There are no strict rules, i.e, there is no way to predict which ending (if any) can be attached to a particular noun, and usage can vary from region to region.
-a short list of diminutive endings-
-ck: bullock, hillock, paddock
-ey/-ie/-y: dearie, doggy, kitty, movie, Mikey, Bobby, Suzy
-ette: diskette, cigarette, kitchenette, suffragette
-let: piglet, chicklet, applet, eyelet, gauntlet, tablet
-ling: duckling, gosling, foundling, darling
-kin: catkin, babykin, munchkin, napkin
Prefixes
mini: minibus, miniskirt, minisub, etc.
micro: microprocessor, microskirt, microscope, etc.
We don't have augmentative suffixes, but there are some prefixes.
mega: megaphone, megalith
over: overseer, overlord, overdrive
super: superman, superdog, superintendent
grand: grandparent, grandmaster
Again, many of the diminutives and augmentatives have become lexalized, i.e, they have become common words that are part of the vocabulary.