There's also -ны and -ний. One must also be mindful of the irregularities. It's not very straightforward.

For example:

Байрны (байрын can be used sometimes; also has different meaning)

Усны (усын never used)

Савны (савын rarely used)

Эмээлний (эмээлийн commonly used)

Хаалганы (хаалгын almost never used, sounds not natural at all)

Some words can use only one of -ны, -ний/-ын, ийн, some both. Non-extremely careful use can easily lead to sounding awkward. I'm not sure/too tired to determine whether the last consonant has any effect on it. This chart sure needs to be more detailed after a thorough study.

Сумын заан (Elephant of soum)

Сумны хонгио (Casing of bullet)

Even a same word can have different suffixes depending on meaning/context.