The Russian word дог is a false cognate: it doesn't mean dog. It means Great Dane. Although regular nominative plurals often end in -ы, here spelling rule 1 interferes so that the plural is доги. The three Russian spelling rules are:

Spelling rule 1:

After г, к, х, and ж, ч, ш, щ — never write ы, instead use и.

Spelling rule 2:

After ж, ч, ш, щ, and ц — never write an unstressed о, instead use е.

Spelling rule 3:

After ж, ч, ш, щ, ц, and г, к, х, — never write я or ю, instead use а and у.