logical deduction – Sushi Sudoku. Yummy!

I love Japanese seafood. When I solve or compose puzzles, I often have with me a 6×6 plate containing 36 sushi pieces, a pot of matcha (Japanese tea) and a dish containing two kinds of wagashi (Japanese pastry), mochi pieces which are sweet rice cakes and yokan pieces which are red bean gelatins. It is a total cultural immersion.

When I look up after a while, I discover that only ten sushi pieces remain, as shown in the diagram below.

S = Scallop     C = Crab       A = Abalone
L = Lobster     O = Oyster     P = Prawn

These are the only kinds of sushi pieces on the plate. I remember that originally, each row and each column contained six different pieces of sushi. Moreover, if the plate is divided into six 2×3 sections or six 3×2 sections, each section also contained six different pieces of sushi. I wonder what kinds of sushi the other pieces were.

Text version of puzzle image:

. . A . . .
. C . L . .
S . . . O .
. O . . . P
. . O . S .
. . . A . . 

Attribution:

Tasty Japanese Morsels in Recreational Mathematics by Yoshiyuki Kotani