I started to make a list of all the "components" (I don't know how to call them) all the joyo kanjis are made of, for me to be easier to memorize all the kanjis (I memorize them by remembering the components the kanjis are made of). At first I thought this list could match the radicals list, but it isn't the case. I was wondering if that list already exists and it has a name, since mine (although it isn't cleaned up) it has over 400 "components", and it isn't sorted in any readable way, so I could improve mine.
Unfortunately, this is a bit of a complicated situation, as there are a few closely related ideas:
- Any piece of a kanji you might recognize as appearing in multiple kanji.
- Any piece of a kanji that either is on the official list of radicals from the Kangxi dictionary, or on a closely related list (such as a slightly edited version of that list for another dictionary).
- The particular piece of a kanji that is used to index it in the Kangxi dictionary (or similar).
1. and 2. are both commonly called "(kanji) components".
2. and 3. are both commonly called "(kanji) radicals".
And 3. alone is called "the main radical".
But a bigger list of components in sense #1 is more rare. A list of 407 (or so) components can be found at Kanshūdō.
The "pieces" are called "radicals", and yes there is a list of all of them. There is a list of 214 radicals used in the Chinese language called the "Kangxi Radicals", located here.
There is a simplified version of these that does away with non-Japanese characters and archaic usages here.
You can also find the main radical for every Joyo Kanji here.
Each radical has a name as well, the full list can be found here.
As you know, the closest concept to your "pieces" is that of radicals.
However, one might use this term to refer to slightly different concepts, varying in strictness:
A radical of a given character is the unique identifier (picked out of a list of 214 radicals) of this character in a dictionary (especially the Kangxi dictionary).
A radical of a character is any part of the character that appears in the list of the 214 Kangxi radicals.
A radical is any part of a given character.
For the jōyō kanji, the indexing radicals are listed on the official list (for example on Wikipedia). Every kanji is either the base character of a radical, or is assigned a radical from the list.
The very first kanji, 亜, already illustrates very nicely why this assignment is not always meaningful when looking at the origin of a character.
From the etymological perspective, 亜 (or rather its traditional form 亞), is a 象形文字 depicting (most likely) the (dark) foundation/basement of a building. The meanings of 悪 and 唖 may be thought of 会意・形声文字 derived from this meaning of "dark".
Being a simple picture, it doesn't have any meaningful components, certainly not any related to 二 "two" which is its indexing radical on the jōyō list.
Moreover, several etymologically distinct radicals have been unified to one form, the most prominent example being the radical ⽉, which represents both 月 (e.g. in 明) and 肉 (e.g. in 腹). Another example is ⺍ which unifies at least the top of 螢 → 蛍 and 學 → 学.
I think it would be interesting to have a list/database of all "pieces", which are based on the characters' actual origins. 亞・亜 would be a "piece" of 亜, 悪 and 唖 and 二 wouldn't be a piece of 亜. Also, 亞・亜 would not be a piece of 壺・壷, because 壺 is itself a 象形文字, and only contains something like 亜 for cosmetic reasons.
In some cases, the origin of a character is unknown, but it's still possible to compile such a list based on a decent kanji etymology dictionary, like 新漢和大辞典. (I also have 漢字源 in my Canon 電子辞書.)
I've made a website that might help. It decomposes complex Kanji into simpler ones.