There's a preconception about the terms "subconscious" coming from it's frequent pairing with the word "mind". The term "subconscious mind" indicates something where there is always a lot subterranean "thinking", going on in some part of our brain that we have no direct access to. That's distinct from something like the activity of the autonomic system, which, although constantly at work outside of our conscious control and awareness, isn't doing something we would characterize as "thinking". Nor would accessing proceedural memories for the fine motor control it takes to hit a key when typing be called "thinking".
I think the OP had the "sunconscious mind" type notion as an assumption: something that can work out problems, form it's own opinions and beliefs, and generally "think" just as the conscious mind does, except behind closed doors This is the classic "subconscious" that comes out of psychiatry, anyway, where people are said to be operating on subconscious beliefs and motivations, and so forth.
I don't think this kind of subconscious probably actually exists.
What Rayne said: "There is a breakdown of memory into implicit memory (procedural) and explicit memory (facts and events). An interesting note that is usually made on the topic is that someone with complete amnesia due to something like hippocampus destruction will retain the ability to, for example, ride a bike or play piano. They can even learn new procedures." is important. There seem to be two distinct kinds of memory. Facts and events are put into, and retrieved from, memory by the action of the hippocampi. Procedural memory is put in place by some other unknown mechanism, but requires, at least, some repetition and practise. You can teach a person with bilateral hippocampal destruction new procedures, and they will remember them. What they will not remember is the facts and events of having learned those procedures. You can teach them to ride a bike on Wednesday. On Thursday, you can ask them to ride a bike for you, and they'll be able to do it. If you ask them when and how they learned to ride a bike, they will honestly admit they have no idea.
So, the distinction between those two kinds of memory is important to bear in mind when you're wondering about memory and what might be called "subconscious". The act of typing is probably a constant back and forth between subtle conscious decisions and automatic following of bits of procedure from procedural memory. The important processing is probably all conscious, but scattered in bits and pieces as little, subtle decisions that are made as needed about what procedures to call up. The better the typist, the fewer decisions you have to consciously make because you're better able to call up the procedures for whole words, and maybe even phrases.
As for laying a problem aside only to find you seem to have arrived at the solution "subconsciously" some time later, my assessment of this is not that there has been some "thinking" going on beneath the level of conscious thought, but that you have cleared extraneous, irrelevant, and perhaps contradictory information from your conscious mind in the meantime, stuff that was interfering with thinking clearly about the problem in the first place. It's probably roughly analagous to having to shut your computer off to clear out your RAM when you've got it so overloaded everything starts going in slow motion. Once you've cleared the conscious thoughts that were interfering with finding the solution, the solution can suddenly be clear the next time you think about the subject.