Use of the concept of subgroup vs field extension
My answer must be partial and idiosyncratic. People may, should, and I hope will, object to my opinions. And correct my misstatement of facts too!
When we look at the smallest possible group, namely the trivial group, to ask for extensions is to ask about all groups! Not an interesting viewpoint, I think. Even when we look at a group like $C_2$, the cyclic group of order $2$, this group is contained in (more exactly, has an injective homomorphism into) every finite group of even order. Even if you demand that $C_2$ should be a normal subgroup of the extended group, you can always look at $C_2\times G$ for any group at all, and many others, containing the group of order $2$.
On the other hand, when we look at a smallest possible field, it’s definitely an interesting question to ask what extensions it has. If your “smallest possible” is $\mathbb F_p=\mathbb Z/(p)$, it’s interesting that there’s only one extension of each finite degree, and these all are normal over $\mathbb F_p$ with cyclic Galois group. If your “smallest” is $\mathbb Q$, then there are extensions of all possible finite degrees, some normal, some not, and yet we don’t yet know whether among the normal extensions, all finite groups occur as Galois group. Interesting facts and questions all.
For substructures, when we look at a group, we can ask about all sorts of properties: abelian or not, soluble or not, nilpotent, etc. And of course we have techniques, extremely well developed. In a way, by looking at the substructures rather than the overlying structures, we have restricted our investigations to more manageable questions.
For substructures of fields, there’s the fact that some fields that are important outside the narrow subject of field theory have no proper subfields, and even for fields that aren’t prime fields, such as $\mathbb C$, there certainly are subfields, but we deal with them in most cases as extensions of $\mathbb Q$ not subfields of $\mathbb C$. A notable exception is the theorem that the only fields $K$ with $[\mathbb C\colon K]<\infty$ have this degree equal to $2$.
The split is not absolute. There are ways of studying situations in which, given a group $G$, we can make statements about groups $X$ that have $G$ as a normal subgroup. Similarly, there are statements about subfields of a given field.