I am reading Dummit & Foote, Abstract Algebra, 3e, p.103ff. We know that the first part of Jordan-Hölder program, the classification of finite simple groups, are finished. But it is not written whether the second part, roughly how to build any group from simple groups [edited] and cyclic groups, are finished.
I say that a finite group $G$ is tangible if: [edited] $G$ is a cyclic group, a finite simple group, or $G$ can be written as a semidirect (possibly direct) product $$ G \cong H_1 \rtimes \dotsc \rtimes H_M $$ with $H_m$ tangible, $m =1,\dotsc,M$. (Tell me if there is an existent terminology.) Note this has included all abelian groups.
They seem to say not all groups are tangible, because some do not have complementary subgroups (D&F p.180). Indeed, the quaternion group $Q_8$ is not tangible (D&F p.181). However, every finite group is the image of a homomorphism from a free group (D&F p.217), and can be presented as such quotient of the free group and some words (D&F p.218)
What are some other groups that are proved not tangible? If not, how can we say that Jordan-Hölder program is finished? Isn't it true that the existence of a composition series still does not concretely describe a group (D&F p.103)?
Edit: System recommends Smallest non-p-group that isnt a semidirect product, which answers my question.