= Solution
A <simple module> is a nonzero module with no proper nonzero <submodule>. For a finite-dimensional simple complex module $S$, any <endomorphism> $\varphi$ has a complex <eigenvalue> $\lambda$. The nonzero <kernel> of $\varphi-\lambda I$ is an $A$-<submodule>, hence all of $S$. Thus \b[$\varphi=\lambda I$], proving <Schur's lemma>.
More generally a nonzero map between two simple modules is an isomorphism, because its <kernel> is zero and its <image> is the whole target. The <endomorphism> <algebra> of $S_1\oplus S_2$ is the block <algebra> with entries $\operatorname{Hom}_A(S_j,S_i)$. For nonisomorphic summands its off-diagonal entries vanish and its diagonal entries are scalars, giving $\mathbb C\times\mathbb C$. For isomorphic summands, choose one fixed isomorphism to identify them; every block is scalar and block composition is matrix multiplication, giving $M_2(\mathbb C)$.
A finite-dimensional <algebra> is a <semisimple algebra> when its <left regular module> is a <direct sum> of simple modules. The complex <Artin–Wedderburn theorem> gives $A\cong\prod_iM_{d_i}(\mathbb C)$, with the multiset of positive integers $d_i$ uniquely determined. Its <dimension> is $\sum_i d_i^2$. For <dimension> eight only sizes one and two can occur; the number of size-two blocks is zero, one, or two. Thus there are \b[exactly three isomorphism types]:
$$
\boxed{\mathbb C^8,\quad M_2(\mathbb C)\times\mathbb C^4,\quad M_2(\mathbb C)\times M_2(\mathbb C).}
$$
They are distinct by the uniqueness clause, or by their numbers of one-dimensional simple modules. If $G$ is nonabelian, $\mathbb CG$ is <noncommutative> because its group elements form a basis, so the first type is excluded. The augmentation $\sum a_gg\mapsto\sum a_g$ provides a one-dimensional <simple module>, excluding the third type. Consequently
$$
\boxed{\mathbb CG\cong\mathbb C^4\times M_2(\mathbb C).}
$$
There is one <simple module> for each matrix block, of <dimension> the block size, so exactly four are one-dimensional. Every nonabelian group of order eight therefore has this same complex <group algebra>.
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