  The generalized $k$-connectivity of a graph $G$, denoted by $\kappa_k(G)$, is
the minimum number of internally edge disjoint $S$-trees for any $S\subseteq
V(G)$ with $|S|=k$. The generalized $k$-connectivity is a natural extension of
the classical connectivity and plays a key role in applications related to the
modern interconnection networks. In this paper, we firstly introduce a family
of regular networks $H_n$ that can be obtained from several subgraphs $G_n^1,
G_n^2, \cdots, G_n^{t_n}$ by adding a matching, where each subgraph $G_n^i$ is
isomorphic to a particular graph $G_n$ ($1\le i\le t_n$). Then we determine the
generalized 3-connectivity of $H_n$. As applications of the main result, the
generalized 3-connectivity of some two-level interconnection networks, such as
the hierarchical star graph $HS_n$, the hierarchical cubic network $HCN_n$ and
the hierarchical folded hypercube $HFQ_n$, are determined directly.
