Designing algorithms that generate networks with a given degree sequence while varying both subgraph composition and distribution of subgraphs around nodes is an important but challenging research problem. Current algorithms lack control of key network parameters, the ability to specify to what subgraphs a node belongs to, come at a considerable complexity cost or, critically, sample from a limited ensemble of networks. To enable controlled investigations of the impact and role of subgraphs, especially for epidemics, neuronal activity or complex contagion, it is essential that the generation process be versatile and the generated networks as diverse as possible. In this paper, we present two new network generation algorithms that use subgraphs as building blocks to construct networks preserving a given degree sequence. Additionally, these algorithms provide control over clustering both at node and global level. In both cases, we show that, despite being constrained by a degree sequence and global clustering, generated networks have markedly different topologies as evidenced by both subgraph prevalence and distribution around nodes, and large-scale network structure metrics such as path length and betweenness measures. Simulations of standard epidemic and complex contagion models on those networks reveal that degree distribution and global clustering do not always accurately predict the outcome of dynamical processes taking place on them. We conclude by discussing the benefits and limitations of both methods.