Multicasting is a fundamental functionality of many network applications, including online conferencing, event monitoring, video streaming, and so on. To ensure reliable, secure, and scalable multicasting, a service chain that consists of network functions (e.g., firewalls, intrusion detection systems, and transcoders) usually is associated with each multicast request. We refer to such a multicast request with service chain requirement as an network function virtualization (NFV)-enabled multicast request. In this paper, we study NFV-enabled multicasting in a software-defined network (SDN) with an aim to maximize network throughput while minimizing the implementation cost of admitted NFV-enabled multicast requests, subject to network resource capacity, where the implementation cost of a request consists of its computing resource consumption cost in servers and its network bandwidth consumption cost when routing and processing its data packets in the network. To this end, we first formulate two NFV-enabled multicasting problems with and without resource capacity constraints and one online NFV-enabled multicasting problem. We then devise two approximation algorithms with an approximation ratio of $2M$ for the NFV-enabled multicasting problems with and without resource capacity constraints, if the number of servers for implementing the service chain of each request is no greater than a constant $M$ (≥1). We also study dynamic admissions of NFV-enabled multicast requests without the knowledge of future request arrivals with the objective to maximize the network throughput, for which we propose an efficient heuristic, and for the special case of dynamic request admissions, we devise an online algorithm with a competitive ratio of $O(\log n)$ for it when $M=1$ , where $n$ is the number of nodes in the network. We finally evaluate the performance of the proposed algorithms through experimental simulations. Experimental results demonstrate that the proposed algorithms are promising and outperform existing heuristics.