Coded Orthogonal frequency division multiplexing with index modulation (OFDM-IM) has been proposed, where channel coding and soft decision are employed to exploit the multipath diversity for the information conveyed by the constellation symbols. However, the performance of coded OFDM-IM is dependent on the detection performance of subcarrier activation pattern (SAP) and the diversity order of the SAPs requires further investigation. In this paper, a lower-bound of LLR detector is derived to show that the maximum achievable diversity order is the minimum Hamming distance between the SAPs for the LLR detector. Based on the analyses, iterative detection is proposed to improve the performance of coded OFDM-IM, where information-assisted LLR detector is proposed to improve the detection performance of SAP by using the soft information of the constellation symbols. Simulation results are provided to corroborate the diversity analyses and show the advantage of the coded OFDM-IM with iterative detection.