A sample of 427 gamma-ray bursts (GRBs) from a database (2002 February-2008 April) of the RHESSI satellite is analyzed statistically. The spectral lags and peak-count rates, which have been calculated for the first time in this paper, are studied, completing an earlier analysis of durations and hardness ratios. The analysis of the RHESSI database has already inferred the existence of a third group with intermediate duration, apart from the so-called short and long groups. The first aim of this article is to discuss the properties of these intermediate-duration bursts in terms of peak-count rates and spectral lags. The second aim is to discuss the number of GRB groups using another statistical method and by also employing the peak-count rates and spectral lags. The standard parametric (model-based clustering) and non-parametric (K-means clustering) statistical tests together with the Kolmogorov-Smirnov and Anderson-Darling tests are used. Two new results are obtained. (1) The intermediate-duration group has properties similar to those of the group of short bursts. Intermediate and long groups appear to be different. (2) The intermediate-duration GRBs in the RHESSI and Swift databases seem to be represented by different phenomena.