It has been well established that Functional Connectomes (FCs), as estimated from functional MRI (fMRI) data, have an individual fingerprint that can be used to identify an individual from a population (subject-identification). Although identification rate is high when using resting-state FCs, other tasks show moderate to low values. Furthermore, identification rate is task-dependent, and is low when distinct cognitive states, as captured by different fMRI tasks, are compared. Here we propose an embedding framework, GEFF (Graph Embedding for Functional Fingerprinting), based on group-level decomposition of FCs into eigenvectors. GEFF creates an eigenspace representation of a group of subjects using one or more task FCs (Learning Stage). In the Identification Stage, we compare new instances of FCs from the Learning subjects within this eigenspace (validation dataset). The validation dataset contains FCs either from the same tasks as the Learning dataset or from the remaining tasks that were not included in Learning. Assessment of validation FCs within the eigenspace results in significantly increased subject-identification rates for all fMRI tasks tested and potentially task-independent fingerprinting process. It is noteworthy that combining resting-state with one fMRI task for GEFF Learning Stage covers most of the cognitive space for subject identification. Thus, while designing an experiment, one could choose a task fMRI to ask a specific question and combine it with resting-state fMRI to extract maximum subject differentiability using GEFF. In addition to subject-identification, GEFF was also used for identification of cognitive states, i.e. to identify the task associated to a given FC, regardless of the subject being already in the Learning dataset or not (subject-independent task-identification). In addition, we also show that eigenvectors from the Learning Stage can be characterized as task- and subject-dominant, subject-dominant or neither, using two-way ANOVA of their corresponding loadings, providing a deeper insight into the extent of variance in functional connectivity across individuals and cognitive states.