1. A storage system comprising: one or more disk devices making up a plurality of storages areas; and a storage controller receiving an input/output (I/O) access from a plurality of computers, wherein the storage controller stores copy group information for allocating groups to selected said storage areas as objects to be copied to other storage areas in other disk devices connected to another storage controller via a communication line, wherein each of the copy groups is made up of sub-copy groups and each of said copy groups is predetermined according to a desired requirement regarding objects to be copied, and the sub-copy groups are defined to correspond to the predetermined part of said storage areas for each of the plurality of computers that make an IO access; wherein each of the copy groups has a consistency assurance level defining an order in which data is updated that is one of sub-group assured in which consistency is assured for all said storage groups in the sub-copy group, completely assured in which consistency is assured for all said storage areas in the copy group or unassured, wherein the storage controller receives, from any of the plurality of computers, an instruction which designates one among the copy groups to perform a copy operation for the designated copy group of the storage areas, and wherein the storage controller: determines whether the copy operation is an asynchronous remote copy operation, and when it is determined that the copy operation is an asynchronous remote copy operation; checks a consistency assurance level for the designated copy group to determine if the designated copy group is completely assured, sub-copy group assured or unassured; assigns a corresponding consistency group number based on the determined consistency assurance level before performing the copy operation; and when a fault occurs in the copy operation of the copy group, suspends the copy operation for all of the storage areas of the designated copy group or suspends the copy operation for all of the storage areas for the sub-copy group associated with the fault in accordance with the consistency group number of the designated copy group.