1. A storage system comprising: one or more disk devices making up a plurality of storage 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 copy 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 the storage controller stores sub-copy group information for allocating at least one sub-copy group to each of the copy groups, and each of said copy groups is predetermined according to a desired requirement regarding objects to be copied, and the at least one sub-copy group is defined for each of the plurality of computers that make an IO access; wherein the storage controller stores storage area list information for allocating at least one storage area list having at least one predetermined storage area of the storage areas to the at least one sub-copy group, and the at least one sub-copy group specifies an asynchronous remote copy operation or a synchronous remote copy operation for a corresponding storage area list; wherein each of the copy groups has a consistency assurance level defining at least three levels of the consistency assurance, wherein said at least three levels comprise sub-copy group assured in which consistency is assured for all storage areas in the at least one sub-copy group, completely assured in which consistency is assured for all storage areas in the copy group, and unassured in which consistency is 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 the asynchronous remote copy operation or the synchronous remote copy operation according to the one of the plurality of computers corresponding to the instruction and the at least one sub-copy group allocated to the designated copy group, when it is determined that the copy operation is the asynchronous remote copy operation, the storage controller: 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 according to the designated copy group, assigns a corresponding consistency group number based on the determined consistency assurance level before performing the copy operation, starts 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 if the consistency assurance level is completely assured, suspends the copy operation for all of the storage areas for the sub-copy group associated with the fault if the consistency assurance level is sub-copy group assured, or suspends the copy operation for the storage area associated with the fault if the consistency assurance level is unassured in accordance with the consistency group number of the designated copy group; and when it is determined that the copy operation is the synchronous remote copy operation, the storage controller: 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 according to the designated copy group, assigns a corresponding consistency group number based on the determined consistency assurance level before performing the copy operation, starts performing the copy operation, and when a fault occurs in the copy operation of the copy group, suspends the transfer of write data into a cache and suspends the copy operation for the storage area of sub-copy group allocated to the designated copy group if the consistency assurance level is completely assured, and suspends the copy operation for the storage area of the sub-copy group allocated to the designated copy group if the consistency assurance level is sub-copy group assured or unassured.