The Ishimura Clan is probably one of the most ancient clans still in existence in 1996.  Throughout the centuries, the Ishimuran gargoyles fought alongside Japanese samurai. Teaching the samurai Bushido probably led the clan to becoming one of the earliest to adopt the use of names.  During the Feudal Period (mid-19th century), a TimeDancing Brooklyn and Fu-Dog spent some time with the Clan.  Katana, a member of the Ishimura Clan, left with them. After peace came over the islands, the clan settled in Ishimura with the human villagers, helping to protect their village from criminals. The humans are on good terms with the gargoyles, and keep them secret from the outside world so that they can go unmolested. As a result, it is very likely that Ishimura is larger than most other clans. 
The gargoyles of Ishimura live in an old temple in the village, perching on its roof in the daytime. They face inwards in their stone sleep, as a gesture of trust towards the humans with whom they live. From time to time, they teach the local humans bushido, the code of honor that they themselves practice.
One such pupil was Taro, but he was a poor student. When he grew up and became a wealthy businessman, he decided to exploit the gargoyles for his own benefit; duping Yama, a dissatisfied member of the clan, into helping him, he abducted the entire clan (alongside Goliath, Angela, and Bronx, who were visiting Ishimura with Elisa on the Avalon World Tour) and took them away to a nearby theme park that he had just built to be put on display there (it is unknown whether or not the eggs from the Ishimura rookery were also moved).  Fortunately, Elisa and Hiroshi, the police constable at Ishimura, rescued the Clan. Afterwards, the humans of Ishimura decided to strengthen their covenant with the gargoyles all the more, and to take regular bushido lessons from them. ("Bushido")
By 2198, the Ishimura Clan will have truly become a clan of teachers. Many gargoyles will go to Ishimura to learn bushido - among them Samson, Delilah, and Zafiro. Humans, including Nick Maza, will also come to Ishimura to study and join the Order of the Guardian.  Though the world will be aware of the Ishimura Clan in 2198, it seems that the secret of Ishimura will remain so for some time and that the Ishimura Clan may be one of the last, if not the last, clans revealed to the world at large. 
It is notable that gargoyles of Ishimura sometimes have fewer digits on their feet or hands, or have more digits on their wings than is found among other gargoyle clans. Besides that, many of the clan's members have thick brow ridges, horns, and spikes, often having more than one pair of horns. Many have spikes, blades, or forked features on the ends of their tails. Some members have beaks, but they appear to be a minority. So far, none of the observed members are hairless or light-haired, none have feather-like wing coverings, and none have their wings attached to their arms and legs.
- Kai - Current Leader
- Yama - Second-In-Command until mid-1996. Currently banished.
- Sora - Second-In-Command from mid-1996 on. 
- Red Male Gargoyle Beast
- Katana - Leaves the clan during Japan's feudal period to join Brooklyn, and ultimately, the Manhattan Clan. 
Real World Background
The decision to create a gargoyle clan in Japan was made partly because the TV series was animated in Japan. 
The nature of the gargoyles of Ishimura is noteworthy, since a race of winged beings named tengu appear in Japanese legends. Tengu are winged humanoids who sometimes teach humans bushido; this evokes the gargoyles of Ishimura enough to suggest that the legend originated - at least partly - with them. (Though the tengu of Japanese legend are also portrayed as behaving like tricksters, with characteristics that sound more like Oberon's Children than like gargoyles.)
In a 2007 interview with COMICON.com about the then-upcoming Gargoyles: Bad Guys mini-series comic, Greg Weisman used the term "tengu" for the Ishimura gargoyles, confirming this connection. The connection became canon at the end of "Strangers", when Hunter called Yama a tengu, and in "The Lost", when Dingo realized that "tengu" is the Japanese word for "gargoyle".