Finite Group Databasebeta

← D9×#237 #11722497

Subgroups — D9×#237 #11722497

Order 864.

532 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#88742C211yesmaximal, characteristic, complemented
2C9×#237C211yesmaximal, characteristic, complemented
3C3×D4×D9C211yesmaximal, characteristic, complemented
4D9×#67C212yesmaximal, complemented
5#88663C212yesmaximal, complemented
6#88658C212yesmaximal, complemented
7C2×Dic3×D9C212yesmaximal, complemented
8C4×D3×D9C211yesmaximal, characteristic, complemented
9#88641C211yesmaximal, characteristic, complemented
10#88634C211yesmaximal, characteristic, complemented
11D9×Dic6C211yesmaximal, characteristic, complemented
12D3×#237—31nomaximal
13#41×D9—31nomaximal
14C22×C3×D9C2212yes—
15C2×D3×D9C2211yescharacteristic
16C3×D4×C9C2211yescharacteristic, complemented
17#6233C2212yescomplemented
18C2×C3⋊Dic9C2212yescomplemented
19C4×C3⋊D9C2211yescharacteristic, complemented
20#6226C2211yescharacteristic, complemented
21C9×#67C2212yescomplemented
22C3×#611C2212yescomplemented
23C2×C9×Dic3C2212yescomplemented
24C4×D3×C9C2211yescharacteristic, complemented
25C3×D36C2211yescharacteristic, complemented
26C3×C4×D9C2211yescharacteristic
27C9×Dic6C2211yescharacteristic, complemented
28#6194C2212yescomplemented
29D3×Dic9C2211yescharacteristic, complemented
30#6192C2212yescomplemented
31(C3⋊D9)⋊C4C2211yescharacteristic, complemented
32Dic3×D9C2212yes—
33Dic3×D9C2211yescharacteristic
34Dic3×D9C2212yescomplemented
35Dic3×D9C2212yescomplemented
36#6189C2212yescomplemented
37#3830—31no—
38C3×#237D311yescharacteristic
39C3×D3×D4—31no—
40D3×#67—32no—
41#3805—32no—
42#3804—32no—
43C2×D3×Dic3—32no—
44C4×D32—31no—
45#3796—31no—
46#3795—31no—
47D3×Dic6—31no—
48C9×#41—31no—
49#3701—31no—
50Q8×D9—31no—
51#3699—31no—
52#3699—32no—
53#3699—61no—
54D4×D9—32no—
55D4×D9D311yescharacteristic, complemented
56#3697—32no—
57#3697—31no—
58C2×C4×D9—32no—
59C2×C4×D9—31no—
60C22×C3×C9C2312yes—
61C2×C3⋊D9C2311yescharacteristic
62C2×D3×C9C2311yescharacteristic
63C2×C3×D9—24no—
64C2×C3×D9C2312yes—
65C2×C3×D9C2311yescharacteristic
66D3×D9—22no—
67C3×C4×C9C2311yescharacteristic
68C3⋊Dic9C2312yes—
69C3⋊Dic9C2311yescharacteristic
70C9×Dic3C2312yes—
71C9×Dic3C2311yescharacteristic
72C3×Dic9C2311yescharacteristic
73D3×#41—91no—
74C2×#237—91no—
75C22×C3×D3—32no—
76C2×D32—31no—
77C32×D4C2×D311yescharacteristic
78#638—32no—
79C2×C3⋊Dic3—32no—
80C4×C3⋊D3—31no—
81#634—31no—
82C3×#67—32no—
83C3×#67C2×D312yes—
84C2×C3×Dic3C2×D312yes—
85C3×D12—31no—
86C3×C4×D3—31no—
87C3×C4×D3C2×D311yescharacteristic
88C3×Dic6C2×D311yescharacteristic
89#627—32no—
90#626—32no—
91#625—32no—
92#624—31no—
93D3×Dic3—31no—
94D3×Dic3—32no—
95D3×Dic3—31no—
96D3×Dic3—32no—
97D3×Dic3—32no—
98C22×D9—31no—
99C22×D9C2×D312yes—
100Q8×C9—31no—
101D4×C9—21no—
102D4×C9—32no—
103D4×C9C2×D311yescharacteristic, complemented
104C2×C4×C9—32no—
105C2×C4×C9—31no—
106#611—32no—
107#611—32no—
108#611—62no—
109#611C2×D312yescomplemented
110C2×Dic9—32no—
111C2×Dic9—31no—
112C2×Dic9—62no—
113D36—32no—
114D36C2×D311yescharacteristic, complemented
115C4×D9—31no—
116C4×D9—32no—
117C4×D9—32no—
118C4×D9—32no—
119C4×D9—31no—
120C4×D9—31no—
121C4×D9—61no—
122C4×D9C2×D311yescharacteristic
123Dic18—31no—
124Dic18—32no—
125Dic18—61no—
126C2×C3×C9—22no—
127C2×C3×C9C2411yescharacteristic
128C3⋊D9—21no—
129D3×C9—21no—
130C3×D9—22no—
131C3×D9#4112yescomplemented
132C3×#41—31no—
133C2×C3×D4—91no—
134C2×#67—92no—
135C22×Dic3—92no—
136#239—91no—
137D3×Q8—91no—
138#237—92no—
139#237—91no—
140#237—91no—
141#237—92no—
142#237—92no—
143#237—91no—
144#237—91no—
145#237—181no—
146#237D911yescharacteristic, direct product factor
147D3×D4—31no—
148D3×D4—92no—
149#235—91no—
150#235—92no—
151C2×C4×D3—91no—
152C2×C4×D3—92no—
153C2×C4×D3—91no—
154C2×Dic6—91no—
155C22×C32C22×D312yes—
156C2×C3⋊D3—31no—
157C2×C3×D3—31no—
158C2×C3×D3—32no—
159C2×C3×D3—64no—
160C2×C3×D3C22×D311yescharacteristic
161D32—62no—
162C32×C4C22×D311yescharacteristic
163C3⋊Dic3—31no—
164C3⋊Dic3—32no—
165C3×Dic3—31no—
166C3×Dic3C22×D311yescharacteristic
167C3×Dic3C22×D312yes—
168C22×C9—22no—
169C22×C9—31no—
170C22×C9C22×D312yes—
171C2×D9—24no—
172C2×D9—31no—
173C2×D9—61no—
174C2×D9—62no—
175C2×D9C22×D312yes—
176C2×D9C22×D311yescharacteristic
177C4×C9—21no—
178C4×C9—32no—
179C4×C9—31no—
180C4×C9C22×D311yescharacteristic
181Dic9—32no—
182Dic9—31no—
183Dic9—62no—
184Dic9—61no—
185Dic9C22×D311yescharacteristic
186C2×#41—271noSylow
187C3×C9C2×#4111yescharacteristic, complemented, Sylow
188C23×C3—92no—
189C22×D3—32no—
190C22×D3—91no—
191C22×D3—91no—
192C3×Q8—31no—
193C3×D4—21no—
194C3×D4—32no—
195C3×D4—92no—
196C3×D4—91no—
197C3×D4C2×D911yescharacteristic, complemented
198C3×D4D3211yescharacteristic
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—91no—
202#67—32no—
203#67—92no—
204#67—92no—
205#67—92no—
206#67—92no—
207#67—92no—
208#67—182no—
209#67C2×D912yescomplemented
210C2×Dic3—91no—
211C2×Dic3—92no—
212C2×Dic3—92no—
213C2×Dic3—92no—
214C2×Dic3—91no—
215C2×Dic3—92no—
216C2×Dic3—92no—
217C2×Dic3—182no—
218C2×Dic3C2×D912yescomplemented
219D12—31no—
220D12—92no—
221C4×D3—31no—
222C4×D3—91no—
223C4×D3—91no—
224C4×D3—92no—
225C4×D3—92no—
226C4×D3—92no—
227C4×D3—91no—
228C4×D3—91no—
229C4×D3—91no—
230C4×D3—91no—
231C4×D3—181no—
232C4×D3C2×D911yescharacteristic, complemented
233Dic6—92no—
234Dic6—91no—
235Dic6—91no—
236Dic6—92no—
237Dic6—181no—
238Dic6C2×D911yescharacteristic, complemented
239C2×C32—22no—
240C2×C32C23×D311yescharacteristic
241C3⋊D3—61no—
242C3×D3—21no—
243C3×D3—32no—
244C3×D3—62no—
245C2×C9—22no—
246C2×C9—21no—
247C2×C9—42no—
248C2×C9—61no—
249C2×C9C23×D311yescharacteristic
250D9—22no—
251D9—61no—
252D9—121no—
253D9#23712yesdirect product factor
254#41—31no—
255#41—272no—
256#41—271no—
257#41—272no—
258#41—271no—
259#41—271no—
260C2×Q8—271no—
261C2×D4—91no—
262C2×D4—272no—
263C22×C4—271no—
264C22×C4—272no—
265C22×C3—22no—
266C22×C3—31no—
267C22×C3—91no—
268C22×C3—92no—
269C22×C3—184no—
270C22×C3C22×D912yes—
271C22×C3C2×D3212yes—
272C2×D3—31no—
273C2×D3—32no—
274C2×D3—64no—
275C2×D3—91no—
276C2×D3—91no—
277C2×D3—181no—
278C2×D3—182no—
279C2×D3—182no—
280C2×D3C22×D911yescharacteristic
281C3×C4—21no—
282C3×C4—32no—
283C3×C4—31no—
284C3×C4—91no—
285C3×C4C22×D911yescharacteristic
286C3×C4C2×D3211yescharacteristic
287Dic3—31no—
288Dic3—92no—
289Dic3—91no—
290Dic3—92no—
291Dic3—91no—
292Dic3—182no—
293Dic3—181no—
294Dic3C22×D912yes—
295Dic3C22×D911yescharacteristic
296C32D3×#4111yescharacteristic
297C9—21no—
298C9C2×#23711yescharacteristic, complemented
299C23—92no—
300C23—271no—
301Q8—31no—
302Q8—272no—
303Q8—271no—
304D4—32no—
305D4—92no—
306D4—91no—
307D4—272no—
308D4—272no—
309D4—272no—
310D4D3×D911yescharacteristic, complemented
311C2×C4—32no—
312C2×C4—31no—
313C2×C4—91no—
314C2×C4—271no—
315C2×C4—271no—
316C2×C4—272no—
317C2×C4—272no—
318C2×C4—272no—
319C2×C4—271no—
320C2×C4—272no—
321C2×C4—271no—
322C2×C3—22no—
323C2×C3—22no—
324C2×C3—21no—
325C2×C3—42no—
326C2×C3—61no—
327C2×C3—92no—
328C2×C3—182no—
329C2×C3C23×D911yescharacteristic
330C2×C3C22×D3211yescharacteristic
331D3—21no—
332D3—32no—
333D3—62no—
334D3—181no—
335D3—181no—
336D3—361no—
337C22—31no—
338C22—91no—
339C22—92no—
340C22—184no—
341C22—271no—
342C22—542no—
343C22C2×D3×D912yes—
344C4—31no—
345C4—32no—
346C4—91no—
347C4—271no—
348C4—272no—
349C4C2×D3×D911yescharacteristic
350C3—21no—
351C3#41×D911yescharacteristic, complemented
352C3D3×#23711yescharacteristic
353C2—22no—
354C2—61no—
355C2—92no—
356C2—182no—
357C2—541no—
358C2C22×D3×D911yescharacteristic
subgroupG/H
centerC2C22×D3×D9
commutator, cosocleC2×C3×C9C24
FrattiniC2×C3C22×D32
FittingC3×D4×C9C22
socleC2×C32C23×D3
hypercenterD4D3×D9
hypocenterC3×C9C2×#41
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #88742C2semidirect
2. C9×#237C2semidirect
3. C3×D4×D9C2semidirect
4. D9×#67C2semidirect
5. #88663C2semidirect
6. #88658C2semidirect
7. C2×Dic3×D9C2semidirect
8. C4×D3×D9C2semidirect
9. #88641C2semidirect
10. #88634C2semidirect
11. D9×Dic6C2semidirect
16. C3×D4×C9C22semidirect
17. #6233C22semidirect
18. C2×C3⋊Dic9C22semidirect
19. C4×C3⋊D9C22semidirect
20. #6226C22semidirect
21. C9×#67C22semidirect
22. C3×#611C22semidirect
23. C2×C9×Dic3C22semidirect
24. C4×D3×C9C22semidirect
25. C3×D36C22semidirect
27. C9×Dic6C22semidirect
28. #6194C22semidirect
29. D3×Dic9C22semidirect
30. #6192C22semidirect
31. (C3⋊D9)⋊C4C22semidirect
34. Dic3×D9C22semidirect
35. Dic3×D9C22semidirect
36. #6189C22semidirect
55. D4×D9D3semidirect
103. D4×C9C2×D3semidirect
109. #611C2×D3semidirect
114. D36C2×D3semidirect
131. C3×D9#41semidirect
146. #237D9direct (N × H)
187. C3×C9C2×#41semidirect
197. C3×D4C2×D9semidirect
209. #67C2×D9semidirect
218. C2×Dic3C2×D9semidirect
232. C4×D3C2×D9semidirect
238. Dic6C2×D9semidirect
253. D9#237direct (N × H)
298. C9C2×#237semidirect
310. D4D3×D9semidirect
351. C3#41×D9semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#887422
C9×#2372
C3×D4×D92
D9×#672
#886632
#886582
C2×Dic3×D92
C4×D3×D92
#886412
#886342
D9×Dic62
D3×#2373
#41×D93