Finite Group Databasebeta

← #11722498

Subgroups — #11722498

Order 864.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
1D4×C3⋊D9C211yesmaximal, characteristic, complemented
2C9×#237C211yesmaximal, characteristic, complemented
3C3×#3699C211yesmaximal, characteristic, complemented
4#88668C212yesmaximal, complemented
5#88662C212yesmaximal, complemented
6C2×(C3⋊D9)⋊C4C212yesmaximal, complemented
7#88658C212yesmaximal, complemented
8C4×D3×D9C211yesmaximal, characteristic, complemented
9#88642C211yesmaximal, characteristic, complemented
10(C3⋊D9)⋊Q8C211yesmaximal, characteristic, complemented
11#88635C211yesmaximal, characteristic, complemented
12#64312—31nomaximal
13#41×D9—31nomaximal
14C22×C3⋊D9C2212yes—
15C2×D3×D9C2211yescharacteristic
16C3×D4×C9C2211yescharacteristic, complemented
17#6233C2212yescomplemented
18C4⋊(C3⋊D9)C2211yescharacteristic, complemented
19C4×C3⋊D9C2211yescharacteristic
20C9×#67C2212yescomplemented
21C3×#611C2212yescomplemented
22C2×C9×Dic3C2212yescomplemented
23C2×C3×Dic9C2212yescomplemented
24C4×D3×C9C2211yescharacteristic, complemented
25C3×C4×D9C2211yescharacteristic, complemented
26C9×Dic6C2211yescharacteristic, complemented
27C3×Dic18C2211yescharacteristic, complemented
28#6195C2212yescomplemented
29D3×Dic9C2211yescharacteristic, complemented
30#6192C2212yescomplemented
31(C3⋊D9)⋊C4C2212yes—
32(C3⋊D9)⋊C4C2211yescharacteristic
33(C3⋊D9)⋊C4C2212yescomplemented
34(C3⋊D9)⋊C4C2212yescomplemented
35Dic3×D9C2211yescharacteristic, complemented
36#6189C2212yescomplemented
37D4×C3⋊D3—31no—
38C3×#237—31no—
39C3×#237D311yescharacteristic
40#3811—32no—
41C2×#624—32no—
42#3804—32no—
43#3804—32no—
44C4×D32—31no—
45#3799—31no—
46#3799—31no—
47#3797—31no—
48C9×#41—31no—
49#3701—31no—
50Q8×D9—31no—
51#3699—32no—
52#3699D311yescharacteristic, complemented
53D4×D9—31no—
54D4×D9—32no—
55D4×D9—61no—
56#3697—32no—
57#3697—31no—
58C2×C4×D9—32no—
59C2×C4×D9—31no—
60C22×C3×C9C2312yes—
61C2×C3⋊D9—24no—
62C2×C3⋊D9C2312yes—
63C2×C3⋊D9C2311yescharacteristic
64C2×D3×C9C2311yescharacteristic
65C2×C3×D9C2311yescharacteristic
66D3×D9—22no—
67C3×C4×C9C2311yescharacteristic
68C3⋊Dic9C2311yescharacteristic
69C9×Dic3C2312yes—
70C9×Dic3C2311yescharacteristic
71C3×Dic9C2312yes—
72C3×Dic9C2311yescharacteristic
73D3×#41—91no—
74D3×#41—91no—
75C22×C3⋊D3—32no—
76C2×D32—31no—
77C32×D4C2×D311yescharacteristic
78#638—32no—
79C4⋊(C3⋊D3)—31no—
80C4×C3⋊D3—31no—
81C3×#67—32no—
82C3×#67C2×D312yes—
83C2×C3×Dic3—32no—
84C2×C3×Dic3C2×D312yes—
85C3×C4×D3—31no—
86C3×C4×D3C2×D311yescharacteristic
87C3×Dic6—31no—
88C3×Dic6C2×D311yescharacteristic
89#627—32no—
90#626—32no—
91#626—32no—
92#624—32no—
93#624—31no—
94#624—32no—
95#624—32no—
96D3×Dic3—31no—
97D3×Dic3—31no—
98C22×D9—32no—
99C22×D9—31no—
100C22×D9—62no—
101Q8×C9—31no—
102D4×C9—21no—
103D4×C9—32no—
104D4×C9C2×D311yescharacteristic, complemented
105C2×C4×C9—32no—
106C2×C4×C9—31no—
107#611—32no—
108#611—32no—
109#611—62no—
110#611C2×D312yescomplemented
111C2×Dic9—31no—
112C2×Dic9C2×D312yescomplemented
113D36—32no—
114D36—31no—
115D36—61no—
116C4×D9—31no—
117C4×D9—31no—
118C4×D9—31no—
119C4×D9—32no—
120C4×D9—32no—
121C4×D9—32no—
122C4×D9—61no—
123C4×D9C2×D311yescharacteristic, complemented
124Dic18—32no—
125Dic18C2×D311yescharacteristic, complemented
126C2×C3×C9—22no—
127C2×C3×C9C2411yescharacteristic
128C3⋊D9—22no—
129C3⋊D9#4112yescomplemented
130D3×C9—21no—
131C3×D9—21no—
132C3×#41—31no—
133C3×#41—91no—
134#239—91no—
135#239—91no—
136D3×Q8—91no—
137D3×Q8—91no—
138#237—31no—
139#237—92no—
140#237—92no—
141#237D911yescharacteristic, complemented
142D3×D4—92no—
143D3×D4—91no—
144D3×D4—91no—
145D3×D4—92no—
146D3×D4—181no—
147#235—91no—
148#235—92no—
149#235—92no—
150#235—91no—
151C2×C4×D3—91no—
152C2×C4×D3—92no—
153C2×C4×D3—92no—
154C2×C4×D3—91no—
155C22×C32C22×D312yes—
156C2×C3⋊D3—31no—
157C2×C3⋊D3—32no—
158C2×C3⋊D3—64no—
159C2×C3×D3—31no—
160C2×C3×D3C22×D311yescharacteristic
161D32—62no—
162C32×C4C22×D311yescharacteristic
163C3⋊Dic3—31no—
164C3×Dic3—31no—
165C3×Dic3—32no—
166C3×Dic3C22×D311yescharacteristic
167C3×Dic3C22×D312yes—
168C22×C9—22no—
169C22×C9—31no—
170C22×C9C22×D312yes—
171C2×D9—31no—
172C2×D9—32no—
173C2×D9—61no—
174C2×D9—62no—
175C2×D9—62no—
176C2×D9—64no—
177C2×D9—124no—
178C2×D9C22×D311yescharacteristic
179C4×C9—21no—
180C4×C9—32no—
181C4×C9—31no—
182C4×C9C22×D311yescharacteristic
183Dic9—31no—
184Dic9—61no—
185Dic9C22×D312yes—
186Dic9C22×D311yescharacteristic
187C2×#41—271noSylow
188C3×C9C2×#4111yescharacteristic, complemented, Sylow
189C22×D3—91no—
190C22×D3—92no—
191C22×D3—92no—
192C22×D3—91no—
193C22×D3—182no—
194C3×Q8—31no—
195C3×Q8—91no—
196C3×D4—21no—
197C3×D4—32no—
198C3×D4—92no—
199C3×D4C2×D911yescharacteristic, complemented
200C3×D4D3211yescharacteristic
201C2×C3×C4—32no—
202C2×C3×C4—31no—
203C2×C3×C4—92no—
204C2×C3×C4—91no—
205#67—32no—
206#67—92no—
207#67—92no—
208#67—92no—
209#67—92no—
210#67—182no—
211#67C2×D912yescomplemented
212C2×Dic3—32no—
213C2×Dic3—91no—
214C2×Dic3—91no—
215C2×Dic3C2×D912yescomplemented
216D12—92no—
217D12—92no—
218D12—91no—
219D12—91no—
220D12—181no—
221C4×D3—31no—
222C4×D3—91no—
223C4×D3—92no—
224C4×D3—92no—
225C4×D3—92no—
226C4×D3—91no—
227C4×D3—91no—
228C4×D3—91no—
229C4×D3—91no—
230C4×D3—91no—
231C4×D3—92no—
232C4×D3—92no—
233C4×D3—92no—
234C4×D3—181no—
235C4×D3C2×D911yescharacteristic, complemented
236Dic6—31no—
237Dic6—92no—
238Dic6—92no—
239Dic6C2×D911yescharacteristic, complemented
240C2×C32—22no—
241C2×C32C23×D311yescharacteristic
242C3⋊D3—32no—
243C3⋊D3—62no—
244C3×D3—21no—
245C3×D3—61no—
246C2×C9—22no—
247C2×C9—21no—
248C2×C9—42no—
249C2×C9—61no—
250C2×C9C23×D311yescharacteristic
251D9—21no—
252D9—32no—
253D9—62no—
254D9—62no—
255D9—122no—
256#41—31no—
257#41—91no—
258#41—272no—
259#41—271no—
260#41—272no—
261#41—271no—
262C2×Q8—271no—
263C2×D4—272no—
264C2×D4—271no—
265C22×C4—272no—
266C22×C4—271no—
267C22×C3—22no—
268C22×C3—31no—
269C22×C3—91no—
270C22×C3C22×D912yes—
271C22×C3C2×D3212yes—
272C2×D3—31no—
273C2×D3—91no—
274C2×D3—91no—
275C2×D3—92no—
276C2×D3—92no—
277C2×D3—181no—
278C2×D3—182no—
279C2×D3—182no—
280C2×D3—182no—
281C2×D3—184no—
282C2×D3—184no—
283C2×D3—364no—
284C2×D3C22×D911yescharacteristic
285C3×C4—21no—
286C3×C4—32no—
287C3×C4—31no—
288C3×C4—92no—
289C3×C4—91no—
290C3×C4C22×D911yescharacteristic
291C3×C4C2×D3211yescharacteristic
292Dic3—32no—
293Dic3—31no—
294Dic3—91no—
295Dic3—91no—
296Dic3—181no—
297Dic3C22×D912yes—
298Dic3C22×D911yescharacteristic
299C32D3×#4111yescharacteristic
300C9—21no—
301C9D3×#4111yescharacteristic, complemented
302C23—271no—
303C23—272no—
304Q8—31no—
305Q8—91no—
306Q8—272no—
307D4—32no—
308D4—92no—
309D4—272no—
310D4—272no—
311D4—272no—
312D4—271no—
313D4D3×D911yescharacteristic, complemented
314C2×C4—32no—
315C2×C4—31no—
316C2×C4—92no—
317C2×C4—91no—
318C2×C4—271no—
319C2×C4—272no—
320C2×C4—272no—
321C2×C4—272no—
322C2×C4—271no—
323C2×C4—271no—
324C2×C4—271no—
325C2×C3—22no—
326C2×C3—22no—
327C2×C3—21no—
328C2×C3—42no—
329C2×C3—61no—
330C2×C3—181no—
331C2×C3C23×D911yescharacteristic
332C2×C3C22×D3211yescharacteristic
333D3—21no—
334D3—61no—
335D3—92no—
336D3—92no—
337D3—182no—
338D3—182no—
339D3—182no—
340D3—362no—
341C22—31no—
342C22—91no—
343C22—271no—
344C22—272no—
345C22—544no—
346C22—542no—
347C22C2×D3×D912yes—
348C4—31no—
349C4—32no—
350C4—91no—
351C4—92no—
352C4—271no—
353C4C2×D3×D911yescharacteristic
354C3—21no—
355C3#41×D911yescharacteristic, complemented
356C3#6431211yescharacteristic
357C2—22no—
358C2—61no—
359C2—181no—
360C2—272no—
361C2—542no—
362C2C22×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. D4×C3⋊D9C2semidirect
2. C9×#237C2semidirect
3. C3×#3699C2semidirect
4. #88668C2semidirect
5. #88662C2semidirect
6. C2×(C3⋊D9)⋊C4C2semidirect
7. #88658C2semidirect
8. C4×D3×D9C2semidirect
9. #88642C2semidirect
10. (C3⋊D9)⋊Q8C2semidirect
11. #88635C2semidirect
16. C3×D4×C9C22semidirect
17. #6233C22semidirect
18. C4⋊(C3⋊D9)C22semidirect
20. C9×#67C22semidirect
21. C3×#611C22semidirect
22. C2×C9×Dic3C22semidirect
23. C2×C3×Dic9C22semidirect
24. C4×D3×C9C22semidirect
25. C3×C4×D9C22semidirect
26. C9×Dic6C22semidirect
27. C3×Dic18C22semidirect
28. #6195C22semidirect
29. D3×Dic9C22semidirect
30. #6192C22semidirect
33. (C3⋊D9)⋊C4C22semidirect
34. (C3⋊D9)⋊C4C22semidirect
35. Dic3×D9C22semidirect
36. #6189C22semidirect
52. #3699D3semidirect
104. D4×C9C2×D3semidirect
110. #611C2×D3semidirect
112. C2×Dic9C2×D3semidirect
123. C4×D9C2×D3semidirect
125. Dic18C2×D3semidirect
129. C3⋊D9#41semidirect
141. #237D9semidirect
188. C3×C9C2×#41semidirect
199. C3×D4C2×D9semidirect
211. #67C2×D9semidirect
215. C2×Dic3C2×D9semidirect
235. C4×D3C2×D9semidirect
239. Dic6C2×D9semidirect
301. C9D3×#41semidirect
313. D4D3×D9semidirect
355. C3#41×D9semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊D92
C9×#2372
C3×#36992
#886682
#886622
C2×(C3⋊D9)⋊C42
#886582
C4×D3×D92
#886422
(C3⋊D9)⋊Q82
#886352
#643123
#41×D93