Finite Group Databasebeta

← #11722500

Subgroups — #11722500

Order 864.

552 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 131 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
2D3×D4×C9C211yesmaximal, characteristic, complemented
3C3×#3699C211yesmaximal, characteristic, complemented
4#88668C212yesmaximal, complemented
5D3×#611C212yesmaximal, complemented
6C2×#6195C212yesmaximal, complemented
7#88662C212yesmaximal, complemented
8D9×D12C211yesmaximal, characteristic, complemented
9#88642C211yesmaximal, characteristic, complemented
10#88640C211yesmaximal, characteristic, complemented
11#88639C211yesmaximal, characteristic, complemented
12#64314—31nomaximal
13#63710—31nomaximal
14C22×C3⋊D9C2212yescomplemented
15C22×D3×C9C2212yescomplemented
16C2×D3×D9C2212yescomplemented
17C3×D4×C9C2211yescharacteristic, complemented
18#6233C2212yescomplemented
19C4⋊(C3⋊D9)C2211yescharacteristic, complemented
20C4×C3⋊D9C2211yescharacteristic, complemented
21C9×#67C2212yescomplemented
22C3×#611C2212yescomplemented
23C2×C3×Dic9C2212yescomplemented
24C9×D12C2211yescharacteristic, complemented
25C4×D3×C9C2211yescharacteristic, complemented
26C3×C4×D9C2211yescharacteristic, complemented
27C3×Dic18C2211yescharacteristic, complemented
28#6195C2212yescomplemented
29#6195C2211yescharacteristic, complemented
30#6195C2212yescomplemented
31#6195C2212yescomplemented
32#6194C2211yescharacteristic, complemented
33D3×Dic9C2212yescomplemented
34#6192C2211yescharacteristic, complemented
35(C3⋊D9)⋊C4C2212yescomplemented
36#6189C2211yescharacteristic, complemented
37D4×C3⋊D3—31no—
38C3×#237—31no—
39C3×D3×D4D311yescharacteristic
40#3811—32no—
41D3×#67—32no—
42C2×#626—32no—
43#3804—32no—
44D3×D12—31no—
45#3799—31no—
46#3798—31no—
47#3796—31no—
48C2×D4×C9—31no—
49C2×#611—32no—
50C2×#611—32no—
51#3699—32no—
52#3699—31no—
53#3699D311yescharacteristic, complemented
54D4×D9—31no—
55D4×D9—32no—
56D4×D9—31no—
57D4×D9—61no—
58#3697—31no—
59#3697—31no—
60C22×C3×C9C2312yes—
61C2×C3⋊D9—42no—
62C2×C3⋊D9C2312yes—
63C2×C3⋊D9C2311yescharacteristic
64C2×D3×C9—42no—
65C2×D3×C9C2312yes—
66C2×D3×C9C2311yescharacteristic
67C2×C3×D9C2311yescharacteristic
68D3×D9—42no—
69C3×C4×C9C2311yescharacteristic
70C3⋊Dic9C2311yescharacteristic
71C9×Dic3C2311yescharacteristic
72C3×Dic9C2311yescharacteristic
73C3×Dic9C2312yes—
74#1009—91no—
75#1004—91no—
76C22×C3⋊D3—32no—
77C22×C3×D3C2×D312yes—
78C2×D32—32no—
79C32×D4C2×D311yescharacteristic
80#638—32no—
81C4⋊(C3⋊D3)—31no—
82C4×C3⋊D3—31no—
83C3×#67—32no—
84C3×#67C2×D312yes—
85C2×C3×Dic3—32no—
86C3×D12C2×D311yescharacteristic
87C3×C4×D3—31no—
88C3×C4×D3C2×D311yescharacteristic
89C3×Dic6—31no—
90#627—31no—
91#626—32no—
92#626—31no—
93#626—32no—
94#626—32no—
95#626—31no—
96#625—31no—
97#624—32no—
98D3×Dic3—32no—
99C23×C9—32no—
100C22×D9—32no—
101C22×D9—32no—
102C22×D9—62no—
103D4×C9—21no—
104D4×C9—32no—
105D4×C9—31no—
106D4×C9C2×D311yescharacteristic, complemented
107C2×C4×C9—31no—
108#611—32no—
109#611—31no—
110#611—31no—
111#611—32no—
112#611—32no—
113#611—32no—
114#611—62no—
115#611C2×D312yescomplemented
116C2×Dic9—32no—
117C2×Dic9C2×D312yescomplemented
118D36—31no—
119D36—31no—
120D36—61no—
121C4×D9—31no—
122C4×D9—32no—
123C4×D9—61no—
124C4×D9C2×D311yescharacteristic, complemented
125Dic18—31no—
126Dic18C2×D311yescharacteristic, complemented
127C2×C3×C9—22no—
128C2×C3×C9C2411yescharacteristic
129C3⋊D9—21no—
130C3⋊D9—22no—
131D3×C9—22no—
132D3×C9—21no—
133C3×D9—21no—
134C3×#41—91no—
135C2×C3×D4—31no—
136C2×#67—92no—
137C2×#67—92no—
138#239—91no—
139#239—91no—
140#237—31no—
141#237—91no—
142#237—92no—
143D3×D4—91no—
144D3×D4—92no—
145D3×D4—92no—
146D3×D4—91no—
147D3×D4—91no—
148D3×D4—92no—
149D3×D4—181no—
150D3×D4D911yescharacteristic, complemented
151#235—91no—
152#235—91no—
153#235—92no—
154#235—91no—
155C2×D12—92no—
156C2×D12—91no—
157C22×C32C22×D312yes—
158C2×C3⋊D3—32no—
159C2×C3⋊D3—31no—
160C2×C3⋊D3—122no—
161C2×C3×D3—31no—
162C2×C3×D3—42no—
163C2×C3×D3C22×D312yes—
164C2×C3×D3C22×D311yescharacteristic
165D32—122no—
166C32×C4C22×D311yescharacteristic
167C3⋊Dic3—31no—
168C3×Dic3—32no—
169C3×Dic3—31no—
170C3×Dic3C22×D311yescharacteristic
171C22×C9—22no—
172C22×C9—32no—
173C22×C9—31no—
174C22×C9—122no—
175C22×C9C22×D312yes—
176C2×D9—32no—
177C2×D9—31no—
178C2×D9—61no—
179C2×D9—62no—
180C2×D9—122no—
181C2×D9—122no—
182C2×D9—124no—
183C2×D9C22×D311yescharacteristic
184C4×C9—21no—
185C4×C9—31no—
186C4×C9C22×D311yescharacteristic
187Dic9—31no—
188Dic9—61no—
189Dic9C22×D311yescharacteristic
190Dic9C22×D312yes—
191H(5,𝔽2)—271noSylow
192C3×C9H(5,𝔽2)11yescharacteristic, complemented, Sylow
193C23×C3—32no—
194C22×D3—92no—
195C22×D3—92no—
196C22×D3—92no—
197C22×D3—92no—
198C22×D3—182no—
199C22×D3C2×D912yescomplemented
200C3×Q8—91no—
201C3×D4—21no—
202C3×D4—32no—
203C3×D4—31no—
204C3×D4—92no—
205C3×D4C2×D911yescharacteristic, complemented
206C3×D4D3211yescharacteristic
207C2×C3×C4—31no—
208C2×C3×C4—92no—
209C2×C3×C4—91no—
210#67—32no—
211#67—92no—
212#67—91no—
213#67—92no—
214#67—92no—
215#67—91no—
216#67—92no—
217#67—92no—
218#67—91no—
219#67—91no—
220#67—182no—
221#67C2×D912yescomplemented
222C2×Dic3—32no—
223C2×Dic3—92no—
224D12—91no—
225D12—92no—
226D12—91no—
227D12—92no—
228D12—92no—
229D12—91no—
230D12—91no—
231D12—181no—
232D12C2×D911yescharacteristic, complemented
233C4×D3—31no—
234C4×D3—92no—
235C4×D3—92no—
236C4×D3—91no—
237C4×D3—91no—
238C4×D3—92no—
239C4×D3—181no—
240C4×D3C2×D911yescharacteristic, complemented
241Dic6—31no—
242Dic6—91no—
243Dic6—91no—
244C2×C32—22no—
245C2×C32C23×D311yescharacteristic
246C3⋊D3—61no—
247C3⋊D3—62no—
248C3×D3—22no—
249C3×D3—21no—
250C3×D3—61no—
251C2×C9—22no—
252C2×C9—21no—
253C2×C9—42no—
254C2×C9—62no—
255C2×C9—61no—
256C2×C9C23×D311yescharacteristic
257D9—21no—
258D9—62no—
259D9—61no—
260D9—62no—
261D9—122no—
262#41—91no—
263#41—271no—
264#41—271no—
265#41—272no—
266#41—271no—
267C2×D4—31no—
268C2×D4—271no—
269C2×D4—272no—
270C2×D4—272no—
271C2×D4—272no—
272C2×D4—271no—
273C22×C3—22no—
274C22×C3—32no—
275C22×C3—31no—
276C22×C3—91no—
277C22×C3—122no—
278C22×C3C22×D912yes—
279C22×C3C2×D3212yes—
280C2×D3—31no—
281C2×D3—42no—
282C2×D3—91no—
283C2×D3—92no—
284C2×D3—91no—
285C2×D3—92no—
286C2×D3—181no—
287C2×D3—182no—
288C2×D3—362no—
289C2×D3—362no—
290C2×D3—362no—
291C2×D3—362no—
292C2×D3—364no—
293C2×D3C22×D912yes—
294C2×D3C22×D911yescharacteristic
295C3×C4—21no—
296C3×C4—31no—
297C3×C4—91no—
298C3×C4—92no—
299C3×C4C22×D911yescharacteristic
300C3×C4C2×D3211yescharacteristic
301Dic3—32no—
302Dic3—31no—
303Dic3—91no—
304Dic3—91no—
305Dic3—181no—
306Dic3C22×D911yescharacteristic
307C32#100411yescharacteristic
308C9—21no—
309C9#100911yescharacteristic, complemented
310C23—32no—
311C23—272no—
312C23—272no—
313Q8—91no—
314Q8—271no—
315D4—32no—
316D4—31no—
317D4—92no—
318D4—271no—
319D4—271no—
320D4—272no—
321D4—272no—
322D4—272no—
323D4—271no—
324D4—272no—
325D4—271no—
326D4D3×D911yescharacteristic, complemented
327C2×C4—31no—
328C2×C4—92no—
329C2×C4—91no—
330C2×C4—272no—
331C2×C4—272no—
332C2×C4—271no—
333C2×C3—22no—
334C2×C3—22no—
335C2×C3—21no—
336C2×C3—42no—
337C2×C3—62no—
338C2×C3—61no—
339C2×C3—181no—
340C2×C3C23×D911yescharacteristic
341C2×C3C22×D3211yescharacteristic
342D3—21no—
343D3—22no—
344D3—61no—
345D3—182no—
346D3—181no—
347D3—182no—
348D3—181no—
349D3—182no—
350D3—362no—
351C22—31no—
352C22—32no—
353C22—91no—
354C22—122no—
355C22—271no—
356C22—272no—
357C22—1082no—
358C22—1082no—
359C22C2×D3×D912yes—
360C4—31no—
361C4—91no—
362C4—92no—
363C4—271no—
364C4C2×D3×D911yescharacteristic
365C3—21no—
366C3#6371011yescharacteristic, complemented
367C3#6431411yescharacteristic
368C2—22no—
369C2—61no—
370C2—62no—
371C2—181no—
372C2—541no—
373C2—542no—
374C2C22×D3×D911yescharacteristic
subgroupG/H
centerC2C22×D3×D9
commutator, cosocleC2×C3×C9C24
FrattiniC2×C3C22×D32
FittingC3×D4×C9C22
socleC2×C32C23×D3
hypercenterD4D3×D9
hypocenterC3×C9H(5,𝔽2)
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. D3×D4×C9C2semidirect
3. C3×#3699C2semidirect
4. #88668C2semidirect
5. D3×#611C2semidirect
6. C2×#6195C2semidirect
7. #88662C2semidirect
8. D9×D12C2semidirect
9. #88642C2semidirect
10. #88640C2semidirect
11. #88639C2semidirect
14. C22×C3⋊D9C22semidirect
15. C22×D3×C9C22semidirect
16. C2×D3×D9C22semidirect
17. C3×D4×C9C22semidirect
18. #6233C22semidirect
19. C4⋊(C3⋊D9)C22semidirect
20. C4×C3⋊D9C22semidirect
21. C9×#67C22semidirect
22. C3×#611C22semidirect
23. C2×C3×Dic9C22semidirect
24. C9×D12C22semidirect
25. C4×D3×C9C22semidirect
26. C3×C4×D9C22semidirect
27. C3×Dic18C22semidirect
28. #6195C22semidirect
29. #6195C22semidirect
30. #6195C22semidirect
31. #6195C22semidirect
32. #6194C22semidirect
33. D3×Dic9C22semidirect
34. #6192C22semidirect
35. (C3⋊D9)⋊C4C22semidirect
36. #6189C22semidirect
53. #3699D3semidirect
106. D4×C9C2×D3semidirect
115. #611C2×D3semidirect
117. C2×Dic9C2×D3semidirect
124. C4×D9C2×D3semidirect
126. Dic18C2×D3semidirect
150. D3×D4D9semidirect
192. C3×C9H(5,𝔽2)semidirect
199. C22×D3C2×D9semidirect
205. C3×D4C2×D9semidirect
221. #67C2×D9semidirect
232. D12C2×D9semidirect
240. C4×D3C2×D9semidirect
309. C9#1009semidirect
326. D4D3×D9semidirect
366. C3#63710semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊D92
D3×D4×C92
C3×#36992
#886682
D3×#6112
C2×#61952
#886622
D9×D122
#886422
#886402
#886392
#643143
#637103