Finite Group Databasebeta

← #11722535

Subgroups — #11722535

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
1C2×#6233C211yesmaximal, characteristic, complemented
2C2×C9×#67C211yesmaximal, characteristic, complemented
3C2×C3×#611C211yesmaximal, characteristic, complemented
4#88668C212yesmaximal, complemented
5D9×#67C212yesmaximal, complemented
6D3×#611C212yesmaximal, complemented
7#88663C212yesmaximal, complemented
8#88662C212yesmaximal, complemented
9#88658C212yesmaximal, complemented
10#64330—31nomaximal
11#63710—31nomaximal
12C23×C3×C9C2211yescharacteristic, complemented
13C22×C3⋊D9C2211yescharacteristic, complemented
14C22×D3×C9C2211yescharacteristic, complemented
15C22×C3×D9C2211yescharacteristic, complemented
16C2×D3×D9C2212yescomplemented
17#6233C2212yescomplemented
18#6233C2212yescomplemented
19C2×C3⋊Dic9C2211yescharacteristic, complemented
20C9×#67C2212yescomplemented
21C9×#67C2212yescomplemented
22C3×#611C2212yescomplemented
23C3×#611C2212yescomplemented
24C2×C9×Dic3C2211yescharacteristic, complemented
25C2×C3×Dic9C2211yescharacteristic, complemented
26#6195C2212yescomplemented
27#6194C2212yescomplemented
28D3×Dic9C2212yescomplemented
29#6192C2212yescomplemented
30(C3⋊D9)⋊C4C2212yescomplemented
31Dic3×D9C2212yescomplemented
32#6189C2212yescomplemented
33C2×#638—31no—
34C2×C3×#67—31no—
35C2×C3×#67D311yescharacteristic
36#3811—32no—
37D3×#67—32no—
38D3×#67—32no—
39#3805—32no—
40#3804—32no—
41#3804—32no—
42C2×D4×C9—31no—
43C2×#611—31no—
44C2×#611—32no—
45C2×#611—61no—
46C2×#611D311yescharacteristic, complemented
47#3699—32no—
48#3699—32no—
49D4×D9—32no—
50D4×D9—32no—
51#3697—32no—
52C22×C3×C9—41no—
53C22×C3×C9C2311yescharacteristic
54C22×C3×C9C2311yescharacteristic
55C22×C3×C9C2311yescharacteristic
56C2×C3⋊D9—41no—
57C2×C3⋊D9C2312yes—
58C2×D3×C9—41no—
59C2×D3×C9C2312yes—
60C2×C3×D9—41no—
61C2×C3×D9C2312yes—
62D3×D9—42no—
63C3⋊Dic9C2312yes—
64C9×Dic3C2312yes—
65C3×Dic9C2312yes—
66#1004—91no—
67#1004—91no—
68C23×C32C2×D311yescharacteristic
69C22×C3⋊D3—31no—
70C22×C3×D3—31no—
71C22×C3×D3C2×D311yescharacteristic
72C2×D32—32no—
73#638—32no—
74#638—32no—
75C2×C3⋊Dic3—31no—
76C3×#67—32no—
77C3×#67—32no—
78C3×#67C2×D312yes—
79C3×#67C2×D312yes—
80C2×C3×Dic3—31no—
81C2×C3×Dic3C2×D311yescharacteristic
82#627—32no—
83#626—32no—
84#626—32no—
85#625—32no—
86#624—32no—
87D3×Dic3—32no—
88D3×Dic3—32no—
89C23×C9—21no—
90C23×C9—31no—
91C23×C9C2×D311yescharacteristic, complemented
92C22×D9—31no—
93C22×D9—32no—
94C22×D9—61no—
95C22×D9C2×D311yescharacteristic, complemented
96D4×C9—32no—
97D4×C9—32no—
98C2×C4×C9—31no—
99#611—32no—
100#611—32no—
101#611—32no—
102#611—32no—
103#611—62no—
104#611—62no—
105#611C2×D312yescomplemented
106#611C2×D312yescomplemented
107C2×Dic9—31no—
108C2×Dic9—32no—
109C2×Dic9—61no—
110C2×Dic9C2×D311yescharacteristic, complemented
111D36—32no—
112C4×D9—32no—
113C4×D9—32no—
114Dic18—32no—
115C2×C3×C9—21no—
116C2×C3×C9—21no—
117C2×C3×C9—21no—
118C2×C3×C9C2411yescharacteristic
119C3⋊D9—22no—
120D3×C9—22no—
121C3×D9—22no—
122C2×C3×D4—31no—
123C2×C3×D4—91no—
124C2×#67—31no—
125C2×#67—92no—
126C2×#67—91no—
127C2×#67—91no—
128C2×#67—92no—
129C2×#67—181no—
130C2×#67D911yescharacteristic, complemented
131#237—92no—
132#237—92no—
133#237—92no—
134#237—92no—
135D3×D4—92no—
136D3×D4—92no—
137D3×D4—92no—
138D3×D4—92no—
139#235—92no—
140#235—92no—
141C22×C32—41no—
142C22×C32C22×D311yescharacteristic
143C22×C32C22×D311yescharacteristic
144C22×C32C22×D311yescharacteristic
145C2×C3⋊D3—32no—
146C2×C3⋊D3—121no—
147C2×C3×D3—32no—
148C2×C3×D3—41no—
149C2×C3×D3—121no—
150C2×C3×D3C22×D312yes—
151D32—122no—
152C3⋊Dic3—32no—
153C3×Dic3—32no—
154C3×Dic3C22×D312yes—
155C22×C9—21no—
156C22×C9—21no—
157C22×C9—21no—
158C22×C9—32no—
159C22×C9—41no—
160C22×C9—42no—
161C22×C9—121no—
162C22×C9C22×D311yescharacteristic
163C22×C9C22×D311yescharacteristic
164C22×C9C22×D311yescharacteristic
165C2×D9—32no—
166C2×D9—41no—
167C2×D9—62no—
168C2×D9—122no—
169C2×D9—121no—
170C2×D9—122no—
171C2×D9C22×D312yes—
172C4×C9—32no—
173Dic9—32no—
174Dic9—62no—
175Dic9C22×D312yes—
176H(5,𝔽2)—271noSylow
177C3×C9H(5,𝔽2)11yescharacteristic, complemented, Sylow
178C23×C3—21no—
179C23×C3—31no—
180C23×C3—91no—
181C23×C3C2×D911yescharacteristic, complemented
182C23×C3D3211yescharacteristic
183C22×D3—31no—
184C22×D3—92no—
185C22×D3—91no—
186C22×D3—91no—
187C22×D3—92no—
188C22×D3—181no—
189C22×D3C2×D911yescharacteristic, complemented
190C3×D4—32no—
191C3×D4—32no—
192C3×D4—92no—
193C3×D4—92no—
194C2×C3×C4—31no—
195C2×C3×C4—91no—
196#67—32no—
197#67—32no—
198#67—92no—
199#67—92no—
200#67—92no—
201#67—92no—
202#67—92no—
203#67—92no—
204#67—92no—
205#67—92no—
206#67—182no—
207#67—182no—
208#67C2×D912yescomplemented
209#67C2×D912yescomplemented
210C2×Dic3—31no—
211C2×Dic3—92no—
212C2×Dic3—91no—
213C2×Dic3—91no—
214C2×Dic3—92no—
215C2×Dic3—181no—
216C2×Dic3C2×D911yescharacteristic, complemented
217D12—92no—
218D12—92no—
219C4×D3—92no—
220C4×D3—92no—
221C4×D3—92no—
222C4×D3—92no—
223Dic6—92no—
224Dic6—92no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32C23×D311yescharacteristic
229C3⋊D3—62no—
230C3×D3—22no—
231C3×D3—62no—
232C2×C9—22no—
233C2×C9—21no—
234C2×C9—21no—
235C2×C9—21no—
236C2×C9—21no—
237C2×C9—41no—
238C2×C9—41no—
239C2×C9—62no—
240C2×C9C23×D311yescharacteristic
241D9—22no—
242D9—62no—
243D9—122no—
244#41—272no—
245#41—272no—
246#41—272no—
247C2×D4—31no—
248C2×D4—91no—
249C2×D4—272no—
250C2×D4—272no—
251C2×D4—272no—
252C2×D4—271no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—32no—
257C22×C3—41no—
258C22×C3—41no—
259C22×C3—42no—
260C22×C3—92no—
261C22×C3—121no—
262C22×C3—361no—
263C22×C3C22×D911yescharacteristic
264C22×C3C22×D911yescharacteristic
265C22×C3C22×D911yescharacteristic
266C22×C3C2×D3211yescharacteristic
267C22×C3C2×D3211yescharacteristic
268C22×C3C2×D3211yescharacteristic
269C2×D3—32no—
270C2×D3—41no—
271C2×D3—92no—
272C2×D3—92no—
273C2×D3—121no—
274C2×D3—182no—
275C2×D3—362no—
276C2×D3—362no—
277C2×D3—361no—
278C2×D3—361no—
279C2×D3—362no—
280C2×D3C22×D912yes—
281C3×C4—32no—
282C3×C4—92no—
283Dic3—32no—
284Dic3—92no—
285Dic3—92no—
286Dic3—182no—
287Dic3C22×D912yes—
288C32#100411yescharacteristic
289C9—21no—
290C9#100411yescharacteristic, complemented
291C23—31no—
292C23—91no—
293C23—272no—
294C23—271no—
295C23D3×D911yescharacteristic, complemented
296Q8—272no—
297D4—32no—
298D4—32no—
299D4—92no—
300D4—92no—
301D4—272no—
302D4—272no—
303D4—272no—
304D4—272no—
305D4—272no—
306C2×C4—31no—
307C2×C4—91no—
308C2×C4—272no—
309C2×C4—272no—
310C2×C4—272no—
311C2×C4—271no—
312C2×C3—22no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—41no—
321C2×C3—41no—
322C2×C3—62no—
323C2×C3—182no—
324C2×C3C23×D911yescharacteristic
325C2×C3C22×D3211yescharacteristic
326D3—22no—
327D3—62no—
328D3—182no—
329D3—182no—
330D3—362no—
331C22—32no—
332C22—41no—
333C22—92no—
334C22—121no—
335C22—272no—
336C22—361no—
337C22—1081no—
338C22—1082no—
339C22C2×D3×D911yescharacteristic
340C22C2×D3×D911yescharacteristic
341C22C2×D3×D911yescharacteristic
342C4—32no—
343C4—92no—
344C4—272no—
345C3—21no—
346C3#6371011yescharacteristic, complemented
347C3#6433011yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—62no—
352C2—182no—
353C2—542no—
354C2C22×D3×D911yescharacteristic
subgroupG/H
centerC2C22×D3×D9
commutator, cosocleC2×C3×C9C24
FrattiniC2×C3C22×D32
FittingC23×C3×C9C22
socleC2×C32C23×D3
hypercenterC23D3×D9
hypocenterC3×C9H(5,𝔽2)
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#6233C2semidirect
2. C2×C9×#67C2semidirect
3. C2×C3×#611C2semidirect
4. #88668C2semidirect
5. D9×#67C2semidirect
6. D3×#611C2semidirect
7. #88663C2semidirect
8. #88662C2semidirect
9. #88658C2semidirect
12. C23×C3×C9C22semidirect
13. C22×C3⋊D9C22semidirect
14. C22×D3×C9C22semidirect
15. C22×C3×D9C22semidirect
16. C2×D3×D9C22semidirect
17. #6233C22semidirect
18. #6233C22semidirect
19. C2×C3⋊Dic9C22semidirect
20. C9×#67C22semidirect
21. C9×#67C22semidirect
22. C3×#611C22semidirect
23. C3×#611C22semidirect
24. C2×C9×Dic3C22semidirect
25. C2×C3×Dic9C22semidirect
26. #6195C22semidirect
27. #6194C22semidirect
28. D3×Dic9C22semidirect
29. #6192C22semidirect
30. (C3⋊D9)⋊C4C22semidirect
31. Dic3×D9C22semidirect
32. #6189C22semidirect
46. C2×#611D3semidirect
91. C23×C9C2×D3semidirect
95. C22×D9C2×D3semidirect
105. #611C2×D3semidirect
106. #611C2×D3semidirect
110. C2×Dic9C2×D3semidirect
130. C2×#67D9semidirect
177. C3×C9H(5,𝔽2)semidirect
181. C23×C3C2×D9semidirect
189. C22×D3C2×D9semidirect
208. #67C2×D9semidirect
209. #67C2×D9semidirect
216. C2×Dic3C2×D9semidirect
290. C9#1004semidirect
295. C23D3×D9semidirect
346. C3#63710semidirect

Maximal subgroups

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

subgroupindex
C2×#62332
C2×C9×#672
C2×C3×#6112
#886682
D9×#672
D3×#6112
#886632
#886622
#886582
#643303
#637103