Finite Group Databasebeta

← #11722499

Subgroups — #11722499

Order 864.

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

normal subgroups only

This group has 538 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
1#88742C211yesmaximal, characteristic, complemented
2D3×D4×C9C211yesmaximal, characteristic, complemented
3C3×D4×D9C211yesmaximal, characteristic, complemented
4D9×#67C212yesmaximal, complemented
5D3×#611C212yesmaximal, complemented
6C2×#6194C212yesmaximal, complemented
7#88663C212yesmaximal, complemented
8#88646C211yesmaximal, characteristic, complemented
9#88641C211yesmaximal, characteristic, complemented
10#88640C211yesmaximal, characteristic, complemented
11#88638C211yesmaximal, characteristic, complemented
12#64313—31nomaximal
13#63710—31nomaximal
14C22×D3×C9C2212yescomplemented
15C22×C3×D9C2212yescomplemented
16C2×D3×D9C2212yescomplemented
17C3×D4×C9C2211yescharacteristic, complemented
18#6233C2212yescomplemented
19C2×C3⋊Dic9C2212yescomplemented
20C4×C3⋊D9C2211yescharacteristic, complemented
21#6226C2211yescharacteristic, complemented
22C9×#67C2212yescomplemented
23C3×#611C2212yescomplemented
24C9×D12C2211yescharacteristic, complemented
25C4×D3×C9C2211yescharacteristic, complemented
26C3×D36C2211yescharacteristic, complemented
27C3×C4×D9C2211yescharacteristic, complemented
28#6195C2211yescharacteristic, complemented
29#6194C2212yescomplemented
30#6194C2211yescharacteristic, complemented
31#6194C2212yescomplemented
32#6194C2212yescomplemented
33D3×Dic9C2212yescomplemented
34#6192C2211yescharacteristic, complemented
35Dic3×D9C2212yescomplemented
36#6189C2211yescharacteristic, complemented
37#3830—31no—
38C3×D3×D4—31no—
39C3×D3×D4D311yescharacteristic
40D3×#67—32no—
41D3×#67—32no—
42C2×#625—32no—
43#3805—32no—
44#3802—31no—
45#3798—31no—
46#3795—31no—
47#3795—31no—
48C2×D4×C9—31no—
49C2×#611—32no—
50C2×#611—32no—
51#3699—31no—
52#3699—32no—
53#3699—31no—
54#3699—61no—
55D4×D9—32no—
56D4×D9—31no—
57D4×D9D311yescharacteristic, complemented
58#3697—31no—
59#3697—31no—
60C22×C3×C9C2312yes—
61C2×C3⋊D9C2311yescharacteristic
62C2×D3×C9—42no—
63C2×D3×C9C2312yes—
64C2×D3×C9C2311yescharacteristic
65C2×C3×D9—42no—
66C2×C3×D9C2312yes—
67C2×C3×D9C2311yescharacteristic
68D3×D9—42no—
69C3×C4×C9C2311yescharacteristic
70C3⋊Dic9C2311yescharacteristic
71C3⋊Dic9C2312yes—
72C9×Dic3C2311yescharacteristic
73C3×Dic9C2311yescharacteristic
74#1004—91no—
75#1004—91no—
76C22×C3×D3—32no—
77C22×C3×D3C2×D312yes—
78C2×D32—32no—
79C32×D4C2×D311yescharacteristic
80#638—32no—
81C2×C3⋊Dic3—32no—
82C4×C3⋊D3—31no—
83#634—31no—
84C3×#67—32no—
85C3×#67C2×D312yes—
86C3×D12—31no—
87C3×D12C2×D311yescharacteristic
88C3×C4×D3—31no—
89C3×C4×D3C2×D311yescharacteristic
90#627—31no—
91#626—31no—
92#626—31no—
93#625—32no—
94#625—32no—
95#625—31no—
96#625—32no—
97D3×Dic3—32no—
98D3×Dic3—32no—
99C23×C9—32no—
100C22×D9—32no—
101C22×D9C2×D312yescomplemented
102D4×C9—21no—
103D4×C9—32no—
104D4×C9—31no—
105D4×C9C2×D311yescharacteristic, complemented
106C2×C4×C9—31no—
107#611—32no—
108#611—32no—
109#611—31no—
110#611—32no—
111#611—32no—
112#611—31no—
113#611—62no—
114#611C2×D312yescomplemented
115C2×Dic9—32no—
116C2×Dic9—32no—
117C2×Dic9—62no—
118D36—31no—
119D36C2×D311yescharacteristic, complemented
120C4×D9—32no—
121C4×D9—31no—
122C4×D9—61no—
123C4×D9C2×D311yescharacteristic, complemented
124Dic18—31no—
125Dic18—31no—
126Dic18—61no—
127C2×C3×C9—22no—
128C2×C3×C9C2411yescharacteristic
129C3⋊D9—21no—
130D3×C9—22no—
131D3×C9—21no—
132C3×D9—22no—
133C3×D9—21no—
134C2×C3×D4—31no—
135C2×C3×D4—91no—
136C2×#67—92no—
137C2×#67—92no—
138C2×#67—92no—
139C2×#67—92no—
140#237—91no—
141#237—92no—
142#237—91no—
143#237—91no—
144#237—92no—
145#237—91no—
146#237—181no—
147D3×D4—31no—
148D3×D4—91no—
149D3×D4—92no—
150D3×D4—92no—
151D3×D4—91no—
152D3×D4D911yescharacteristic, complemented
153#235—91no—
154#235—91no—
155#235—91no—
156#235—91no—
157C22×C32C22×D312yes—
158C2×C3⋊D3—31no—
159C2×C3×D3—32no—
160C2×C3×D3—31no—
161C2×C3×D3—42no—
162C2×C3×D3—122no—
163C2×C3×D3C22×D312yes—
164C2×C3×D3C22×D311yescharacteristic
165D32—122no—
166C32×C4C22×D311yescharacteristic
167C3⋊Dic3—32no—
168C3⋊Dic3—31no—
169C3×Dic3—31no—
170C3×Dic3C22×D311yescharacteristic
171C22×C9—22no—
172C22×C9—32no—
173C22×C9—31no—
174C22×C9—122no—
175C22×C9C22×D312yes—
176C2×D9—31no—
177C2×D9—42no—
178C2×D9—61no—
179C2×D9—122no—
180C2×D9C22×D312yes—
181C2×D9C22×D311yescharacteristic
182C4×C9—21no—
183C4×C9—31no—
184C4×C9C22×D311yescharacteristic
185Dic9—31no—
186Dic9—32no—
187Dic9—62no—
188Dic9—61no—
189Dic9C22×D311yescharacteristic
190H(5,𝔽2)—271noSylow
191C3×C9H(5,𝔽2)11yescharacteristic, complemented, Sylow
192C23×C3—32no—
193C23×C3—92no—
194C22×D3—32no—
195C22×D3—92no—
196C22×D3—92no—
197C22×D3C2×D912yescomplemented
198C3×D4—21no—
199C3×D4—32no—
200C3×D4—31no—
201C3×D4—92no—
202C3×D4—91no—
203C3×D4C2×D911yescharacteristic, complemented
204C3×D4D3211yescharacteristic
205C2×C3×C4—31no—
206C2×C3×C4—91no—
207#67—32no—
208#67—91no—
209#67—91no—
210#67—92no—
211#67—92no—
212#67—92no—
213#67—92no—
214#67—92no—
215#67—92no—
216#67—91no—
217#67—92no—
218#67—92no—
219#67—91no—
220#67—182no—
221#67C2×D912yescomplemented
222C2×Dic3—92no—
223C2×Dic3—92no—
224C2×Dic3—92no—
225C2×Dic3—92no—
226C2×Dic3—182no—
227D12—31no—
228D12—91no—
229D12—91no—
230D12C2×D911yescharacteristic, complemented
231C4×D3—31no—
232C4×D3—92no—
233C4×D3—92no—
234C4×D3—91no—
235C4×D3—91no—
236C4×D3—181no—
237C4×D3C2×D911yescharacteristic, complemented
238Dic6—91no—
239Dic6—91no—
240Dic6—91no—
241Dic6—91no—
242Dic6—181no—
243C2×C32—22no—
244C2×C32C23×D311yescharacteristic
245C3⋊D3—61no—
246C3×D3—22no—
247C3×D3—21no—
248C3×D3—62no—
249C3×D3—61no—
250C2×C9—22no—
251C2×C9—21no—
252C2×C9—42no—
253C2×C9—62no—
254C2×C9—61no—
255C2×C9C23×D311yescharacteristic
256D9—21no—
257D9—22no—
258D9—61no—
259D9—121no—
260#41—271no—
261#41—271no—
262#41—271no—
263#41—272no—
264#41—271no—
265C2×D4—31no—
266C2×D4—91no—
267C2×D4—271no—
268C2×D4—272no—
269C2×D4—272no—
270C2×D4—272no—
271C22×C3—22no—
272C22×C3—32no—
273C22×C3—31no—
274C22×C3—92no—
275C22×C3—91no—
276C22×C3—122no—
277C22×C3—362no—
278C22×C3C22×D912yes—
279C22×C3C2×D3212yes—
280C2×D3—31no—
281C2×D3—32no—
282C2×D3—42no—
283C2×D3—91no—
284C2×D3—91no—
285C2×D3—122no—
286C2×D3—181no—
287C2×D3—362no—
288C2×D3—362no—
289C2×D3C22×D912yes—
290C2×D3C22×D911yescharacteristic
291C3×C4—21no—
292C3×C4—31no—
293C3×C4—91no—
294C3×C4C22×D911yescharacteristic
295C3×C4C2×D3211yescharacteristic
296Dic3—31no—
297Dic3—91no—
298Dic3—92no—
299Dic3—92no—
300Dic3—91no—
301Dic3—182no—
302Dic3—181no—
303Dic3C22×D911yescharacteristic
304C32#100411yescharacteristic
305C9—21no—
306C9#100411yescharacteristic, complemented
307C23—32no—
308C23—92no—
309C23—272no—
310Q8—271no—
311Q8—271no—
312D4—32no—
313D4—31no—
314D4—92no—
315D4—91no—
316D4—271no—
317D4—271no—
318D4—271no—
319D4—272no—
320D4—272no—
321D4—272no—
322D4—272no—
323D4D3×D911yescharacteristic, complemented
324C2×C4—31no—
325C2×C4—91no—
326C2×C4—272no—
327C2×C4—272no—
328C2×C4—272no—
329C2×C4—271no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—42no—
334C2×C3—62no—
335C2×C3—61no—
336C2×C3—182no—
337C2×C3—181no—
338C2×C3C23×D911yescharacteristic
339C2×C3C22×D3211yescharacteristic
340D3—21no—
341D3—22no—
342D3—61no—
343D3—62no—
344D3—181no—
345D3—181no—
346D3—361no—
347C22—31no—
348C22—32no—
349C22—91no—
350C22—92no—
351C22—122no—
352C22—271no—
353C22—362no—
354C22—1082no—
355C22C2×D3×D912yes—
356C4—31no—
357C4—91no—
358C4—272no—
359C4—271no—
360C4C2×D3×D911yescharacteristic
361C3—21no—
362C3#6371011yescharacteristic, complemented
363C3#6431311yescharacteristic
364C2—22no—
365C2—61no—
366C2—62no—
367C2—181no—
368C2—182no—
369C2—541no—
370C2C22×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. #88742C2semidirect
2. D3×D4×C9C2semidirect
3. C3×D4×D9C2semidirect
4. D9×#67C2semidirect
5. D3×#611C2semidirect
6. C2×#6194C2semidirect
7. #88663C2semidirect
8. #88646C2semidirect
9. #88641C2semidirect
10. #88640C2semidirect
11. #88638C2semidirect
14. C22×D3×C9C22semidirect
15. C22×C3×D9C22semidirect
16. C2×D3×D9C22semidirect
17. C3×D4×C9C22semidirect
18. #6233C22semidirect
19. C2×C3⋊Dic9C22semidirect
20. C4×C3⋊D9C22semidirect
21. #6226C22semidirect
22. C9×#67C22semidirect
23. C3×#611C22semidirect
24. C9×D12C22semidirect
25. C4×D3×C9C22semidirect
26. C3×D36C22semidirect
27. C3×C4×D9C22semidirect
28. #6195C22semidirect
29. #6194C22semidirect
30. #6194C22semidirect
31. #6194C22semidirect
32. #6194C22semidirect
33. D3×Dic9C22semidirect
34. #6192C22semidirect
35. Dic3×D9C22semidirect
36. #6189C22semidirect
57. D4×D9D3semidirect
101. C22×D9C2×D3semidirect
105. D4×C9C2×D3semidirect
114. #611C2×D3semidirect
119. D36C2×D3semidirect
123. C4×D9C2×D3semidirect
152. D3×D4D9semidirect
191. C3×C9H(5,𝔽2)semidirect
197. C22×D3C2×D9semidirect
203. C3×D4C2×D9semidirect
221. #67C2×D9semidirect
230. D12C2×D9semidirect
237. C4×D3C2×D9semidirect
306. C9#1004semidirect
323. D4D3×D9semidirect
362. C3#63710semidirect

Maximal subgroups

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

subgroupindex
#887422
D3×D4×C92
C3×D4×D92
D9×#672
D3×#6112
C2×#61942
#886632
#886462
#886412
#886402
#886382
#643133
#637103