Finite Group Databasebeta

← #22572028

Subgroups — #22572028

Order 8576.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864104C211yesmaximal, characteristic, complemented
2#16864030C211yesmaximal, characteristic
3#16863741C211yesmaximal, characteristic, complemented
4#16863716C211yesmaximal, characteristic, complemented
5#16863712C211yesmaximal, characteristic, complemented
6#16863685C211yesmaximal, characteristic, complemented
7#16863628C211yesmaximal, characteristic, complemented
8#16863594C211yesmaximal, characteristic, complemented
9#16863546C211yesmaximal, characteristic, complemented
10#16863545C211yesmaximal, characteristic, complemented
11#16863481C211yesmaximal, characteristic, complemented
12C8×C22⋊D67C211yesmaximal, characteristic, complemented
13C2×#14523965C211yesmaximal, characteristic, complemented
14#102×D67C211yesmaximal, characteristic, complemented
15#465×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C2×D4×D67—21no—
18C2×D4×D67C2211yescharacteristic
19C2×#11761615C2211yescharacteristic
20C2×#11761615C2211yescharacteristic, complemented
21C2×C8×D67C2211yescharacteristic, complemented
22#14524042C2211yescharacteristic, complemented
23#14524039C2211yescharacteristic
24#14524033—21no—
25#14524023—21no—
26#14524020—21no—
27#14524019C2211yescharacteristic, complemented
28#14524018C2211yescharacteristic
29#14524016—21no—
30#14524013—21no—
31#14524011—21no—
32Q8×Dic67C2211yescharacteristic, complemented
33#14524005C2211yescharacteristic
34C4×C22⋊D67C2211yescharacteristic
35(C22⋊D67)⋊C4—21no—
36C2×C4⋊Dic67C2211yescharacteristic, complemented
37C4⋊C4×D67C2211yescharacteristic, complemented
38#14523974C2211yescharacteristic, complemented
39#14523974C2211yescharacteristic, complemented
40Dic67⋊C8C2211yescharacteristic, complemented
41#14523965C2211yescharacteristic
42#14523965C2211yescharacteristic
43#14523965C2211yescharacteristic, complemented
44#14523965C2211yescharacteristic, complemented
45#14523964C2211yescharacteristic, complemented
46#14523964C2211yescharacteristic, complemented
47#14523963C2211yescharacteristic
48#14523963C2211yescharacteristic, complemented
49C22⋊(C67⋊C8)C2211yescharacteristic, complemented
50(C2×D67)⋊C8C2211yescharacteristic, complemented
51C8×Dic67C2211yescharacteristic, complemented
52C2×SD16×C67C2211yescharacteristic, complemented
53C22×C8×C67C2211yescharacteristic, complemented
54C22⋊Q8×C67C2211yescharacteristic, complemented
55#121×C67C2211yescharacteristic
56#106×C67C2211yescharacteristic, complemented
57#103×C67C2211yescharacteristic, complemented
58#102×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60C23×D67—21no—
61C2×Dic134—21no—
62C2×D268—22no—
63C2×D268—21no—
64C2×D268—21no—
65C2×D268—41no—
66C2×D268C2311yescharacteristic
67C2×D268C2311yescharacteristic
68C2×C22⋊D67—21no—
69C2×C22⋊D67—21no—
70C2×C22⋊D67—21no—
71C2×C22⋊D67D411yescharacteristic
72D4×D67—22no—
73D4×D67—41no—
74D4×D67—44no—
75C22×Dic67—21no—
76C2×C4×D67—21no—
77C2×C4×D67—21no—
78C2×C4×D67C2311yescharacteristic
79#11761615—22no—
80#11761615—41no—
81C2×C67⋊C8C2311yescharacteristic
82C2×C67⋊C8C2311yescharacteristic
83C8×D67—41no—
84C4⋊Dic67—21no—
85C4⋊Dic67—21no—
86C4⋊Dic67C2311yescharacteristic
87C4⋊Dic67C2311yescharacteristic
88Dic67⋊C4—21no—
89Dic67⋊C4—21no—
90Dic67⋊C4—21no—
91Dic67⋊C4D411yescharacteristic
92C22⋊Dic67D411yescharacteristic
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96(C2×D67)⋊C4D411yescharacteristic
97C4×Dic67—21no—
98C4×Dic67C2311yescharacteristic
99C2×Q8×C67C2311yescharacteristic
100C2×D4×C67—21no—
101C2×D4×C67C2311yescharacteristic
102C22×C4×C67C2311yescharacteristic
103SD16×C67—41no—
104C2×C8×C67—41no—
105C2×C8×C67C2311yescharacteristic
106C2×C8×C67C2311yescharacteristic
107C4⋊C4×C67—21no—
108C4⋊C4×C67C2311yescharacteristic
109C4⋊C4×C67C2311yescharacteristic
110C22⋊C4×C67—21no—
111C22⋊C4×C67—21no—
112C23×C67—21no—
113C23×C67C2×D411yescharacteristic
114C22×D67—21no—
115C22×D67—21no—
116C22×D67—42no—
117C22×D67—42no—
118C22×D67—42no—
119C22×D67—42no—
120C22×D67—81no—
121C22×D67—81no—
122C22×D67C2×D411yescharacteristic
123Q8×C67—21no—
124Q8×C67—21no—
125D4×C67—21no—
126D4×C67—21no—
127D4×C67—42no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67C2×D411yescharacteristic
134C2×C4×C67C2411yescharacteristic
135C22⋊D67—42no—
136C22⋊D67—42no—
137C22⋊D67—81no—
138C22⋊D67SD1614yescomplemented
139C2×Dic67—21no—
140C2×Dic67—21no—
141C2×Dic67—81no—
142C2×Dic67C2×D411yescharacteristic
143C2×Dic67C2×D411yescharacteristic
144C2×Dic67C2×D411yescharacteristic
145D268—21no—
146D268—21no—
147D268—21no—
148D268—42no—
149D268—42no—
150D268#4112yes—
151C4×D67—21no—
152C4×D67—21no—
153C4×D67—42no—
154C4×D67—81no—
155Dic134—81no—
156C8×C67—21no—
157C8×C67—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C22×C67—21no—
161C22×C67—21no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67C2×SD1612yes—
166C22×C67C22×D411yescharacteristic
167C2×D67—21no—
168C2×D67—21no—
169C2×D67—22no—
170C2×D67—41no—
171C2×D67—41no—
172C2×D67—41no—
173C2×D67—41no—
174C2×D67—41no—
175C2×D67—84no—
176C2×D67—84no—
177C2×D67C2×SD1612yes—
178C4×C67—21no—
179C4×C67—41no—
180C4×C67—41no—
181C4×C67—41no—
182C4×C67C2×#4111yescharacteristic
183C4×C67H(5,𝔽2)11yescharacteristic
184Dic67—21no—
185Dic67—21no—
186Dic67—41no—
187Dic67—41no—
188Dic67C2×SD1612yes—
189C2×C67—22no—
190C2×C67—81no—
191C2×C67#54611yescharacteristic
192C2×C67C22×SD1611yescharacteristic
193C2×C67#57611yescharacteristic
194D67—22no—
195D67—42no—
196D67—81no—
197#3295—671nomaximal, Sylow
198C67#329511yescharacteristic, complemented, Sylow
199#554—671no—
200D42—671no—
201#486—671no—
202#481—671no—
203#481—671no—
204#476—671no—
205#465—671no—
206#465D6711yescharacteristic, complemented
207#460—671no—
208#450—671no—
209#450—671no—
210C4×SD16—671no—
211C8×D4—671no—
212C2×#102—671no—
213C2×#102—671no—
214C22×D4—671no—
215C22×D4—671no—
216C22×D4—1341no—
217C2×SD16—671no—
218C2×SD16—671no—
219C2×SD16C2×D6711yescharacteristic, complemented
220C22×C8—671no—
221C22×C8C2×D6711yescharacteristic, complemented
222C4⋊Q8—671no—
223#127—671no—
224#125—1341no—
225C22⋊Q8—671no—
226C22⋊Q8—1341no—
227C22⋊Q8—1341no—
228C22⋊Q8C2×D6711yescharacteristic, complemented
229#121—671no—
230#121—1341no—
231#121C2×D6711yescharacteristic
232#120—1341no—
233#120—1341no—
234C4×Q8—671no—
235C4×D4—671no—
236C4×D4—671no—
237C4×D4—1341no—
238C2×C4⋊C4—671no—
239C2×C4⋊C4—671no—
240#106—671no—
241#106—671no—
242#106C2×D6711yescharacteristic, complemented
243C4⋊C8—671no—
244#103C2×D6711yescharacteristic, complemented
245#102—671no—
246#102—671no—
247#102—671no—
248#102—671no—
249#102—671no—
250#102—671no—
251#102—671no—
252#102—671no—
253#102C2×D6711yescharacteristic, complemented
254C22⋊C8—671no—
255C22⋊C8—671no—
256C4×C8—671no—
257C24—1341no—
258C24—1341no—
259C2×Q8—1341no—
260C2×Q8C22×D6711yescharacteristic
261C2×D4—21no—
262C2×D4—2681no—
263C2×D4—2684no—
264C2×D4—2681no—
265C2×D4—671no—
266C2×D4—671no—
267C2×D4—671no—
268C2×D4—1341no—
269C2×D4—1342no—
270C2×D4—1341no—
271C2×D4—1341no—
272C2×D4—1341no—
273C2×D4—1342no—
274C2×D4—1341no—
275C2×D4C22×D6711yescharacteristic
276C22×C4—671no—
277C22×C4—1341no—
278C22×C4—1341no—
279C22×C4—1341no—
280C22×C4C22×D6711yescharacteristic
281SD16—41no—
282SD16—2681no—
283SD16—1342no—
284C2×C8—41no—
285C2×C8—2681no—
286C2×C8—671no—
287C2×C8—671no—
288C2×C8C22×D6711yescharacteristic
289C2×C8C22×D6711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—671no—
292C4⋊C4—671no—
293C4⋊C4—671no—
294C4⋊C4—1341no—
295C4⋊C4—1341no—
296C4⋊C4—1341no—
297C4⋊C4—1341no—
298C4⋊C4—1341no—
299C4⋊C4C22×D6711yescharacteristic
300C4⋊C4C22×D6711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—671no—
304C22⋊C4—671no—
305C22⋊C4—1341no—
306C22⋊C4—1341no—
307C22⋊C4—1341no—
308C42—671no—
309C42—1341no—
310C23—21no—
311C23—2682no—
312C23—2682no—
313C23—2682no—
314C23—2682no—
315C23—5361no—
316C23—5361no—
317C23—671no—
318C23—1341no—
319C23—1341no—
320C23D4×D6711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—5361no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—2682no—
328D4—2682no—
329D4—2682no—
330D4—2682no—
331D4—5361no—
332D4—674no—
333D4—672no—
334D4—1341no—
335D4—1341no—
336D4—1341no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—2682no—
343C2×C4—5361no—
344C2×C4—5361no—
345C2×C4—671no—
346C2×C4—671no—
347C2×C4—671no—
348C2×C4—1341no—
349C2×C4—1341no—
350C2×C4—1341no—
351C2×C4—1341no—
352C2×C4D4×D6711yescharacteristic
353C2×C4C23×D6711yescharacteristic
354C8—21no—
355C8—21no—
356C8—1341no—
357C8—1341no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—2681no—
364C22—2681no—
365C22—2681no—
366C22—2681no—
367C22—2681no—
368C22—5364no—
369C22—5364no—
370C22—672no—
371C22—1342no—
372C22—1341no—
373C22—1341no—
374C22SD16×D6712yes—
375C22C2×D4×D6711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2681no—
381C4—2681no—
382C4—672no—
383C4—1341no—
384C4—1341no—
385C4#41×D6711yescharacteristic
386C4#1452408811yescharacteristic
387C2—22no—
388C2—81no—
389C2—2682no—
390C2—5361no—
391C2—1342no—
392C2#1686403511yescharacteristic
393C2C2×SD16×D6711yescharacteristic
394C2#1686419811yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#465×C67C2
socleC22×C67C22×D4
hypercenter#465D67
hypocenterC67#3295
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #16864104C2semidirect
3. #16863741C2semidirect
4. #16863716C2semidirect
5. #16863712C2semidirect
6. #16863685C2semidirect
7. #16863628C2semidirect
8. #16863594C2semidirect
9. #16863546C2semidirect
10. #16863545C2semidirect
11. #16863481C2semidirect
12. C8×C22⋊D67C2semidirect
13. C2×#14523965C2semidirect
14. #102×D67C2semidirect
15. #465×C67C2semidirect
20. C2×#11761615C22semidirect
21. C2×C8×D67C22semidirect
22. #14524042C22semidirect
27. #14524019C22semidirect
32. Q8×Dic67C22semidirect
36. C2×C4⋊Dic67C22semidirect
37. C4⋊C4×D67C22semidirect
38. #14523974C22semidirect
39. #14523974C22semidirect
40. Dic67⋊C8C22semidirect
43. #14523965C22semidirect
44. #14523965C22semidirect
45. #14523964C22semidirect
46. #14523964C22semidirect
48. #14523963C22semidirect
49. C22⋊(C67⋊C8)C22semidirect
50. (C2×D67)⋊C8C22semidirect
51. C8×Dic67C22semidirect
52. C2×SD16×C67C22semidirect
53. C22×C8×C67C22semidirect
54. C22⋊Q8×C67C22semidirect
56. #106×C67C22semidirect
57. #103×C67C22semidirect
58. #102×C67C22semidirect
138. C22⋊D67SD16semidirect
198. C67#3295semidirect
206. #465D67semidirect
219. C2×SD16C2×D67semidirect
221. C22×C8C2×D67semidirect
228. C22⋊Q8C2×D67semidirect
242. #106C2×D67semidirect
244. #103C2×D67semidirect
253. #102C2×D67semidirect