Finite Group Databasebeta

← #22572046

Subgroups — #22572046

Order 8576.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864086C211yesmaximal, characteristic, complemented
2#16864042C211yesmaximal, characteristic
3#16863751C211yesmaximal, characteristic, complemented
4#16863727C211yesmaximal, characteristic, complemented
5#16863721C211yesmaximal, characteristic, complemented
6#16863697C211yesmaximal, characteristic, complemented
7#16863637C211yesmaximal, characteristic, complemented
8#16863625C211yesmaximal, characteristic, complemented
9#16863619C211yesmaximal, characteristic, complemented
10#16863560C211yesmaximal, characteristic, complemented
11#16863549C211yesmaximal, characteristic, complemented
12C4×#11761617C211yesmaximal, characteristic, complemented
13#16863393C211yesmaximal, characteristic, complemented
14C2×#14523965C211yesmaximal, characteristic, complemented
15C8×D4×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×D268C2211yescharacteristic
18C2×Dic268C2211yescharacteristic, complemented
19C2×#11761617C2211yescharacteristic
20C2×#11761617C2211yescharacteristic
21#14524036—21no—
22#14524035C2211yescharacteristic, complemented
23#14524033—21no—
24#14524031C2211yescharacteristic
25#14524028—21no—
26#14524024—21no—
27#14524022C2211yescharacteristic
28#14524022C2211yescharacteristic
29#14524022C2211yescharacteristic, complemented
30#14524017C2211yescharacteristic
31#14524016—21no—
32#14524011—21no—
33C4×Dic134C2211yescharacteristic, complemented
34C4×D268C2211yescharacteristic
35#14524002C2211yescharacteristic, complemented
36#14523999—21no—
37C2×C4⋊Dic67C2211yescharacteristic
38C22⋊C4×D67—21no—
39#14523977C2211yescharacteristic, complemented
40#14523975C2211yescharacteristic, complemented
41#14523975C2211yescharacteristic, complemented
42#14523968C2211yescharacteristic, complemented
43#14523968C2211yescharacteristic, complemented
44#14523968C2211yescharacteristic, complemented
45#14523968C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic
47#14523965C2211yescharacteristic
48#14523965C2211yescharacteristic
49#14523965C2211yescharacteristic
50#14523965C2211yescharacteristic
51#14523965C2211yescharacteristic
52C22×C8×C67C2211yescharacteristic
53C22×C8×C67C2211yescharacteristic, complemented
54C4×D4×C67C2211yescharacteristic
55C4⋊C8×C67C2211yescharacteristic, complemented
56C22⋊C8×C67C2211yescharacteristic
57C22⋊C8×C67C2211yescharacteristic, complemented
58C4×C8×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60#11761628—22no—
61#11761628—41no—
62C2×Dic134C2311yescharacteristic
63C2×Dic134C2311yescharacteristic
64C2×D268—22no—
65C2×D268—21no—
66C2×D268—41no—
67C2×D268C2311yescharacteristic
68C2×D268C2311yescharacteristic
69C2×C22⋊D67—21no—
70C2×C22⋊D67—21no—
71C22×Dic67—21no—
72C2×C4×D67—21no—
73C2×C4×D67—21no—
74Dic268—41no—
75#11761617—22no—
76#11761617—41no—
77C4⋊Dic67—21no—
78C4⋊Dic67C2311yescharacteristic
79C4⋊Dic67C2311yescharacteristic
80C4⋊Dic67C2311yescharacteristic
81C4⋊Dic67C2311yescharacteristic
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4—21no—
88C22⋊Dic67—21no—
89C22⋊Dic67—21no—
90C22⋊Dic67—21no—
91C22⋊Dic67—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96C4×Dic67—21no—
97C4×Dic67—21no—
98C2×D4×C67D411yescharacteristic
99C22×C4×C67C2311yescharacteristic
100C22×C4×C67C2311yescharacteristic
101C2×C8×C67—41no—
102C2×C8×C67—41no—
103C2×C8×C67C2311yescharacteristic
104C2×C8×C67C2311yescharacteristic
105C2×C8×C67C2311yescharacteristic
106C2×C8×C67C2311yescharacteristic
107C4⋊C4×C67D411yescharacteristic
108C22⋊C4×C67D411yescharacteristic
109C22⋊C4×C67D411yescharacteristic
110C42×C67C2311yescharacteristic
111C23×C67C2×D411yescharacteristic
112C23×C67C2×D411yescharacteristic
113C22×D67—21no—
114C22×D67—21no—
115C22×D67—42no—
116C22×D67—42no—
117C22×D67—81no—
118D4×C67—22no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2×D411yescharacteristic
127C2×C4×C67C2411yescharacteristic
128C22⋊D67—42no—
129C22⋊D67—81no—
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67—21no—
136C2×Dic67—42no—
137D268—21no—
138D268—21no—
139D268—21no—
140D268—42no—
141D268#4112yes—
142C4×D67—42no—
143C4×D67—81no—
144Dic134—21no—
145Dic134—21no—
146Dic134—21no—
147Dic134—21no—
148C8×C67—21no—
149C8×C67—21no—
150C8×C67—21no—
151C8×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67#13512yes—
158C22×C67C22×D411yescharacteristic
159C2×D67—22no—
160C2×D67—41no—
161C2×D67—41no—
162C2×D67—41no—
163C2×D67—41no—
164C2×D67—41no—
165C2×D67—84no—
166C4×C67—21no—
167C4×C67—21no—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67C2×#4111yescharacteristic
171C4×C67H(5,𝔽2)11yescharacteristic
172Dic67—41no—
173Dic67—41no—
174Dic67—41no—
175Dic67—41no—
176Dic67—41no—
177Dic67—41no—
178C2×C67—22no—
179C2×C67—41no—
180C2×C67#54611yescharacteristic
181C2×C67C2×#13511yescharacteristic
182C2×C67#57711yescharacteristic
183D67—42no—
184D67—81no—
185#3296—671nomaximal, Sylow
186C67#329611yescharacteristic, complemented, Sylow
187#552—671no—
188#546—671no—
189#488—671no—
190#483—671no—
191#482—671no—
192#478—671no—
193#467—671no—
194#465—671no—
195#464—671no—
196#452—671no—
197#450—671no—
198C4×SD16—671no—
199C8×D4D6711yescharacteristic, complemented
200#416—671no—
201C2×#102—671no—
202C2×#41—671no—
203C22×D4—671no—
204C2×Q16—671no—
205C2×SD16—671no—
206C2×SD16—671no—
207C22×C8C2×D6711yescharacteristic
208C22×C8C2×D6711yescharacteristic, complemented
209#126—1341no—
210#125—671no—
211#125—1341no—
212#124—671no—
213#123—1341no—
214#123—1341no—
215C22⋊Q8—671no—
216C22⋊Q8—671no—
217C22⋊Q8—671no—
218#121—671no—
219#121—1341no—
220#120—1341no—
221C4×Q8—671no—
222C4×D4—671no—
223C4×D4C2×D6711yescharacteristic
224#117—671no—
225#117—1341no—
226C2×C4⋊C4—671no—
227C2×C22⋊C4—1341no—
228#107—671no—
229#106—671no—
230#106—671no—
231C4⋊C8C2×D6711yescharacteristic, complemented
232#103—671no—
233#103—671no—
234#103—671no—
235#103—671no—
236#102—671no—
237#102—671no—
238#102—671no—
239#102—671no—
240#102—671no—
241#102—671no—
242C22⋊C8C2×D6711yescharacteristic
243C22⋊C8C2×D6711yescharacteristic, complemented
244C4×C8C2×D6711yescharacteristic, complemented
245C24—1341no—
246#41—2681no—
247#41—1342no—
248C2×Q8—671no—
249C2×Q8—671no—
250C2×D4—2681no—
251C2×D4—671no—
252C2×D4—671no—
253C2×D4—1342no—
254C2×D4—1341no—
255C2×D4—1341no—
256C2×D4—1341no—
257C2×D4D26811yescharacteristic
258C22×C4—1341no—
259C22×C4—1341no—
260C22×C4—1341no—
261C22×C4C22×D6711yescharacteristic
262C22×C4C22×D6711yescharacteristic
263Q16—2681no—
264SD16—2681no—
265SD16—1342no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D6711yescharacteristic
269C2×C8C22×D6711yescharacteristic
270C2×C8C22×D6711yescharacteristic
271C2×C8C22×D6711yescharacteristic
272C4⋊C4—671no—
273C4⋊C4—671no—
274C4⋊C4—671no—
275C4⋊C4—671no—
276C4⋊C4—1341no—
277C4⋊C4—1341no—
278C4⋊C4—1341no—
279C4⋊C4—1341no—
280C4⋊C4—1341no—
281C4⋊C4—1341no—
282C4⋊C4—1341no—
283C4⋊C4D26811yescharacteristic
284C22⋊C4—1341no—
285C22⋊C4—1341no—
286C22⋊C4—1341no—
287C22⋊C4—1341no—
288C22⋊C4—1341no—
289C22⋊C4—1341no—
290C22⋊C4—1341no—
291C22⋊C4—1341no—
292C22⋊C4D26811yescharacteristic
293C22⋊C4D26811yescharacteristic
294C42—1341no—
295C42—1341no—
296C42C22×D6711yescharacteristic
297C23—2682no—
298C23—2682no—
299C23—5361no—
300C23—1341no—
301C23—1341no—
302C23C2×D26811yescharacteristic
303C23C2×D26811yescharacteristic
304Q8—1341no—
305Q8—1341no—
306Q8—1341no—
307Q8—1341no—
308D4—22no—
309D4—2682no—
310D4—2682no—
311D4—5361no—
312D4—672no—
313D4—1341no—
314D4—1341no—
315D4—1341no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—2682no—
321C2×C4—2682no—
322C2×C4—5361no—
323C2×C4—1341no—
324C2×C4—1341no—
325C2×C4—1341no—
326C2×C4—1341no—
327C2×C4—1341no—
328C2×C4—1341no—
329C2×C4C2×D26811yescharacteristic
330C2×C4C2×D26811yescharacteristic
331C2×C4C2×D26811yescharacteristic
332C2×C4C2×D26811yescharacteristic
333C2×C4C23×D6711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—2681no—
344C22—2681no—
345C22—2681no—
346C22—2681no—
347C22—2681no—
348C22—5364no—
349C22—1342no—
350C22#1452406512yes—
351C22C22×D26811yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—2681no—
357C4—2681no—
358C4—2681no—
359C4—2681no—
360C4—2681no—
361C4—2681no—
362C4#41×D6711yescharacteristic
363C4#1452408811yescharacteristic
364C2—22no—
365C2—41no—
366C2—2682no—
367C2—5361no—
368C2#1686404111yescharacteristic
369C2C2×#1452406511yescharacteristic
370C2#1686420811yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
FittingC8×D4×C67C2
socleC22×C67C22×D4
hypercenterC8×D4D67
hypocenterC67#3296
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #16864086C2semidirect
3. #16863751C2semidirect
4. #16863727C2semidirect
5. #16863721C2semidirect
6. #16863697C2semidirect
7. #16863637C2semidirect
8. #16863625C2semidirect
9. #16863619C2semidirect
10. #16863560C2semidirect
11. #16863549C2semidirect
12. C4×#11761617C2semidirect
13. #16863393C2semidirect
14. C2×#14523965C2semidirect
15. C8×D4×C67C2semidirect
18. C2×Dic268C22semidirect
22. #14524035C22semidirect
29. #14524022C22semidirect
33. C4×Dic134C22semidirect
35. #14524002C22semidirect
39. #14523977C22semidirect
40. #14523975C22semidirect
41. #14523975C22semidirect
42. #14523968C22semidirect
43. #14523968C22semidirect
44. #14523968C22semidirect
45. #14523968C22semidirect
53. C22×C8×C67C22semidirect
55. C4⋊C8×C67C22semidirect
57. C22⋊C8×C67C22semidirect
58. C4×C8×C67C22semidirect
186. C67#3296semidirect
199. C8×D4D67semidirect
208. C22×C8C2×D67semidirect
231. C4⋊C8C2×D67semidirect
243. C22⋊C8C2×D67semidirect
244. C4×C8C2×D67semidirect