Finite Group Databasebeta

← #22572038

Subgroups — #22572038

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#16864042C211yesmaximal, characteristic
2#16863753C211yesmaximal, characteristic, complemented
3#16863726C211yesmaximal, characteristic, complemented
4#16863720C211yesmaximal, characteristic, complemented
5#16863693C211yesmaximal, characteristic, complemented
6#16863635C211yesmaximal, characteristic, complemented
7#16863627C211yesmaximal, characteristic, complemented
8#16863618C211yesmaximal, characteristic, complemented
9#16863556C211yesmaximal, characteristic, complemented
10#16863546C211yesmaximal, characteristic, complemented
11C4×#11761615C211yesmaximal, characteristic, complemented
12D4×C67⋊C8C211yesmaximal, characteristic, complemented
13#16863390C211yesmaximal, characteristic, complemented
14C2×#14523963C211yesmaximal, characteristic, complemented
15#552×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×D268C2211yescharacteristic
18C2×#11761618C2211yescharacteristic, complemented
19C2×#11761615C2211yescharacteristic
20C2×#11761615C2211yescharacteristic
21C22×C67⋊C8C2211yescharacteristic
22C22×C67⋊C8C2211yescharacteristic, complemented
23#14524031C2211yescharacteristic
24#14524028—21no—
25#14524022C2211yescharacteristic
26#14524017C2211yescharacteristic
27#14524016—21no—
28#14524011—21no—
29C4×D268C2211yescharacteristic
30C22⋊C4×D67—21no—
31#14523976C2211yescharacteristic, complemented
32#14523974C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35#14523967C2211yescharacteristic, complemented
36#14523966C2211yescharacteristic, complemented
37#14523966C2211yescharacteristic, complemented
38#14523966C2211yescharacteristic, complemented
39#14523963C2211yescharacteristic
40#14523963C2211yescharacteristic
41#14523963C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44#14523963C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic, complemented
47C4×C67⋊C8C2211yescharacteristic, complemented
48#126×C67—21no—
49#125×C67—21no—
50#125×C67C2211yescharacteristic, complemented
51#123×C67—21no—
52C22⋊Q8×C67C2211yescharacteristic
53C22⋊Q8×C67C2211yescharacteristic, complemented
54C4×Q8×C67C2211yescharacteristic, complemented
55C4×D4×C67C2211yescharacteristic
56#117×C67—21no—
57#117×C67C2211yescharacteristic, complemented
58C2×C4⋊C4×C67C2211yescharacteristic
59C23×D67—21no—
60#11761628—22no—
61#11761628—41no—
62C2×Dic134C2311yescharacteristic
63C2×D268—22no—
64C2×D268—21no—
65C2×D268—41no—
66C2×D268C2311yescharacteristic
67C2×D268C2311yescharacteristic
68C2×C22⋊D67—21no—
69C2×C22⋊D67—21no—
70C2×C4×D67—21no—
71C2×C4×D67—21no—
72#11761618—41no—
73#11761615—22no—
74#11761615—41no—
75C2×C67⋊C8—41no—
76C2×C67⋊C8—41no—
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C2×C67⋊C8C2311yescharacteristic
80C2×C67⋊C8C2311yescharacteristic
81C4⋊Dic67C2311yescharacteristic
82Dic67⋊C4—21no—
83C22⋊Dic67—21no—
84(C2×D67)⋊C4—21no—
85(C2×D67)⋊C4—21no—
86(C2×D67)⋊C4—21no—
87(C2×D67)⋊C4—21no—
88C2×Q8×C67C2311yescharacteristic
89C2×D4×C67D411yescharacteristic
90C22×C4×C67—21no—
91C22×C4×C67C2311yescharacteristic
92C22×C4×C67C2311yescharacteristic
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67—21no—
97C4⋊C4×C67—21no—
98C4⋊C4×C67—21no—
99C4⋊C4×C67D411yescharacteristic
100C4⋊C4×C67C2311yescharacteristic
101C4⋊C4×C67C2311yescharacteristic
102C4⋊C4×C67C2311yescharacteristic
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67D411yescharacteristic
107C22⋊C4×C67D411yescharacteristic
108C42×C67—21no—
109C42×C67—21no—
110C42×C67C2311yescharacteristic
111C23×C67C2×D411yescharacteristic
112C23×C67C2×D411yescharacteristic
113C22×D67—21no—
114C22×D67—21no—
115C22×D67—42no—
116C22×D67—42no—
117C22×D67—81no—
118Q8×C67—21no—
119Q8×C67—21no—
120D4×C67—22no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—42no—
130C2×C4×C67C2×D411yescharacteristic
131C2×C4×C67C2×D411yescharacteristic
132C2×C4×C67C2×D411yescharacteristic
133C2×C4×C67C2×D411yescharacteristic
134C2×C4×C67C2411yescharacteristic
135C22⋊D67—42no—
136C22⋊D67—81no—
137C2×Dic67—21no—
138C2×Dic67—21no—
139D268—21no—
140D268—21no—
141D268—21no—
142D268—42no—
143D268#4112yes—
144C4×D67—42no—
145C4×D67—81no—
146Dic134—21no—
147Dic134—21no—
148C67⋊C8—21no—
149C67⋊C8—21no—
150C67⋊C8—21no—
151C67⋊C8—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×C67—41no—
171C4×C67—41no—
172C4×C67—41no—
173C4×C67—41no—
174C4×C67C2×#4111yescharacteristic
175C4×C67H(5,𝔽2)11yescharacteristic
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#552D6711yescharacteristic, complemented
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×D4—671no—
200#416—671no—
201C2×#102—671no—
202C2×#41—671no—
203C22×D4—671no—
204C2×Q16—671no—
205C2×SD16—671no—
206C2×SD16—671no—
207C22×C8—671no—
208C22×C8—671no—
209#126—21no—
210#125—21no—
211#125C2×D6711yescharacteristic, complemented
212#124—671no—
213#123—21no—
214#123—1341no—
215C22⋊Q8—671no—
216C22⋊Q8C2×D6711yescharacteristic
217C22⋊Q8C2×D6711yescharacteristic, complemented
218#121—671no—
219#121—1341no—
220#120—1341no—
221C4×Q8C2×D6711yescharacteristic, complemented
222C4×D4—671no—
223C4×D4C2×D6711yescharacteristic
224#117—21no—
225#117C2×D6711yescharacteristic, complemented
226C2×C4⋊C4C2×D6711yescharacteristic
227C2×C22⋊C4—1341no—
228#107—671no—
229#106—671no—
230#106—671no—
231C4⋊C8—671no—
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⋊C8—671no—
243C22⋊C8—671no—
244C4×C8—671no—
245C24—1341no—
246#41—2681no—
247#41—1342no—
248C2×Q8—671no—
249C2×Q8C22×D6711yescharacteristic
250C2×D4—2681no—
251C2×D4—671no—
252C2×D4—671no—
253C2×D4—1342no—
254C2×D4—1341no—
255C2×D4—1341no—
256C2×D4—1341no—
257C2×D4C22⋊D6711yescharacteristic
258C22×C4—21no—
259C22×C4—1341no—
260C22×C4—1341no—
261C22×C4C22×D6711yescharacteristic
262C22×C4C22×D6711yescharacteristic
263Q16—2681no—
264SD16—2681no—
265SD16—1342no—
266C2×C8—2681no—
267C2×C8—2681no—
268C2×C8—671no—
269C2×C8—671no—
270C2×C8—671no—
271C2×C8—671no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—671no—
279C4⋊C4—1341no—
280C4⋊C4C22⋊D6711yescharacteristic
281C4⋊C4C22×D6711yescharacteristic
282C4⋊C4C22×D6711yescharacteristic
283C4⋊C4C22×D6711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1341no—
288C22⋊C4—1341no—
289C22⋊C4—1341no—
290C22⋊C4—1341no—
291C22⋊C4—1341no—
292C22⋊C4C22⋊D6711yescharacteristic
293C22⋊C4C22⋊D6711yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D6711yescharacteristic
297C23—2682no—
298C23—2682no—
299C23—5361no—
300C23—1341no—
301C23—1341no—
302C23C2×C22⋊D6711yescharacteristic
303C23C2×C22⋊D6711yescharacteristic
304Q8—21no—
305Q8—21no—
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—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—2682no—
326C2×C4—5361no—
327C2×C4—1341no—
328C2×C4—1341no—
329C2×C4C2×C22⋊D6711yescharacteristic
330C2×C4C2×C22⋊D6711yescharacteristic
331C2×C4C2×C22⋊D6711yescharacteristic
332C2×C4C2×C22⋊D6711yescharacteristic
333C2×C4C23×D6711yescharacteristic
334C8—1341no—
335C8—1341no—
336C8—1341no—
337C8—1341no—
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#1452406312yes—
351C22C22×C22⋊D6711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2681no—
361C4—2681no—
362C4#41×D6711yescharacteristic
363C4#1452408911yescharacteristic
364C2—22no—
365C2—41no—
366C2—2682no—
367C2—5361no—
368C2#1686403411yescharacteristic
369C2C2×#1452406311yescharacteristic
370C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#552×C67C2
socleC22×C67C22×D4
hypercenter#552D67
hypocenterC67#3296
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #16863753C2semidirect
3. #16863726C2semidirect
4. #16863720C2semidirect
5. #16863693C2semidirect
6. #16863635C2semidirect
7. #16863627C2semidirect
8. #16863618C2semidirect
9. #16863556C2semidirect
10. #16863546C2semidirect
11. C4×#11761615C2semidirect
12. D4×C67⋊C8C2semidirect
13. #16863390C2semidirect
14. C2×#14523963C2semidirect
15. #552×C67C2semidirect
18. C2×#11761618C22semidirect
22. C22×C67⋊C8C22semidirect
31. #14523976C22semidirect
32. #14523974C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. #14523967C22semidirect
36. #14523966C22semidirect
37. #14523966C22semidirect
38. #14523966C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. C4×C67⋊C8C22semidirect
50. #125×C67C22semidirect
53. C22⋊Q8×C67C22semidirect
54. C4×Q8×C67C22semidirect
57. #117×C67C22semidirect
186. C67#3296semidirect
187. #552D67semidirect
211. #125C2×D67semidirect
217. C22⋊Q8C2×D67semidirect
221. C4×Q8C2×D67semidirect
225. #117C2×D67semidirect