Finite Group Databasebeta

← #22570457

Subgroups — #22570457

Order 8576.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864006C211yesmaximal, characteristic, complemented
2#16863870C211yesmaximal, characteristic
3#16863737C211yesmaximal, characteristic, complemented
4#16863699C211yesmaximal, characteristic, complemented
5#16863695C211yesmaximal, characteristic, complemented
6#16863600C211yesmaximal, characteristic, complemented
7#16863600C211yesmaximal, characteristic, complemented
8#16863549C211yesmaximal, characteristic
9#16863546C211yesmaximal, characteristic
10#16863499C211yesmaximal, characteristic, complemented
11#16863495C211yesmaximal, characteristic, complemented
12#16863483C211yesmaximal, characteristic, complemented
13#16863481C211yesmaximal, characteristic, complemented
14#16863454C211yesmaximal, characteristic, complemented
15#485×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C2×#11761617C2211yescharacteristic
18C2×#11761617C2211yescharacteristic
19C2×#11761615C2211yescharacteristic
20C2×#11761615C2211yescharacteristic
21#14524041C2211yescharacteristic, complemented
22#14524036—21no—
23#14524036—21no—
24#14524034—21no—
25#14524034C2211yescharacteristic, complemented
26#14524033—21no—
27#14524022C2211yescharacteristic
28Q8×Dic67C2211yescharacteristic, complemented
29#14524008C2211yescharacteristic, complemented
30C4×D268—21no—
31C4×D268—21no—
32#14524005C2211yescharacteristic
33#14524005C2211yescharacteristic
34#14524005C2211yescharacteristic
35#14524005C2211yescharacteristic
36#14524002C2211yescharacteristic
37C22⋊C4×D67—21no—
38C22⋊C4×D67—21no—
39#14523976C2211yescharacteristic, complemented
40#14523974C2211yescharacteristic, complemented
41Dic67⋊C8C2211yescharacteristic, complemented
42Dic67⋊C8C2211yescharacteristic, complemented
43#14523966C2211yescharacteristic, complemented
44#14523966C2211yescharacteristic, complemented
45#14523965C2211yescharacteristic
46#14523965C2211yescharacteristic
47#14523963C2211yescharacteristic
48#14523963C2211yescharacteristic
49C22⋊(C67⋊C8)C2211yescharacteristic
50#14523953C2211yescharacteristic, complemented
51C8×Dic67C2211yescharacteristic, complemented
52C22⋊Q8×C67C2211yescharacteristic
53#117×C67C2211yescharacteristic
54#107×C67C2211yescharacteristic, complemented
55#106×C67C2211yescharacteristic, complemented
56#103×C67C2211yescharacteristic, complemented
57#103×C67C2211yescharacteristic, complemented
58C22⋊C8×C67C2211yescharacteristic
59C23×D67—21no—
60C2×Dic134C2311yescharacteristic
61C2×D268—21no—
62C2×D268—22no—
63C2×D268—22no—
64C2×D268C2311yescharacteristic
65C2×D268C2311yescharacteristic
66C2×C4×D67—21no—
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70#11761617—22no—
71#11761617—22no—
72#11761615—22no—
73#11761615—22no—
74C2×C67⋊C8C2311yescharacteristic
75C2×C67⋊C8C2311yescharacteristic
76C4⋊Dic67—21no—
77C4⋊Dic67—21no—
78C4⋊Dic67D411yescharacteristic
79C4⋊Dic67D411yescharacteristic
80C4⋊Dic67C2311yescharacteristic
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86C22⋊Dic67—21no—
87C22⋊Dic67D411yescharacteristic
88C22⋊Dic67D411yescharacteristic
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93C4×Dic67—21no—
94C4×Dic67—21no—
95C4×Dic67C2311yescharacteristic
96C4×Dic67C2311yescharacteristic
97C2×Q8×C67C2311yescharacteristic
98C22×C4×C67C2311yescharacteristic
99C2×C8×C67C2311yescharacteristic
100C2×C8×C67C2311yescharacteristic
101C4⋊C4×C67—21no—
102C4⋊C4×C67C2311yescharacteristic
103C4⋊C4×C67C2311yescharacteristic
104C4⋊C4×C67C2311yescharacteristic
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C42×C67—21no—
108C23×C67C2×D411yescharacteristic
109C22×D67—21no—
110C22×D67—21no—
111C22×D67—24no—
112C22×D67—81no—
113C22×D67—81no—
114Q8×C67—21no—
115Q8×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67C2×D411yescharacteristic
123C2×C4×C67C2411yescharacteristic
124C2×Dic67—21no—
125C2×Dic67—21no—
126C2×Dic67C2×D411yescharacteristic
127C2×Dic67C2×D411yescharacteristic
128C2×Dic67C2×D411yescharacteristic
129C2×Dic67C2×D411yescharacteristic
130D268—24no—
131D268—21no—
132D268—21no—
133D268#4112yes—
134D268#4112yes—
135C4×D67—81no—
136C4×D67—81no—
137C4×D67—81no—
138C4×D67—81no—
139Dic134—21no—
140Dic134—21no—
141C8×C67—21no—
142C8×C67—21no—
143C67⋊C8—21no—
144C67⋊C8—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—21no—
148C22×C67C22×D411yescharacteristic
149C2×D67—22no—
150C2×D67—22no—
151C2×D67—41no—
152C2×D67—41no—
153C2×D67—41no—
154C2×D67—41no—
155C2×D67—84no—
156C4×C67—21no—
157C4×C67—41no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67C2×#4111yescharacteristic
162C4×C67C2×#4111yescharacteristic
163Dic67—21no—
164Dic67—21no—
165Dic67—21no—
166Dic67—21no—
167Dic67—41no—
168Dic67—41no—
169C2×C67—41no—
170C2×C67#52511yescharacteristic
171C2×C67#57711yescharacteristic
172C2×C67#57711yescharacteristic
173D67—42no—
174D67—42no—
175#3123—671nomaximal, Sylow
176C67#312311yescharacteristic, complemented, Sylow
177#541—671no—
178#518—671no—
179#485—671no—
180#485D6711yescharacteristic, complemented
181#478—671no—
182#478—671no—
183#461—671no—
184#461—671no—
185#450—671no—
186#450—671no—
187#440—671no—
188#440—671no—
189C4×SD16—671no—
190C4×SD16—671no—
191#433—671no—
192C22×D4—671no—
193C2×SD16—671no—
194C2×SD16—671no—
195C2×SD16—671no—
196C2×SD16—671no—
197C4⋊Q8—671no—
198#126—1341no—
199#126—1341no—
200#125—671no—
201#125—1341no—
202#125—1341no—
203C22⋊Q8—671no—
204C22⋊Q8C2×D6711yescharacteristic
205C4×Q8—671no—
206C4×Q8—671no—
207C4×D4—671no—
208C4×D4—671no—
209C4×D4—671no—
210C4×D4—671no—
211C4×D4—1341no—
212C4×D4—1341no—
213#117—671no—
214#117C2×D6711yescharacteristic
215C2×C22⋊C4—1341no—
216C2×C22⋊C4—1341no—
217#107—671no—
218#107C2×D6711yescharacteristic, complemented
219#106—671no—
220#106C2×D6711yescharacteristic, complemented
221C4⋊C8—671no—
222C4⋊C8—671no—
223#103—671no—
224#103—671no—
225#103C2×D6711yescharacteristic, complemented
226#103C2×D6711yescharacteristic, complemented
227#102—671no—
228#102—671no—
229#102—671no—
230#102—671no—
231C22⋊C8—671no—
232C22⋊C8C2×D6711yescharacteristic
233#97—671no—
234C4×C8—671no—
235C24—1341no—
236C2×Q8—671no—
237C2×Q8C22×D6711yescharacteristic
238C2×D4—671no—
239C2×D4—671no—
240C2×D4—1342no—
241C2×D4—1342no—
242C2×D4—1341no—
243C22×C4—1341no—
244C22×C4—1341no—
245C22×C4—1341no—
246C22×C4—1341no—
247C22×C4C22×D6711yescharacteristic
248SD16—1342no—
249SD16—1342no—
250SD16—1342no—
251SD16—1342no—
252C2×C8—671no—
253C2×C8—671no—
254C2×C8C22×D6711yescharacteristic
255C2×C8C22×D6711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—671no—
258C4⋊C4—671no—
259C4⋊C4—671no—
260C4⋊C4—1341no—
261C4⋊C4—1341no—
262C4⋊C4—1341no—
263C4⋊C4—1341no—
264C4⋊C4—1341no—
265C4⋊C4—1341no—
266C4⋊C4—1341no—
267C4⋊C4C22×D6711yescharacteristic
268C4⋊C4C22×D6711yescharacteristic
269C4⋊C4C22×D6711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—671no—
273C22⋊C4—671no—
274C22⋊C4—1341no—
275C22⋊C4—1341no—
276C22⋊C4—1341no—
277C22⋊C4—1341no—
278C22⋊C4—1341no—
279C42—21no—
280C42—671no—
281C42—671no—
282C42—1341no—
283C42—1341no—
284C23—5361no—
285C23—5361no—
286C23—1344no—
287C23—1341no—
288C23—1341no—
289C23D4×D6711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—1341no—
293Q8—1341no—
294D4—672no—
295D4—672no—
296D4—1344no—
297D4—1341no—
298D4—1341no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—5361no—
306C2×C4—5361no—
307C2×C4—5361no—
308C2×C4—5361no—
309C2×C4—671no—
310C2×C4—671no—
311C2×C4—671no—
312C2×C4—671no—
313C2×C4—1341no—
314C2×C4—1341no—
315C2×C4D4×D6711yescharacteristic
316C2×C4C23×D6711yescharacteristic
317C8—21no—
318C8—21no—
319C8—1341no—
320C8—1341no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—2681no—
325C22—2681no—
326C22—2681no—
327C22—2681no—
328C22—5364no—
329C22—1342no—
330C22—1342no—
331C22C2×D4×D6711yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2681no—
338C4—2681no—
339C4—1341no—
340C4—1341no—
341C4—1341no—
342C4—1341no—
343C4#41×D6711yescharacteristic
344C4#41×D6711yescharacteristic
345C2—41no—
346C2—2682no—
347C2—2682no—
348C2#1686390411yescharacteristic
349C2#1686420311yescharacteristic
350C2#1686420511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#485×C67C2
socleC22×C67C22×D4
hypercenter#485D67
hypocenterC67#3123
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #16864006C2semidirect
3. #16863737C2semidirect
4. #16863699C2semidirect
5. #16863695C2semidirect
6. #16863600C2semidirect
7. #16863600C2semidirect
10. #16863499C2semidirect
11. #16863495C2semidirect
12. #16863483C2semidirect
13. #16863481C2semidirect
14. #16863454C2semidirect
15. #485×C67C2semidirect
21. #14524041C22semidirect
25. #14524034C22semidirect
28. Q8×Dic67C22semidirect
29. #14524008C22semidirect
39. #14523976C22semidirect
40. #14523974C22semidirect
41. Dic67⋊C8C22semidirect
42. Dic67⋊C8C22semidirect
43. #14523966C22semidirect
44. #14523966C22semidirect
50. #14523953C22semidirect
51. C8×Dic67C22semidirect
54. #107×C67C22semidirect
55. #106×C67C22semidirect
56. #103×C67C22semidirect
57. #103×C67C22semidirect
176. C67#3123semidirect
180. #485D67semidirect
218. #107C2×D67semidirect
220. #106C2×D67semidirect
225. #103C2×D67semidirect
226. #103C2×D67semidirect