Finite Group Databasebeta

← #22569255

Subgroups — #22569255

Order 8576.

412 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#16863908C211yesmaximal, characteristic
2#16863475C211yesmaximal, characteristic, complemented
3#16863473C211yesmaximal, characteristic, complemented
4#16863469C211yesmaximal, characteristic, complemented
5#16863469C211yesmaximal, characteristic, complemented
6#16863466C211yesmaximal, characteristic, complemented
7#16863466C211yesmaximal, characteristic, complemented
8C8×D268C211yesmaximal, characteristic, complemented
9#16863461C211yesmaximal, characteristic, complemented
10#16863457C211yesmaximal, characteristic, complemented
11#16863361C211yesmaximal, characteristic, complemented
12#16863361C211yesmaximal, characteristic, complemented
13#16863358C211yesmaximal, characteristic, complemented
14C22⋊C8×D67C211yesmaximal, characteristic, complemented
15#433×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×#11761611C2211yescharacteristic
20C2×#11761611C2211yescharacteristic, complemented
21C2×#11761611C2211yescharacteristic, complemented
22C2×C8×D67C2211yescharacteristic
23C2×C8×D67C2211yescharacteristic
24C2×C8×D67C2211yescharacteristic, complemented
25C2×C8×D67C2211yescharacteristic, complemented
26#14524027C411yescharacteristic
27#14524027C411yescharacteristic
28#14524023C411yescharacteristic
29#14524023C411yescharacteristic
30#14524018C411yescharacteristic
31#14524018C411yescharacteristic
32#14524013C411yescharacteristic
33#14524013C411yescharacteristic
34C4×D268C2211yescharacteristic
35C4×D268C2211yescharacteristic
36C4×D268C2211yescharacteristic
37C4×D268C2211yescharacteristic
38C4⋊(C67⋊C8)C2211yescharacteristic, complemented
39C4⋊(C67⋊C8)C2211yescharacteristic, complemented
40C22⋊(C67⋊C8)C2211yescharacteristic
41C22⋊(C67⋊C8)C2211yescharacteristic
42(C2×D67)⋊C8C2211yescharacteristic
43(C2×D67)⋊C8C2211yescharacteristic
44(C2×D67)⋊C8C2211yescharacteristic
45(C2×D67)⋊C8C2211yescharacteristic
46(C2×D67)⋊C8C2211yescharacteristic, complemented
47(C2×D67)⋊C8C2211yescharacteristic, complemented
48(C2×D67)⋊C8C2211yescharacteristic, complemented
49(C2×D67)⋊C8C2211yescharacteristic, complemented
50#14523954C2211yescharacteristic, complemented
51C4×C67⋊C8C2211yescharacteristic, complemented
52#117×C67C2211yescharacteristic
53C4⋊C8×C67C2211yescharacteristic, complemented
54C4⋊C8×C67C2211yescharacteristic, complemented
55C22⋊C8×C67C2211yescharacteristic
56C22⋊C8×C67C2211yescharacteristic
57#97×C67C2211yescharacteristic, complemented
58C4×C8×C67C2211yescharacteristic, complemented
59C23×D67C2×C411yescharacteristic
60#11761628—41no—
61#11761628—41no—
62C2×Dic134C2×C411yescharacteristic
63C2×D268C2×C411yescharacteristic
64C2×D268C2×C411yescharacteristic
65C2×D268C2×C411yescharacteristic
66C2×D268C2×C411yescharacteristic
67C2×D268C2×C411yescharacteristic
68C2×C22⋊D67C2×C411yescharacteristic
69C2×C22⋊D67C2×C411yescharacteristic
70C22×Dic67C2×C411yescharacteristic
71C2×C4×D67—41no—
72C2×C4×D67—41no—
73C2×C4×D67C2×C411yescharacteristic
74C2×C4×D67C2×C411yescharacteristic
75C2×C4×D67C2311yescharacteristic
76C2×C4×D67C2311yescharacteristic
77C2×C4×D67C2311yescharacteristic
78C2×C4×D67C2311yescharacteristic
79#11761611—41no—
80#11761611—41no—
81#11761611—41no—
82#11761611—41no—
83C2×C67⋊C8C2311yescharacteristic
84C2×C67⋊C8C2311yescharacteristic
85C2×C67⋊C8C2311yescharacteristic
86C2×C67⋊C8C2311yescharacteristic
87C8×D67—41no—
88C8×D67—41no—
89C8×D67—41no—
90C8×D67—41no—
91C4⋊Dic67C2×C411yescharacteristic
92C4⋊Dic67C2×C411yescharacteristic
93C4⋊Dic67C2×C411yescharacteristic
94C4⋊Dic67C2×C411yescharacteristic
95(C2×D67)⋊C4C2×C411yescharacteristic
96(C2×D67)⋊C4C2×C411yescharacteristic
97(C2×D67)⋊C4C2×C411yescharacteristic
98(C2×D67)⋊C4C2×C411yescharacteristic
99(C2×D67)⋊C4C2×C411yescharacteristic
100(C2×D67)⋊C4C2×C411yescharacteristic
101(C2×D67)⋊C4C2×C411yescharacteristic
102(C2×D67)⋊C4C2×C411yescharacteristic
103C22×C4×C67C2311yescharacteristic
104C2×C8×C67C2311yescharacteristic
105C2×C8×C67C2311yescharacteristic
106C2×C8×C67C2311yescharacteristic
107C2×C8×C67C2311yescharacteristic
108C4⋊C4×C67C2×C411yescharacteristic
109C4⋊C4×C67C2×C411yescharacteristic
110C22⋊C4×C67C2×C411yescharacteristic
111C22⋊C4×C67C2×C411yescharacteristic
112C42×C67C2311yescharacteristic
113C42×C67C2311yescharacteristic
114C23×C67C22×C411yescharacteristic
115C22×D67—22no—
116C22×D67—41no—
117C22×D67—41no—
118C22×D67C22×C411yescharacteristic
119C22×D67C22×C411yescharacteristic
120C22×D67C22×C411yescharacteristic
121C22×D67C22×C411yescharacteristic
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67C22×C411yescharacteristic
125C2×C4×C67C22×C411yescharacteristic
126C2×C4×C67C22×C411yescharacteristic
127C2×C4×C67C22×C411yescharacteristic
128C2×C4×C67C22×C411yescharacteristic
129C2×C4×C67C2411yescharacteristic
130C22⋊D67—41no—
131C22⋊D67—41no—
132C2×Dic67—41no—
133C2×Dic67C22×C411yescharacteristic
134C2×Dic67C22×C411yescharacteristic
135C2×Dic67C22×C411yescharacteristic
136C2×Dic67C22×C411yescharacteristic
137D268—22no—
138D268—22no—
139D268—41no—
140D268—41no—
141D268—41no—
142C4×D67—21no—
143C4×D67—21no—
144C4×D67—21no—
145C4×D67—21no—
146C4×D67—21no—
147C4×D67—21no—
148C4×D67—21no—
149C4×D67—21no—
150C4×D67—41no—
151C4×D67—41no—
152Dic134—41no—
153C8×C67—21no—
154C8×C67—21no—
155C8×C67—21no—
156C8×C67—21no—
157C67⋊C8—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C67⋊C8—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67C23×C411yescharacteristic
165C2×D67—21no—
166C2×D67—21no—
167C2×D67—21no—
168C2×D67—21no—
169C2×D67—21no—
170C2×D67—21no—
171C2×D67—21no—
172C2×D67—21no—
173C2×D67—21no—
174C2×D67—21no—
175C2×D67—21no—
176C2×D67—44no—
177C2×D67#13112yes—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67—21no—
181C4×C67—21no—
182C4×C67—21no—
183C4×C67H(5,𝔽2)11yescharacteristic
184C4×C67H(5,𝔽2)11yescharacteristic
185Dic67—21no—
186Dic67—21no—
187Dic67—21no—
188Dic67—21no—
189C2×C67—41no—
190C2×C67#51811yescharacteristic
191C2×C67C2×#13111yescharacteristic
192C2×C67#56811yescharacteristic
193D67—22no—
194D67—41no—
195D67—41no—
196D67—41no—
197#2976—671nomaximal, Sylow
198C67#297611yescharacteristic, complemented, Sylow
199#525—671no—
200#436—671no—
201#436—671no—
202#435—671no—
203#435—671no—
204#435—671no—
205#435—671no—
206C8×D4—671no—
207C8×D4—671no—
208#433—671no—
209#433D6711yescharacteristic, complemented
210#408—671no—
211#408—671no—
212#407—671no—
213C2×C22⋊C8—671no—
214C2×#41—671no—
215C23×C4—671no—
216C2×M16—671no—
217C2×M16—671no—
218C2×M16—671no—
219C2×M16—671no—
220C22×C8—671no—
221C22×C8—671no—
222C22×C8—671no—
223C22×C8—671no—
224#123—671no—
225#123—671no—
226C22⋊Q8—671no—
227C22⋊Q8—671no—
228#121—671no—
229#121—671no—
230#120—671no—
231#120—671no—
232C4×D4—671no—
233C4×D4—671no—
234C4×D4—671no—
235C4×D4—671no—
236#117C2×D6711yescharacteristic
237C4⋊C8—671no—
238C4⋊C8—671no—
239C4⋊C8C2×D6711yescharacteristic, complemented
240C4⋊C8C2×D6711yescharacteristic, complemented
241C22⋊C8—671no—
242C22⋊C8—671no—
243C22⋊C8—671no—
244C22⋊C8—671no—
245C22⋊C8—671no—
246C22⋊C8—671no—
247C22⋊C8—671no—
248C22⋊C8—671no—
249C22⋊C8—671no—
250C22⋊C8—671no—
251C22⋊C8C2×D6711yescharacteristic
252C22⋊C8C2×D6711yescharacteristic
253#97—671no—
254#97C2×D6711yescharacteristic, complemented
255C4×C8—671no—
256C4×C8C2×D6711yescharacteristic, complemented
257C24—671no—
258#41—2681no—
259#41—2681no—
260C2×Q8—671no—
261C2×D4—671no—
262C2×D4—671no—
263C2×D4—671no—
264C2×D4—671no—
265C2×D4—671no—
266C2×D4—671no—
267C2×D4—671no—
268C22×C4—2681no—
269C22×C4—2681no—
270C22×C4—671no—
271C22×C4—671no—
272C22×C4—671no—
273C22×C4—671no—
274C22×C4—671no—
275C22×C4—671no—
276C22×C4—671no—
277C22×C4C22×D6711yescharacteristic
278M16—2681no—
279M16—2681no—
280M16—2681no—
281M16—2681no—
282C2×C8—2681no—
283C2×C8—2681no—
284C2×C8—2681no—
285C2×C8—2681no—
286C2×C8—671no—
287C2×C8—671no—
288C2×C8—671no—
289C2×C8—671no—
290C2×C8C22×D6711yescharacteristic
291C2×C8C22×D6711yescharacteristic
292C2×C8C22×D6711yescharacteristic
293C2×C8C22×D6711yescharacteristic
294C4⋊C4—671no—
295C4⋊C4—671no—
296C4⋊C4—671no—
297C4⋊C4—671no—
298C4⋊C4C4×D6711yescharacteristic
299C4⋊C4C4×D6711yescharacteristic
300C22⋊C4—671no—
301C22⋊C4—671no—
302C22⋊C4—671no—
303C22⋊C4—671no—
304C22⋊C4—671no—
305C22⋊C4—671no—
306C22⋊C4—671no—
307C22⋊C4—671no—
308C22⋊C4C4×D6711yescharacteristic
309C22⋊C4C4×D6711yescharacteristic
310C42C22×D6711yescharacteristic
311C42C22×D6711yescharacteristic
312C23—2681no—
313C23—2681no—
314C23—671no—
315C23—671no—
316C23—671no—
317C23—671no—
318C23—1342no—
319C23C2×C4×D6711yescharacteristic
320Q8—2681no—
321D4—2681no—
322D4—2681no—
323D4—2681no—
324D4—2681no—
325D4—2681no—
326D4—1342no—
327D4—1342no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—2681no—
331C2×C4—2681no—
332C2×C4—2681no—
333C2×C4—671no—
334C2×C4—671no—
335C2×C4—671no—
336C2×C4—671no—
337C2×C4—1341no—
338C2×C4—1341no—
339C2×C4—1341no—
340C2×C4—1341no—
341C2×C4—1341no—
342C2×C4—1341no—
343C2×C4—1341no—
344C2×C4—1341no—
345C2×C4C2×C4×D6711yescharacteristic
346C2×C4C2×C4×D6711yescharacteristic
347C2×C4C2×C4×D6711yescharacteristic
348C2×C4C2×C4×D6711yescharacteristic
349C2×C4C2×C4×D6711yescharacteristic
350C2×C4C23×D6711yescharacteristic
351C8—21no—
352C8—21no—
353C8—21no—
354C8—21no—
355C8—1341no—
356C8—1341no—
357C8—1341no—
358C8—1341no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—2684no—
363C22—672no—
364C22—1341no—
365C22—1341no—
366C22—1341no—
367C22—1341no—
368C22—1341no—
369C22—1341no—
370C22—1341no—
371C22—1341no—
372C22—1341no—
373C22—1341no—
374C22—1341no—
375C22C22×C4×D6711yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—1341no—
382C4—1341no—
383C4—1341no—
384C4—1341no—
385C4#1452408911yescharacteristic
386C4#1452408911yescharacteristic
387C2—41no—
388C2—2681no—
389C2—2681no—
390C2—2681no—
391C2—1342no—
392C2#1686387011yescharacteristic
393C2#131×D6711yescharacteristic
394C2#1686416111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D67
commutator, socleC22×C67C23×C4
Fitting#433×C67C2
hypercenter#433D67
hypocenterC67#2976
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #16863475C2semidirect
3. #16863473C2semidirect
4. #16863469C2semidirect
5. #16863469C2semidirect
6. #16863466C2semidirect
7. #16863466C2semidirect
8. C8×D268C2semidirect
9. #16863461C2semidirect
10. #16863457C2semidirect
11. #16863361C2semidirect
12. #16863361C2semidirect
13. #16863358C2semidirect
14. C22⋊C8×D67C2semidirect
15. #433×C67C2semidirect
20. C2×#11761611C22semidirect
21. C2×#11761611C22semidirect
24. C2×C8×D67C22semidirect
25. C2×C8×D67C22semidirect
38. C4⋊(C67⋊C8)C22semidirect
39. C4⋊(C67⋊C8)C22semidirect
46. (C2×D67)⋊C8C22semidirect
47. (C2×D67)⋊C8C22semidirect
48. (C2×D67)⋊C8C22semidirect
49. (C2×D67)⋊C8C22semidirect
50. #14523954C22semidirect
51. C4×C67⋊C8C22semidirect
53. C4⋊C8×C67C22semidirect
54. C4⋊C8×C67C22semidirect
57. #97×C67C22semidirect
58. C4×C8×C67C22semidirect
198. C67#2976semidirect
209. #433D67semidirect
239. C4⋊C8C2×D67semidirect
240. C4⋊C8C2×D67semidirect
254. #97C2×D67semidirect
256. C4×C8C2×D67semidirect