Finite Group Databasebeta

← #22572053

Subgroups — #22572053

Order 8576.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864042C211yesmaximal, characteristic, complemented
2#16863753C211yesmaximal, characteristic, complemented
3#16863731C211yesmaximal, characteristic, complemented
4#16863720C211yesmaximal, characteristic, complemented
5#16863693C211yesmaximal, characteristic, complemented
6#16863630C211yesmaximal, characteristic, complemented
7#16863623C211yesmaximal, characteristic, complemented
8#16863611C211yesmaximal, characteristic, complemented
9#16863539C211yesmaximal, characteristic, complemented
10#16863529C211yesmaximal, characteristic, complemented
11C4×#11761613C211yesmaximal, characteristic, complemented
12D4×C67⋊C8C211yesmaximal, characteristic, complemented
13#16863390C211yesmaximal, characteristic, complemented
14C2×#14523963C211yesmaximal, characteristic, complemented
15#553×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C22×D268C2211yescharacteristic
18C2×#11761616C2211yescharacteristic, complemented
19C2×#11761613C2211yescharacteristic
20C2×#11761613C2211yescharacteristic
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#14523976C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35#14523966C2211yescharacteristic, complemented
36#14523966C2211yescharacteristic, complemented
37#14523966C2211yescharacteristic, complemented
38#14523964C2211yescharacteristic, 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×C67C2211yescharacteristic, complemented
50#123×C67—21no—
51#121×C67—21no—
52#121×C67C2211yescharacteristic
53#121×C67C2211yescharacteristic, complemented
54C4×D4×C67—21no—
55C4×D4×C67C2211yescharacteristic
56C4×D4×C67C2211yescharacteristic, complemented
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#11761616—41no—
73#11761613—22no—
74#11761613—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×D4×C67—21no—
89C2×D4×C67—21no—
90C2×D4×C67D411yescharacteristic
91C2×D4×C67C2311yescharacteristic
92C22×C4×C67—21no—
93C22×C4×C67—21no—
94C22×C4×C67C2311yescharacteristic
95C22×C4×C67C2311yescharacteristic
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×C67—21no—
107C22⋊C4×C67D411yescharacteristic
108C22⋊C4×C67D411yescharacteristic
109C42×C67—21no—
110C42×C67C2311yescharacteristic
111C23×C67—21no—
112C23×C67C2×D411yescharacteristic
113C23×C67C2×D411yescharacteristic
114C22×D67—21no—
115C22×D67—21no—
116C22×D67—42no—
117C22×D67—42no—
118C22×D67—81no—
119D4×C67—22no—
120D4×C67—21no—
121D4×C67—21no—
122D4×C67—42no—
123D4×C67—81no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—81no—
132C2×C4×C67—81no—
133C2×C4×C67C2×D411yescharacteristic
134C2×C4×C67C2×D411yescharacteristic
135C2×C4×C67C2×D411yescharacteristic
136C2×C4×C67C2×D411yescharacteristic
137C2×C4×C67C2411yescharacteristic
138C22⋊D67—42no—
139C22⋊D67—81no—
140C2×Dic67—21no—
141C2×Dic67—21no—
142D268—21no—
143D268—21no—
144D268—21no—
145D268—42no—
146D268#4112yes—
147C4×D67—42no—
148C4×D67—81no—
149Dic134—21no—
150Dic134—21no—
151C67⋊C8—21no—
152C67⋊C8—21no—
153C67⋊C8—21no—
154C67⋊C8—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67#13512yes—
164C22×C67C22×D411yescharacteristic
165C2×D67—22no—
166C2×D67—41no—
167C2×D67—41no—
168C2×D67—41no—
169C2×D67—41no—
170C2×D67—41no—
171C2×D67—84no—
172C4×C67—21no—
173C4×C67—21no—
174C4×C67—21no—
175C4×C67—21no—
176C4×C67—41no—
177C4×C67—41no—
178C4×C67—41no—
179C4×C67C2×#4111yescharacteristic
180C4×C67H(5,𝔽2)11yescharacteristic
181Dic67—41no—
182Dic67—41no—
183C2×C67—22no—
184C2×C67—41no—
185C2×C67—81no—
186C2×C67#54611yescharacteristic
187C2×C67C2×#13511yescharacteristic
188C2×C67#57611yescharacteristic
189D67—42no—
190D67—81no—
191#3297—671nomaximal, Sylow
192C67#329711yescharacteristic, complemented, Sylow
193#553D6711yescharacteristic, complemented
194#546—671no—
195#488—671no—
196#484—671no—
197#482—671no—
198#478—671no—
199#466—671no—
200#465—671no—
201#463—671no—
202#449—671no—
203#447—671no—
204C4×D8—671no—
205C8×D4—671no—
206#416—671no—
207C2×#102—671no—
208C2×#41—671no—
209C22×D4—671no—
210C2×SD16—671no—
211C2×D8—671no—
212C2×D8—671no—
213C22×C8—671no—
214C22×C8—671no—
215#126—21no—
216#125C2×D6711yescharacteristic, complemented
217#124—671no—
218#123—21no—
219#123—1341no—
220C22⋊Q8—671no—
221#121—21no—
222#121—671no—
223#121—1341no—
224#121C2×D6711yescharacteristic
225#121C2×D6711yescharacteristic, complemented
226#120—1341no—
227C4×D4—21no—
228C4×D4—671no—
229C4×D4C2×D6711yescharacteristic
230C4×D4C2×D6711yescharacteristic, complemented
231#117C2×D6711yescharacteristic, complemented
232C2×C4⋊C4C2×D6711yescharacteristic
233C2×C22⋊C4—1341no—
234#107—671no—
235#107—671no—
236#106—671no—
237C4⋊C8—671no—
238#103—671no—
239#103—671no—
240#103—671no—
241#102—671no—
242#102—671no—
243#102—671no—
244#102—671no—
245#102—671no—
246#102—671no—
247#102—671no—
248C22⋊C8—671no—
249C22⋊C8—671no—
250C4×C8—671no—
251C24—1341no—
252#41—2681no—
253#41—1342no—
254C2×Q8—671no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2681no—
258C2×D4—671no—
259C2×D4—671no—
260C2×D4—1342no—
261C2×D4—1341no—
262C2×D4—1341no—
263C2×D4—1341no—
264C2×D4C22⋊D6711yescharacteristic
265C2×D4C22×D6711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—1341no—
269C22×C4—1341no—
270C22×C4C22×D6711yescharacteristic
271C22×C4C22×D6711yescharacteristic
272SD16—2681no—
273D8—2681no—
274D8—1342no—
275C2×C8—2681no—
276C2×C8—2681no—
277C2×C8—671no—
278C2×C8—671no—
279C2×C8—671no—
280C2×C8—671no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—671no—
285C4⋊C4—1341no—
286C4⋊C4C22⋊D6711yescharacteristic
287C4⋊C4C22×D6711yescharacteristic
288C4⋊C4C22×D6711yescharacteristic
289C4⋊C4C22×D6711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—1341no—
295C22⋊C4—1341no—
296C22⋊C4—1341no—
297C22⋊C4—1341no—
298C22⋊C4—1341no—
299C22⋊C4C22⋊D6711yescharacteristic
300C22⋊C4C22⋊D6711yescharacteristic
301C42—21no—
302C42C22×D6711yescharacteristic
303C23—21no—
304C23—2682no—
305C23—2682no—
306C23—5361no—
307C23—1341no—
308C23—1341no—
309C23C2×C22⋊D6711yescharacteristic
310C23C2×C22⋊D6711yescharacteristic
311Q8—1341no—
312Q8—1341no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—2682no—
319D4—2682no—
320D4—5361no—
321D4—672no—
322D4—1341no—
323D4—1341no—
324D4—1341no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—2682no—
335C2×C4—5361no—
336C2×C4—1341no—
337C2×C4—1341no—
338C2×C4C2×C22⋊D6711yescharacteristic
339C2×C4C2×C22⋊D6711yescharacteristic
340C2×C4C2×C22⋊D6711yescharacteristic
341C2×C4C2×C22⋊D6711yescharacteristic
342C2×C4C23×D6711yescharacteristic
343C8—1341no—
344C8—1341no—
345C8—1341no—
346C8—1341no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—2681no—
356C22—2681no—
357C22—2681no—
358C22—2681no—
359C22—2681no—
360C22—5364no—
361C22—1342no—
362C22#1452406312yes—
363C22C22×C22⋊D6711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—2681no—
372C4—2681no—
373C4#41×D6711yescharacteristic
374C4#1452408911yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—2682no—
379C2—5361no—
380C2#1686403411yescharacteristic
381C2C2×#1452406311yescharacteristic
382C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#553×C67C2
socleC22×C67C22×D4
hypercenter#553D67
hypocenterC67#3297
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #16864042C2semidirect
2. #16863753C2semidirect
3. #16863731C2semidirect
4. #16863720C2semidirect
5. #16863693C2semidirect
6. #16863630C2semidirect
7. #16863623C2semidirect
8. #16863611C2semidirect
9. #16863539C2semidirect
10. #16863529C2semidirect
11. C4×#11761613C2semidirect
12. D4×C67⋊C8C2semidirect
13. #16863390C2semidirect
14. C2×#14523963C2semidirect
15. #553×C67C2semidirect
18. C2×#11761616C22semidirect
22. C22×C67⋊C8C22semidirect
31. #14523976C22semidirect
32. #14523976C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. #14523966C22semidirect
36. #14523966C22semidirect
37. #14523966C22semidirect
38. #14523964C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. C4×C67⋊C8C22semidirect
49. #125×C67C22semidirect
53. #121×C67C22semidirect
56. C4×D4×C67C22semidirect
57. #117×C67C22semidirect
192. C67#3297semidirect
193. #553D67semidirect
216. #125C2×D67semidirect
225. #121C2×D67semidirect
230. C4×D4C2×D67semidirect
231. #117C2×D67semidirect