Finite Group Databasebeta

← #22572059

Subgroups — #22572059

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#16864092C211yesmaximal, characteristic, complemented
2#16863750C211yesmaximal, characteristic, complemented
3#16863729C211yesmaximal, characteristic, complemented
4#16863717C211yesmaximal, characteristic, complemented
5#16863696C211yesmaximal, characteristic, complemented
6#16863630C211yesmaximal, characteristic, complemented
7#16863627C211yesmaximal, characteristic, complemented
8#16863608C211yesmaximal, characteristic, complemented
9#16863541C211yesmaximal, characteristic, complemented
10#16863530C211yesmaximal, characteristic, complemented
11C4×#11761613C211yesmaximal, characteristic, complemented
12D4×C67⋊C8C211yesmaximal, characteristic, complemented
13#16863392C211yesmaximal, characteristic, complemented
14C2×#14523964C211yesmaximal, characteristic, complemented
15#546×C67C211yesmaximal, characteristic, complemented
16C2×#11761615C2211yescharacteristic, complemented
17C2×#11761613C2211yescharacteristic
18C2×#11761613C2211yescharacteristic
19C22×C67⋊C8C2211yescharacteristic
20C22×C67⋊C8C2211yescharacteristic, complemented
21#14524038—21no—
22#14524035C2211yescharacteristic, complemented
23#14524025—21no—
24#14524017C2211yescharacteristic
25#14524017C2211yescharacteristic, complemented
26#14524015—21no—
27C4×D268C2211yescharacteristic, complemented
28(C22⋊D67)⋊C4—21no—
29#14524002C2211yescharacteristic, complemented
30C2×C4⋊Dic67C2211yescharacteristic
31#14523976C2211yescharacteristic, complemented
32#14523976C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35#14523967C2211yescharacteristic, complemented
36#14523967C2211yescharacteristic, complemented
37#14523967C2211yescharacteristic, complemented
38#14523964C2211yescharacteristic
39#14523964C2211yescharacteristic
40#14523964C2211yescharacteristic
41#14523964C2211yescharacteristic
42#14523964C2211yescharacteristic
43#14523964C2211yescharacteristic
44#14523963C2211yescharacteristic, complemented
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic, complemented
47C4×C67⋊C8C2211yescharacteristic, complemented
48C2×#41×C67C2211yescharacteristic
49C22×D4×C67C2211yescharacteristic
50#124×C67C2211yescharacteristic
51#123×C67—21no—
52C22⋊Q8×C67C2211yescharacteristic
53#121×C67—21no—
54#121×C67C2211yescharacteristic
55#120×C67—21no—
56C4×D4×C67C2211yescharacteristic
57C4×D4×C67C2211yescharacteristic
58C2×C22⋊C4×C67—21no—
59C2×D268C2311yescharacteristic
60C2×C22⋊D67—21no—
61C2×C22⋊D67—21no—
62C22×Dic67—21no—
63C2×C4×D67—21no—
64#11761615—41no—
65#11761613—22no—
66#11761613—41no—
67C2×C67⋊C8—41no—
68C2×C67⋊C8—41no—
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C2×C67⋊C8C2311yescharacteristic
72C2×C67⋊C8C2311yescharacteristic
73C4⋊Dic67—21no—
74C4⋊Dic67C2311yescharacteristic
75C4⋊Dic67C2311yescharacteristic
76C4⋊Dic67C2311yescharacteristic
77Dic67⋊C4—21no—
78Dic67⋊C4—21no—
79C22⋊Dic67—21no—
80(C2×D67)⋊C4—21no—
81(C2×D67)⋊C4—21no—
82(C2×D67)⋊C4—21no—
83C4×Dic67—21no—
84C24×C67—21no—
85#41×C67—22no—
86#41×C67—41no—
87C2×Q8×C67C2311yescharacteristic
88C2×D4×C67—22no—
89C2×D4×C67—21no—
90C2×D4×C67—21no—
91C2×D4×C67—21no—
92C2×D4×C67—41no—
93C2×D4×C67D411yescharacteristic
94C2×D4×C67C2311yescharacteristic
95C2×D4×C67C2311yescharacteristic
96C22×C4×C67—21no—
97C22×C4×C67—21no—
98C22×C4×C67C2311yescharacteristic
99C22×C4×C67C2311yescharacteristic
100C4⋊C4×C67—21no—
101C4⋊C4×C67D411yescharacteristic
102C4⋊C4×C67C2311yescharacteristic
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C22⋊C4×C67D411yescharacteristic
109C22⋊C4×C67D411yescharacteristic
110C42×C67C2311yescharacteristic
111C23×C67—21no—
112C23×C67—21no—
113C23×C67—42no—
114C23×C67—42no—
115C23×C67—81no—
116C23×C67C2×D411yescharacteristic
117C23×C67C2×D411yescharacteristic
118C22×D67—21no—
119Q8×C67—21no—
120Q8×C67—21no—
121D4×C67—22no—
122D4×C67—21no—
123D4×C67—21no—
124D4×C67—21no—
125D4×C67—42no—
126D4×C67—42no—
127D4×C67—81no—
128D4×C67#4112yes—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—21no—
135C2×C4×C67—42no—
136C2×C4×C67—81no—
137C2×C4×C67C2×D411yescharacteristic
138C2×C4×C67C2×D411yescharacteristic
139C2×C4×C67C2×D411yescharacteristic
140C2×C4×C67C2×D411yescharacteristic
141C2×C4×C67C2411yescharacteristic
142C22⋊D67—42no—
143C22⋊D67—81no—
144C2×Dic67—21no—
145C2×Dic67—21no—
146C2×Dic67—21no—
147C2×Dic67—81no—
148D268—21no—
149D268—21no—
150C4×D67—81no—
151C67⋊C8—21no—
152C67⋊C8—21no—
153C67⋊C8—21no—
154C67⋊C8—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—22no—
159C22×C67—21no—
160C22×C67—21no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—41no—
166C22×C67—84no—
167C22×C67#13512yes—
168C22×C67C22×D411yescharacteristic
169C2×D67—41no—
170C2×D67—41no—
171C2×D67—41no—
172C4×C67—21no—
173C4×C67—21no—
174C4×C67—21no—
175C4×C67—21no—
176C4×C67—41no—
177C4×C67—41no—
178C4×C67C2×#4111yescharacteristic
179C4×C67H(5,𝔽2)11yescharacteristic
180Dic67—41no—
181Dic67—41no—
182Dic67—41no—
183C2×C67—22no—
184C2×C67—41no—
185C2×C67—42no—
186C2×C67—81no—
187C2×C67#54611yescharacteristic
188C2×C67C2×#13511yescharacteristic
189C2×C67#57611yescharacteristic
190D67—81no—
191#3297—671nomaximal, Sylow
192C67#329711yescharacteristic, complemented, Sylow
193#553—671no—
194#546D6711yescharacteristic, complemented
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×#41C2×D6711yescharacteristic
209C22×D4C2×D6711yescharacteristic
210C2×SD16—671no—
211C2×D8—671no—
212C2×D8—671no—
213C22×C8—671no—
214C22×C8—671no—
215#126—1341no—
216#125—671no—
217#124C2×D6711yescharacteristic
218#123—21no—
219#123—1341no—
220C22⋊Q8C2×D6711yescharacteristic
221#121—21no—
222#121—671no—
223#121—671no—
224#121—1341no—
225#121C2×D6711yescharacteristic
226#120—21no—
227C4×D4—671no—
228C4×D4—1341no—
229C4×D4C2×D6711yescharacteristic
230C4×D4C2×D6711yescharacteristic
231#117—671no—
232C2×C4⋊C4—671no—
233C2×C22⋊C4—21no—
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—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D6711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—671no—
261C2×D4—1341no—
262C2×D4—1341no—
263C2×D4C22⋊D6711yescharacteristic
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—671no—
283C4⋊C4—671no—
284C4⋊C4—671no—
285C4⋊C4—1341no—
286C4⋊C4—1341no—
287C4⋊C4—1341no—
288C4⋊C4C22⋊D6711yescharacteristic
289C4⋊C4C22×D6711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—1341no—
296C22⋊C4—1341no—
297C22⋊C4—1341no—
298C22⋊C4—1341no—
299C22⋊C4C22⋊D6711yescharacteristic
300C22⋊C4C22⋊D6711yescharacteristic
301C42—1341no—
302C42C22×D6711yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—1341no—
309C23C2×C22⋊D6711yescharacteristic
310C23C2×C22⋊D6711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—2682no—
321D4—5361no—
322D4—1341no—
323D4—1341no—
324D4#1176162712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—5361no—
334C2×C4—5361no—
335C2×C4—1341no—
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—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—2681no—
360C22—2681no—
361C22—2681no—
362C22#1452406312yes—
363C22C22×C22⋊D6711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—2681no—
371C4—2681no—
372C4—2681no—
373C4C2×#1176162711yescharacteristic
374C4#1452408811yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—5361no—
380C2#1686404011yescharacteristic
381C2C2×#1452406311yescharacteristic
382C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#546×C67C2
socleC22×C67C22×D4
hypercenter#546D67
hypocenterC67#3297
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #16864092C2semidirect
2. #16863750C2semidirect
3. #16863729C2semidirect
4. #16863717C2semidirect
5. #16863696C2semidirect
6. #16863630C2semidirect
7. #16863627C2semidirect
8. #16863608C2semidirect
9. #16863541C2semidirect
10. #16863530C2semidirect
11. C4×#11761613C2semidirect
12. D4×C67⋊C8C2semidirect
13. #16863392C2semidirect
14. C2×#14523964C2semidirect
15. #546×C67C2semidirect
16. C2×#11761615C22semidirect
20. C22×C67⋊C8C22semidirect
22. #14524035C22semidirect
25. #14524017C22semidirect
27. C4×D268C22semidirect
29. #14524002C22semidirect
31. #14523976C22semidirect
32. #14523976C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. #14523967C22semidirect
36. #14523967C22semidirect
37. #14523967C22semidirect
44. #14523963C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. C4×C67⋊C8C22semidirect
192. C67#3297semidirect
194. #546D67semidirect