Finite Group Databasebeta

← #22572425

Subgroups — #22572425

Order 8576.

402 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#16864086C211yesmaximal, characteristic, complemented
2#16863750C211yesmaximal, characteristic, complemented
3#16863734C211yesmaximal, characteristic, complemented
4#16863712C211yesmaximal, characteristic, complemented
5#16863696C211yesmaximal, characteristic, complemented
6#16863652C211yesmaximal, characteristic, complemented
7#16863640C211yesmaximal, characteristic, complemented
8#16863615C211yesmaximal, characteristic, complemented
9#16863558C211yesmaximal, characteristic, complemented
10#16863547C211yesmaximal, characteristic, complemented
11C4×#11761616C211yesmaximal, characteristic, complemented
12#16863474C211yesmaximal, characteristic, complemented
13#16863401C211yesmaximal, characteristic, complemented
14#16863397C211yesmaximal, characteristic, complemented
15#546×C67C211yesmaximal, characteristic
16C2×#11761618C2211yescharacteristic, complemented
17C2×#11761616C2211yescharacteristic
18C2×#11761616C2211yescharacteristic
19C2×#11761612C2211yescharacteristic
20C2×#11761612C2211yescharacteristic, complemented
21#14524036—21no—
22#14524035C2211yescharacteristic, complemented
23#14524033—21no—
24#14524024—21no—
25#14524022C2211yescharacteristic
26#14524022C2211yescharacteristic, complemented
27C4×Dic134C2211yescharacteristic, complemented
28#14524002C2211yescharacteristic
29#14523999—21no—
30C2×C4⋊Dic67C2211yescharacteristic, complemented
31#14523976C2211yescharacteristic, complemented
32#14523974C2211yescharacteristic, complemented
33#14523974C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35#14523967C2211yescharacteristic, complemented
36#14523967C2211yescharacteristic, complemented
37#14523967C2211yescharacteristic, complemented
38#14523966C2211yescharacteristic, complemented
39#14523964C2211yescharacteristic
40#14523964C2211yescharacteristic
41#14523964C2211yescharacteristic
42#14523964C2211yescharacteristic
43#14523964C2211yescharacteristic
44#14523964C2211yescharacteristic
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×Dic134C2311yescharacteristic
60C22×Dic67—21no—
61#11761618—41no—
62#11761616—22no—
63#11761616—41no—
64#11761612—41no—
65#11761612—41no—
66C2×C67⋊C8C2311yescharacteristic
67C2×C67⋊C8C2311yescharacteristic
68C2×C67⋊C8C2311yescharacteristic
69C2×C67⋊C8C2311yescharacteristic
70C4⋊Dic67—21no—
71C4⋊Dic67C2311yescharacteristic
72C4⋊Dic67C2311yescharacteristic
73C4⋊Dic67C2311yescharacteristic
74Dic67⋊C4—21no—
75Dic67⋊C4—21no—
76Dic67⋊C4—21no—
77Dic67⋊C4—21no—
78Dic67⋊C4—21no—
79C22⋊Dic67—21no—
80C22⋊Dic67—21no—
81C22⋊Dic67—21no—
82C4×Dic67—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
118Q8×C67—21no—
119Q8×C67—21no—
120D4×C67—21no—
121D4×C67—21no—
122D4×C67—21no—
123D4×C67—41no—
124D4×C67—42no—
125D4×C67—42no—
126D4×C67—81no—
127D4×C67#4112yes—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—42no—
135C2×C4×C67—81no—
136C2×C4×C67C2×D411yescharacteristic
137C2×C4×C67C2×D411yescharacteristic
138C2×C4×C67C2×D411yescharacteristic
139C2×C4×C67C2×D411yescharacteristic
140C2×C4×C67C2411yescharacteristic
141C2×Dic67—21no—
142C2×Dic67—21no—
143C2×Dic67—21no—
144C2×Dic67—21no—
145C2×Dic67—42no—
146Dic134—21no—
147Dic134—21no—
148C67⋊C8—21no—
149C67⋊C8—21no—
150C67⋊C8—21no—
151C67⋊C8—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—22no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—21no—
159C22×C67—41no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—84no—
165C22×C67C22×D411yescharacteristic
166C4×C67—21no—
167C4×C67—21no—
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—41no—
171C4×C67—41no—
172C4×C67C2×#4111yescharacteristic
173C4×C67H(5,𝔽2)11yescharacteristic
174Dic67—41no—
175Dic67—41no—
176Dic67—41no—
177Dic67—41no—
178C2×C67—41no—
179C2×C67—41no—
180C2×C67—42no—
181C2×C67—81no—
182C2×C67#54611yescharacteristic
183C2×C67#57511yescharacteristic
184C2×C67#57711yescharacteristic
185#3323—671nomaximal, Sylow
186C67#332311yescharacteristic, complemented, Sylow
187#552—671no—
188#546D6711yescharacteristic
189#488—671no—
190#485—671no—
191#481—671no—
192#478—671no—
193#470—671no—
194#468—671no—
195#464—671no—
196#452—671no—
197#450—671no—
198C4×SD16—671no—
199#436—671no—
200#418—671no—
201#417—671no—
202C2×#41C2×D6711yescharacteristic
203C22×D4C2×D6711yescharacteristic
204C2×Q16—671no—
205C2×SD16—671no—
206C2×SD16—671no—
207C2×M16—671no—
208C2×M16—671no—
209#126—1341no—
210#125—671no—
211#125—1341no—
212#124C2×D6711yescharacteristic
213#123—21no—
214#123—1341no—
215C22⋊Q8—671no—
216C22⋊Q8—671no—
217C22⋊Q8C2×D6711yescharacteristic
218#121—21no—
219#121C2×D6711yescharacteristic
220#120—21no—
221C4×Q8—671no—
222C4×D4C2×D6711yescharacteristic
223C4×D4C2×D6711yescharacteristic
224#117—671no—
225#117—1341no—
226C2×C4⋊C4—671no—
227C2×C22⋊C4—21no—
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—21no—
246#41—22no—
247#41—41no—
248C2×Q8—671no—
249C2×Q8C22×D6711yescharacteristic
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—41no—
255C2×D4C22⋊D6711yescharacteristic
256C2×D4C22×D6711yescharacteristic
257C2×D4C22×D6711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1341no—
261C22×C4C22×D6711yescharacteristic
262C22×C4C22×D6711yescharacteristic
263Q16—2681no—
264SD16—2681no—
265SD16—1342no—
266M16—2681no—
267M16—2681no—
268C2×C8—671no—
269C2×C8—671no—
270C2×C8—671no—
271C2×C8—671no—
272C4⋊C4—21no—
273C4⋊C4—671no—
274C4⋊C4—671no—
275C4⋊C4—671no—
276C4⋊C4—1341no—
277C4⋊C4—1341no—
278C4⋊C4—1341no—
279C4⋊C4—1341no—
280C4⋊C4—1341no—
281C4⋊C4—1341no—
282C4⋊C4C22⋊D6711yescharacteristic
283C4⋊C4C22×D6711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—1341no—
290C22⋊C4—1341no—
291C22⋊C4—1341no—
292C22⋊C4C22⋊D6711yescharacteristic
293C22⋊C4C22⋊D6711yescharacteristic
294C42—1341no—
295C42—1341no—
296C42C22×D6711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×C22⋊D6711yescharacteristic
303C23C2×C22⋊D6711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1341no—
307Q8—1341no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—41no—
312D4—42no—
313D4—42no—
314D4—81no—
315D4#1176162712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—2682no—
325C2×C4—1341no—
326C2×C4—1341no—
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—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×C22⋊D6711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—2681no—
359C4—2681no—
360C4—2681no—
361C4—2681no—
362C4C2×#1176162711yescharacteristic
363C4#1452408811yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#1686404011yescharacteristic
369C2#1686419411yescharacteristic
370C2#1686420711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#546×C67C2
socleC22×C67C22×D4
hypercenter#546D67
hypocenterC67#3323
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. #16864086C2semidirect
2. #16863750C2semidirect
3. #16863734C2semidirect
4. #16863712C2semidirect
5. #16863696C2semidirect
6. #16863652C2semidirect
7. #16863640C2semidirect
8. #16863615C2semidirect
9. #16863558C2semidirect
10. #16863547C2semidirect
11. C4×#11761616C2semidirect
12. #16863474C2semidirect
13. #16863401C2semidirect
14. #16863397C2semidirect
16. C2×#11761618C22semidirect
20. C2×#11761612C22semidirect
22. #14524035C22semidirect
26. #14524022C22semidirect
27. C4×Dic134C22semidirect
30. C2×C4⋊Dic67C22semidirect
31. #14523976C22semidirect
32. #14523974C22semidirect
33. #14523974C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. #14523967C22semidirect
36. #14523967C22semidirect
37. #14523967C22semidirect
38. #14523966C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. C4×C67⋊C8C22semidirect
186. C67#3323semidirect