Finite Group Databasebeta

← #22572241

Subgroups — #22572241

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#16864114C211yesmaximal, characteristic, complemented
2#16864034C211yesmaximal, characteristic
3#16863757C211yesmaximal, characteristic, complemented
4#16863735C211yesmaximal, characteristic, complemented
5#16863720C211yesmaximal, characteristic, complemented
6#16863684C211yesmaximal, characteristic, complemented
7#16863653C211yesmaximal, characteristic, complemented
8#16863625C211yesmaximal, characteristic, complemented
9#16863621C211yesmaximal, characteristic, complemented
10#16863558C211yesmaximal, characteristic, complemented
11#16863548C211yesmaximal, characteristic, complemented
12#16863493C211yesmaximal, characteristic, complemented
13#16863467C211yesmaximal, characteristic, complemented
14#16863399C211yesmaximal, characteristic, complemented
15#416×C67C211yesmaximal, characteristic, complemented
16C2×D4×D67C2211yescharacteristic
17C2×#11761618C2211yescharacteristic, complemented
18C2×#11761617C2211yescharacteristic
19C2×#11761616C2211yescharacteristic
20C2×#11761611C2211yescharacteristic
21#14524042C2211yescharacteristic, complemented
22#14524037—21no—
23#14524036—21no—
24#14524032C2211yescharacteristic
25#14524029—21no—
26#14524026—21no—
27#14524023C2211yescharacteristic
28#14524022C2211yescharacteristic, complemented
29#14524019C2211yescharacteristic
30#14524017C2211yescharacteristic
31#14524015—21no—
32#14524011—21no—
33C4×Dic134—21no—
34#14524008C2211yescharacteristic, complemented
35D4×Dic67C2211yescharacteristic
36C4×C22⋊D67C2211yescharacteristic
37#14524001C2211yescharacteristic
38C2×(C2×D67)⋊C4—21no—
39#14523976C2211yescharacteristic, complemented
40#14523975C2211yescharacteristic, complemented
41#14523974C2211yescharacteristic, complemented
42Dic67⋊C8C2211yescharacteristic, complemented
43#14523968C2211yescharacteristic, complemented
44#14523967C2211yescharacteristic, complemented
45#14523965C2211yescharacteristic
46#14523964C2211yescharacteristic
47#14523963C2211yescharacteristic
48#14523963C2211yescharacteristic
49C22⋊(C67⋊C8)C2211yescharacteristic, complemented
50(C2×D67)⋊C8C2211yescharacteristic
51#14523953C2211yescharacteristic, complemented
52C2×#41×C67C2211yescharacteristic
53C22×C8×C67C2211yescharacteristic, complemented
54#117×C67C2211yescharacteristic, complemented
55#103×C67C2211yescharacteristic, complemented
56#103×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic
58#102×C67C2211yescharacteristic
59C23×D67—21no—
60C2×Dic134—21no—
61C2×Dic134C2311yescharacteristic
62C2×D268C2311yescharacteristic
63C2×C22⋊D67—21no—
64C2×C22⋊D67—21no—
65C2×C22⋊D67D411yescharacteristic
66D4×D67—22no—
67D4×D67—41no—
68C22×Dic67—21no—
69C2×C4×D67C2311yescharacteristic
70#11761618—41no—
71#11761617—41no—
72#11761616—22no—
73#11761611—41no—
74C2×C67⋊C8C2311yescharacteristic
75C2×C67⋊C8C2311yescharacteristic
76C4⋊Dic67—21no—
77C4⋊Dic67C2311yescharacteristic
78C4⋊Dic67C2311yescharacteristic
79Dic67⋊C4—21no—
80Dic67⋊C4—21no—
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83Dic67⋊C4D411yescharacteristic
84C22⋊Dic67—21no—
85C22⋊Dic67—21no—
86C22⋊Dic67D411yescharacteristic
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4D411yescharacteristic
94C4×Dic67—21no—
95C4×Dic67—21no—
96C4×Dic67C2311yescharacteristic
97#41×C67—22no—
98#41×C67—41no—
99C2×Q8×C67C2311yescharacteristic
100C2×D4×C67—21no—
101C2×D4×C67C2311yescharacteristic
102C22×C4×C67—21no—
103C22×C4×C67C2311yescharacteristic
104C2×C8×C67—41no—
105C2×C8×C67C2311yescharacteristic
106C2×C8×C67C2311yescharacteristic
107C4⋊C4×C67C2311yescharacteristic
108C4⋊C4×C67C2311yescharacteristic
109C22⋊C4×C67—21no—
110C42×C67—21no—
111C23×C67—21no—
112C23×C67C2×D411yescharacteristic
113C22×D67—21no—
114C22×D67—42no—
115C22×D67—42no—
116C22×D67—81no—
117C22×D67C2×D411yescharacteristic
118Q8×C67—21no—
119Q8×C67—21no—
120D4×C67—21no—
121D4×C67—42no—
122D4×C67#4112yes—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—42no—
129C2×C4×C67C2×D411yescharacteristic
130C2×C4×C67C2411yescharacteristic
131C22⋊D67—41no—
132C22⋊D67—42no—
133C22⋊D67—81no—
134C2×Dic67—21no—
135C2×Dic67—21no—
136C2×Dic67—21no—
137C2×Dic67—81no—
138C2×Dic67C2×D411yescharacteristic
139C2×Dic67C2×D411yescharacteristic
140C2×Dic67C2×D411yescharacteristic
141D268—21no—
142D268—21no—
143C4×D67—21no—
144C4×D67—21no—
145Dic134—21no—
146Dic134—21no—
147Dic134—42no—
148C8×C67—21no—
149C8×C67—21no—
150C67⋊C8—21no—
151C67⋊C8—21no—
152C22×C67—22no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67C22×D411yescharacteristic
159C2×D67—21no—
160C2×D67—21no—
161C2×D67—21no—
162C2×D67—41no—
163C2×D67—41no—
164C2×D67—41no—
165C2×D67—84no—
166C4×C67—21no—
167C4×C67—41no—
168C4×C67—41no—
169C4×C67—41no—
170C4×C67C2×#4111yescharacteristic
171C4×C67H(5,𝔽2)11yescharacteristic
172Dic67—21no—
173Dic67—21no—
174Dic67—21no—
175Dic67—41no—
176Dic67—41no—
177Dic67—41no—
178C2×C67—42no—
179C2×C67—41no—
180C2×C67#54611yescharacteristic
181C2×C67#57511yescharacteristic
182C2×C67#57711yescharacteristic
183D67—41no—
184D67—81no—
185#3311—671nomaximal, Sylow
186C67#331111yescharacteristic, complemented, Sylow
187#556—671no—
188#546—671no—
189#489—671no—
190#485—671no—
191#482—671no—
192#476—671no—
193#470—671no—
194#465—671no—
195#464—671no—
196#452—671no—
197#450—671no—
198#440—671no—
199#435—671no—
200#418—671no—
201#416D6711yescharacteristic, complemented
202C2×#41C2×D6711yescharacteristic
203C22×D4—671no—
204C2×Q16—671no—
205C2×SD16—671no—
206C2×SD16—671no—
207C2×M16—671no—
208C22×C8C2×D6711yescharacteristic, complemented
209C4⋊Q8—671no—
210#126—1341no—
211#126—1341no—
212#124—671no—
213#124—1341no—
214#123—1341no—
215C22⋊Q8—671no—
216C22⋊Q8—671no—
217C22⋊Q8—671no—
218#121—671no—
219#121—1341no—
220#120—1341no—
221C4×Q8—671no—
222C4×Q8—1341no—
223C4×D4—671no—
224C4×D4—671no—
225#117—671no—
226#117C2×D6711yescharacteristic, complemented
227C2×C22⋊C4—1341no—
228#107—671no—
229#106—671no—
230#106—671no—
231C4⋊C8—671no—
232#103—671no—
233#103—671no—
234#103C2×D6711yescharacteristic, complemented
235#103C2×D6711yescharacteristic, complemented
236#102—671no—
237#102—671no—
238#102—671no—
239#102—671no—
240#102C2×D6711yescharacteristic
241#102C2×D6711yescharacteristic
242C22⋊C8—671no—
243C22⋊C8—671no—
244#97—671no—
245C24—1341no—
246#41—22no—
247#41—41no—
248C2×Q8—671no—
249C2×Q8—1341no—
250C2×Q8C22×D6711yescharacteristic
251C2×D4—21no—
252C2×D4—2681no—
253C2×D4—671no—
254C2×D4—671no—
255C2×D4—1341no—
256C2×D4—1342no—
257C2×D4—1341no—
258C2×D4C22×D6711yescharacteristic
259C22×C4—21no—
260C22×C4—671no—
261C22×C4—1341no—
262C22×C4C22×D6711yescharacteristic
263Q16—2681no—
264SD16—2681no—
265SD16—1342no—
266M16—2681no—
267C2×C8—41no—
268C2×C8—671no—
269C2×C8—671no—
270C2×C8C22×D6711yescharacteristic
271C2×C8C22×D6711yescharacteristic
272C4⋊C4—671no—
273C4⋊C4—671no—
274C4⋊C4—671no—
275C4⋊C4—1341no—
276C4⋊C4—1341no—
277C4⋊C4—1341no—
278C4⋊C4—1341no—
279C4⋊C4—1341no—
280C4⋊C4C22×D6711yescharacteristic
281C4⋊C4C22×D6711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—671no—
284C22⋊C4—671no—
285C22⋊C4—1341no—
286C22⋊C4—1341no—
287C22⋊C4—1341no—
288C22⋊C4—1341no—
289C22⋊C4—1341no—
290C22⋊C4—1341no—
291C22⋊C4—1341no—
292C22⋊C4—1341no—
293C42—21no—
294C42—671no—
295C42—1341no—
296C42—1341no—
297C23—21no—
298C23—2682no—
299C23—2682no—
300C23—5361no—
301C23—671no—
302C23—1341no—
303C23D4×D6711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—2682no—
307Q8—1341no—
308Q8—1341no—
309D4—21no—
310D4—42no—
311D4—2682no—
312D4—2681no—
313D4—5361no—
314D4—1341no—
315D4—1341no—
316D4#1176162812yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—5361no—
324C2×C4—671no—
325C2×C4—671no—
326C2×C4—671no—
327C2×C4—1341no—
328C2×C4—1341no—
329C2×C4—1341no—
330C2×C4—1341no—
331C2×C4—1341no—
332C2×C4D4×D6711yescharacteristic
333C2×C4C23×D6711yescharacteristic
334C8—21no—
335C8—21no—
336C8—1341no—
337C8—1341no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—2681no—
345C22—2681no—
346C22—2681no—
347C22—5364no—
348C22—1341no—
349C22—1341no—
350C22—1341no—
351C22C2×D4×D6711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—2681no—
357C4—2681no—
358C4—2681no—
359C4—1341no—
360C4—1341no—
361C4—1341no—
362C4C2×#1176162811yescharacteristic
363C4#1452408811yescharacteristic
364C2—41no—
365C2—42no—
366C2—2681no—
367C2—5361no—
368C2#1686403611yescharacteristic
369C2#1686419311yescharacteristic
370C2#1686420311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#416×C67C2
socleC22×C67C22×D4
hypercenter#416D67
hypocenterC67#3311
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #16864114C2semidirect
3. #16863757C2semidirect
4. #16863735C2semidirect
5. #16863720C2semidirect
6. #16863684C2semidirect
7. #16863653C2semidirect
8. #16863625C2semidirect
9. #16863621C2semidirect
10. #16863558C2semidirect
11. #16863548C2semidirect
12. #16863493C2semidirect
13. #16863467C2semidirect
14. #16863399C2semidirect
15. #416×C67C2semidirect
17. C2×#11761618C22semidirect
21. #14524042C22semidirect
28. #14524022C22semidirect
34. #14524008C22semidirect
39. #14523976C22semidirect
40. #14523975C22semidirect
41. #14523974C22semidirect
42. Dic67⋊C8C22semidirect
43. #14523968C22semidirect
44. #14523967C22semidirect
49. C22⋊(C67⋊C8)C22semidirect
51. #14523953C22semidirect
53. C22×C8×C67C22semidirect
54. #117×C67C22semidirect
55. #103×C67C22semidirect
56. #103×C67C22semidirect
186. C67#3311semidirect
201. #416D67semidirect
208. C22×C8C2×D67semidirect
226. #117C2×D67semidirect
234. #103C2×D67semidirect
235. #103C2×D67semidirect