Finite Group Databasebeta

← #22572293

Subgroups — #22572293

Order 8576.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864101C211yesmaximal, characteristic, complemented
2#16864043C211yesmaximal, characteristic
3#16863754C211yesmaximal, characteristic, complemented
4#16863733C211yesmaximal, characteristic, complemented
5#16863675C211yesmaximal, characteristic, complemented
6#16863640C211yesmaximal, characteristic, complemented
7#16863638C211yesmaximal, characteristic, complemented
8#16863617C211yesmaximal, characteristic, complemented
9#16863560C211yesmaximal, characteristic, complemented
10#16863548C211yesmaximal, characteristic, complemented
11#16863495C211yesmaximal, characteristic, complemented
12#16863470C211yesmaximal, characteristic, complemented
13#16863395C211yesmaximal, characteristic, complemented
14#102×D67C211yesmaximal, characteristic, complemented
15#480×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C2×D4×D67C2211yescharacteristic
18C2×Dic268C2211yescharacteristic, complemented
19C2×#11761617C2211yescharacteristic
20C2×#11761616C2211yescharacteristic
21C2×#11761612C2211yescharacteristic, complemented
22C2×C8×D67C2211yescharacteristic
23#14524042C2211yescharacteristic, complemented
24#14524033—21no—
25#14524030C2211yescharacteristic
26#14524025—21no—
27#14524023C2211yescharacteristic
28#14524023C2211yescharacteristic
29#14524022—21no—
30#14524022C2211yescharacteristic, complemented
31#14524021—21no—
32#14524014—21no—
33#14524011—21no—
34#14524008C2211yescharacteristic, complemented
35#14524005C2211yescharacteristic
36C4×C22⋊D67—21no—
37C4×C22⋊D67C2211yescharacteristic
38C4⋊C4×D67C2211yescharacteristic
39C22⋊C4×D67—21no—
40#14523976C2211yescharacteristic, complemented
41Dic67⋊C8C2211yescharacteristic, complemented
42#14523968C2211yescharacteristic, complemented
43#14523966C2211yescharacteristic, complemented
44#14523966C2211yescharacteristic, complemented
45#14523966C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic
47#14523964C2211yescharacteristic
48#14523963C2211yescharacteristic
49#14523963C2211yescharacteristic
50(C2×D67)⋊C8C2211yescharacteristic
51#14523953C2211yescharacteristic, complemented
52#121×C67C2211yescharacteristic
53C2×C4⋊C4×C67C2211yescharacteristic, complemented
54#107×C67C2211yescharacteristic, complemented
55#107×C67C2211yescharacteristic, complemented
56#102×C67C2211yescharacteristic
57#102×C67C2211yescharacteristic
58C22⋊C8×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60#11761628—22no—
61#11761628—41no—
62C2×Dic134—21no—
63C2×Dic134C2311yescharacteristic
64C2×Dic134C2311yescharacteristic
65C2×D268C2311yescharacteristic
66C2×C22⋊D67—21no—
67C2×C22⋊D67—21no—
68C2×C22⋊D67D411yescharacteristic
69D4×D67—22no—
70D4×D67—41no—
71C2×C4×D67—21no—
72C2×C4×D67—21no—
73C2×C4×D67—21no—
74C2×C4×D67C2311yescharacteristic
75Dic268—41no—
76#11761617—22no—
77#11761616—41no—
78#11761612—41no—
79C2×C67⋊C8C2311yescharacteristic
80C2×C67⋊C8C2311yescharacteristic
81C8×D67—41no—
82C4⋊Dic67—21no—
83C4⋊Dic67—21no—
84C4⋊Dic67C2311yescharacteristic
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4—21no—
88Dic67⋊C4—21no—
89Dic67⋊C4D411yescharacteristic
90C22⋊Dic67—21no—
91C22⋊Dic67—21no—
92C22⋊Dic67—21no—
93C22⋊Dic67D411yescharacteristic
94(C2×D67)⋊C4—21no—
95(C2×D67)⋊C4—21no—
96(C2×D67)⋊C4—21no—
97(C2×D67)⋊C4D411yescharacteristic
98C4×Dic67—21no—
99C4×Dic67C2311yescharacteristic
100C2×D4×C67—21no—
101C2×D4×C67C2311yescharacteristic
102C22×C4×C67—21no—
103C22×C4×C67C2311yescharacteristic
104C2×C8×C67C2311yescharacteristic
105C2×C8×C67C2311yescharacteristic
106C4⋊C4×C67—21no—
107C4⋊C4×C67C2311yescharacteristic
108C4⋊C4×C67C2311yescharacteristic
109C4⋊C4×C67C2311yescharacteristic
110C22⋊C4×C67—21no—
111C23×C67—21no—
112C23×C67C2×D411yescharacteristic
113C22×D67—21no—
114C22×D67—42no—
115C22×D67—42no—
116C22×D67—81no—
117C22×D67C2×D411yescharacteristic
118D4×C67—21no—
119D4×C67—21no—
120D4×C67—81no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—81no—
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2411yescharacteristic
129C22⋊D67—22no—
130C22⋊D67—42no—
131C22⋊D67—42no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67C2×D411yescharacteristic
136C2×Dic67C2×D411yescharacteristic
137C2×Dic67C2×D411yescharacteristic
138D268—21no—
139D268#4112yes—
140C4×D67—21no—
141C4×D67—21no—
142C4×D67—42no—
143C4×D67—81no—
144C4×D67—81no—
145Dic134—21no—
146Dic134—21no—
147Dic134—21no—
148Dic134—21no—
149Dic134—81no—
150C8×C67—21no—
151C8×C67—21no—
152C67⋊C8—21no—
153C67⋊C8—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—41no—
158C22×C67—41no—
159C22×C67—41no—
160C22×C67C22×D411yescharacteristic
161C2×D67—21no—
162C2×D67—21no—
163C2×D67—22no—
164C2×D67—41no—
165C2×D67—41no—
166C2×D67—84no—
167C2×D67#13712yes—
168C4×C67—21no—
169C4×C67—41no—
170C4×C67—41no—
171C4×C67—41no—
172C4×C67C2×#4111yescharacteristic
173C4×C67H(5,𝔽2)11yescharacteristic
174Dic67—21no—
175Dic67—21no—
176Dic67—21no—
177Dic67—41no—
178Dic67—41no—
179Dic67—41no—
180C2×C67—41no—
181C2×C67—81no—
182C2×C67#54611yescharacteristic
183C2×C67C2×#13711yescharacteristic
184C2×C67#57611yescharacteristic
185D67—22no—
186D67—42no—
187#3314—671nomaximal, Sylow
188C67#331411yescharacteristic, complemented, Sylow
189#554—671no—
190#546—671no—
191#489—671no—
192#484—671no—
193#480D6711yescharacteristic, complemented
194#474—671no—
195#468—671no—
196#467—671no—
197#464—671no—
198#452—671no—
199#450—671no—
200#440—671no—
201#435—671no—
202#417—671no—
203C2×#102—671no—
204C2×#41—671no—
205C22×D4—671no—
206C2×Q16—671no—
207C2×SD16—671no—
208C2×SD16—671no—
209C2×M16—671no—
210C22×C8—671no—
211C4⋊Q8—671no—
212#125—1341no—
213#124—671no—
214#123—1341no—
215C22⋊Q8—671no—
216C22⋊Q8—671no—
217C22⋊Q8—671no—
218C22⋊Q8—1341no—
219C22⋊Q8—1341no—
220#121—1341no—
221#121C2×D6711yescharacteristic
222#120—1341no—
223C4×Q8—671no—
224C4×D4—671no—
225C4×D4—671no—
226C4×D4—1341no—
227C2×C4⋊C4—671no—
228C2×C4⋊C4C2×D6711yescharacteristic, complemented
229C2×C22⋊C4—1341no—
230#107—671no—
231#107C2×D6711yescharacteristic, complemented
232#107C2×D6711yescharacteristic, complemented
233C4⋊C8—671no—
234#103—671no—
235#103—671no—
236#103—671no—
237#103—671no—
238#102—671no—
239#102—671no—
240#102—671no—
241#102—671no—
242#102C2×D6711yescharacteristic
243#102C2×D6711yescharacteristic
244C22⋊C8—671no—
245C22⋊C8C2×D6711yescharacteristic, complemented
246#97—671no—
247C24—1341no—
248#41—2681no—
249#41—1342no—
250C2×Q8—671no—
251C2×Q8—671no—
252C2×Q8—1341no—
253C2×D4—21no—
254C2×D4—2681no—
255C2×D4—671no—
256C2×D4—671no—
257C2×D4—1341no—
258C2×D4—1342no—
259C2×D4—1341no—
260C2×D4C22×D6711yescharacteristic
261C22×C4—21no—
262C22×C4—671no—
263C22×C4—1341no—
264C22×C4—1341no—
265C22×C4—1341no—
266C22×C4C22×D6711yescharacteristic
267Q16—2681no—
268SD16—2681no—
269SD16—1342no—
270M16—2681no—
271C2×C8—2681no—
272C2×C8—671no—
273C2×C8—671no—
274C2×C8C22×D6711yescharacteristic
275C2×C8C22×D6711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—671no—
278C4⋊C4—671no—
279C4⋊C4—1341no—
280C4⋊C4—1341no—
281C4⋊C4—1341no—
282C4⋊C4—1341no—
283C4⋊C4—1341no—
284C4⋊C4—1341no—
285C4⋊C4C22×D6711yescharacteristic
286C4⋊C4C22×D6711yescharacteristic
287C4⋊C4C22×D6711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—671no—
290C22⋊C4—671no—
291C22⋊C4—1341no—
292C22⋊C4—1341no—
293C22⋊C4—1341no—
294C22⋊C4—1341no—
295C22⋊C4—1341no—
296C22⋊C4—1341no—
297C42—671no—
298C42—1341no—
299C23—21no—
300C23—2682no—
301C23—2682no—
302C23—5361no—
303C23—671no—
304C23—1341no—
305C23D4×D6711yescharacteristic
306Q8—5361no—
307Q8—1341no—
308Q8—1341no—
309Q8—1341no—
310Q8—1341no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—2682no—
315D4—2682no—
316D4—672no—
317D4—1342no—
318D4—1341no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—81no—
325C2×C4—2682no—
326C2×C4—5361no—
327C2×C4—5361no—
328C2×C4—671no—
329C2×C4—671no—
330C2×C4—671no—
331C2×C4—1341no—
332C2×C4—1341no—
333C2×C4—1341no—
334C2×C4—1341no—
335C2×C4—1341no—
336C2×C4D4×D6711yescharacteristic
337C2×C4C23×D6711yescharacteristic
338C8—21no—
339C8—21no—
340C8—1341no—
341C8—1341no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—2681no—
349C22—2681no—
350C22—5364no—
351C22—672no—
352C22—1342no—
353C22—1341no—
354C22—1341no—
355C22C2×D4×D6711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—2681no—
361C4—2681no—
362C4—2681no—
363C4—1341no—
364C4—1341no—
365C4—1341no—
366C4#41×D6711yescharacteristic
367C4#1452408911yescharacteristic
368C2—41no—
369C2—81no—
370C2—2682no—
371C2—1342no—
372C2#1686403811yescharacteristic
373C2#137×D6711yescharacteristic
374C2#1686419711yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#480×C67C2
socleC22×C67C22×D4
hypercenter#480D67
hypocenterC67#3314
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #16864101C2semidirect
3. #16863754C2semidirect
4. #16863733C2semidirect
5. #16863675C2semidirect
6. #16863640C2semidirect
7. #16863638C2semidirect
8. #16863617C2semidirect
9. #16863560C2semidirect
10. #16863548C2semidirect
11. #16863495C2semidirect
12. #16863470C2semidirect
13. #16863395C2semidirect
14. #102×D67C2semidirect
15. #480×C67C2semidirect
18. C2×Dic268C22semidirect
21. C2×#11761612C22semidirect
23. #14524042C22semidirect
30. #14524022C22semidirect
34. #14524008C22semidirect
40. #14523976C22semidirect
41. Dic67⋊C8C22semidirect
42. #14523968C22semidirect
43. #14523966C22semidirect
44. #14523966C22semidirect
45. #14523966C22semidirect
51. #14523953C22semidirect
53. C2×C4⋊C4×C67C22semidirect
54. #107×C67C22semidirect
55. #107×C67C22semidirect
58. C22⋊C8×C67C22semidirect
188. C67#3314semidirect
193. #480D67semidirect
228. C2×C4⋊C4C2×D67semidirect
231. #107C2×D67semidirect
232. #107C2×D67semidirect
245. C22⋊C8C2×D67semidirect