Finite Group Databasebeta

← #22572291

Subgroups — #22572291

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#16864034C211yesmaximal, characteristic
3#16863757C211yesmaximal, characteristic, complemented
4#16863730C211yesmaximal, characteristic, complemented
5#16863709C211yesmaximal, characteristic, complemented
6#16863674C211yesmaximal, characteristic, complemented
7#16863642C211yesmaximal, characteristic, complemented
8#16863636C211yesmaximal, characteristic, complemented
9#16863621C211yesmaximal, characteristic, complemented
10#16863558C211yesmaximal, characteristic, complemented
11#16863548C211yesmaximal, characteristic, complemented
12#16863493C211yesmaximal, characteristic, complemented
13#16863465C211yesmaximal, characteristic, complemented
14#102×D67C211yesmaximal, characteristic, complemented
15#417×C67C211yesmaximal, characteristic, complemented
16C2×D4×D67C2211yescharacteristic
17C2×#11761618C2211yescharacteristic, complemented
18C2×#11761617C2211yescharacteristic
19C2×#11761616C2211yescharacteristic
20C2×C8×D67C2211yescharacteristic
21#14524042C2211yescharacteristic, complemented
22#14524033—21no—
23#14524032C2211yescharacteristic
24#14524026—21no—
25#14524023C2211yescharacteristic
26#14524022—21no—
27#14524022C2211yescharacteristic, complemented
28#14524021—21no—
29#14524019C2211yescharacteristic
30#14524017C2211yescharacteristic
31#14524015—21no—
32#14524011—21no—
33#14524008C2211yescharacteristic, complemented
34D4×Dic67C2211yescharacteristic
35C4×C22⋊D67—21no—
36C4×C22⋊D67C2211yescharacteristic
37C4⋊C4×D67C2211yescharacteristic
38C2×(C2×D67)⋊C4—21no—
39#14523977C2211yescharacteristic, complemented
40#14523976C2211yescharacteristic, complemented
41#14523976C2211yescharacteristic, 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
53C2×M16×C67C2211yescharacteristic, complemented
54C2×C4⋊C4×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×D67—21no—
70C2×C4×D67C2311yescharacteristic
71#11761618—41no—
72#11761617—41no—
73#11761616—22no—
74C2×C67⋊C8C2311yescharacteristic
75C2×C67⋊C8C2311yescharacteristic
76C8×D67—41no—
77C4⋊Dic67—21no—
78C4⋊Dic67C2311yescharacteristic
79C4⋊Dic67C2311yescharacteristic
80Dic67⋊C4—21no—
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4D411yescharacteristic
86C22⋊Dic67—21no—
87C22⋊Dic67—21no—
88C22⋊Dic67D411yescharacteristic
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94(C2×D67)⋊C4D411yescharacteristic
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×C67—21no—
104C22×C4×C67C2311yescharacteristic
105M16×C67—41no—
106C2×C8×C67C2311yescharacteristic
107C2×C8×C67C2311yescharacteristic
108C4⋊C4×C67—21no—
109C4⋊C4×C67C2311yescharacteristic
110C4⋊C4×C67C2311yescharacteristic
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×C67—81no—
130C2×C4×C67C2×D411yescharacteristic
131C2×C4×C67C2411yescharacteristic
132C22⋊D67—22no—
133C22⋊D67—42no—
134C22⋊D67—81no—
135C2×Dic67—21no—
136C2×Dic67—21no—
137C2×Dic67—21no—
138C2×Dic67—81no—
139C2×Dic67C2×D411yescharacteristic
140C2×Dic67C2×D411yescharacteristic
141C2×Dic67C2×D411yescharacteristic
142D268—21no—
143D268—21no—
144C4×D67—21no—
145C4×D67—21no—
146C4×D67—81no—
147Dic134—21no—
148Dic134—21no—
149Dic134—81no—
150C8×C67—21no—
151C8×C67—21no—
152C67⋊C8—21no—
153C67⋊C8—21no—
154C22×C67—22no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—21no—
158C22×C67—41no—
159C22×C67—41no—
160C22×C67C22×D411yescharacteristic
161C2×D67—21no—
162C2×D67—21no—
163C2×D67—41no—
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—42no—
181C2×C67—41no—
182C2×C67#54611yescharacteristic
183C2×C67C2×#13711yescharacteristic
184C2×C67#57611yescharacteristic
185D67—22no—
186D67—81no—
187#3314—671nomaximal, Sylow
188C67#331411yescharacteristic, complemented, Sylow
189#554—671no—
190#546—671no—
191#489—671no—
192#484—671no—
193#480—671no—
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#417D6711yescharacteristic, complemented
203C2×#102—671no—
204C2×#41C2×D6711yescharacteristic
205C22×D4—671no—
206C2×Q16—671no—
207C2×SD16—671no—
208C2×SD16—671no—
209C2×M16C2×D6711yescharacteristic, complemented
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—671no—
221#121—1341no—
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#107—671no—
232#107—671no—
233C4⋊C8—671no—
234#103—671no—
235#103—671no—
236#103C2×D6711yescharacteristic, complemented
237#103C2×D6711yescharacteristic, complemented
238#102—671no—
239#102—671no—
240#102—671no—
241#102—671no—
242#102C2×D6711yescharacteristic
243#102C2×D6711yescharacteristic
244C22⋊C8—671no—
245C22⋊C8—671no—
246#97—671no—
247C24—1341no—
248#41—22no—
249#41—41no—
250C2×Q8—671no—
251C2×Q8—1341no—
252C2×Q8C22×D6711yescharacteristic
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—21no—
263C22×C4—671no—
264C22×C4—1341no—
265C22×C4—1341no—
266C22×C4C22×D6711yescharacteristic
267Q16—2681no—
268SD16—2681no—
269SD16—1342no—
270M16—41no—
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—671no—
280C4⋊C4—1341no—
281C4⋊C4—1341no—
282C4⋊C4—1341no—
283C4⋊C4—1341no—
284C4⋊C4—1341no—
285C4⋊C4—1341no—
286C4⋊C4C22×D6711yescharacteristic
287C4⋊C4C22×D6711yescharacteristic
288C22⋊C4—671no—
289C22⋊C4—671no—
290C22⋊C4—1341no—
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—21no—
307Q8—21no—
308Q8—5361no—
309Q8—1341no—
310Q8—1341no—
311D4—21no—
312D4—42no—
313D4—2682no—
314D4—5361no—
315D4—1342no—
316D4—1341no—
317D4—1341no—
318D4#1176162812yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
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—22no—
344C22—21no—
345C22—21no—
346C22—41no—
347C22—41no—
348C22—2681no—
349C22—2681no—
350C22—2681no—
351C22—5364no—
352C22—672no—
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—
366C4C2×#1176162811yescharacteristic
367C4#1452408811yescharacteristic
368C2—41no—
369C2—42no—
370C2—5361no—
371C2—1342no—
372C2#1686403611yescharacteristic
373C2#137×D6711yescharacteristic
374C2#1686420011yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#417×C67C2
socleC22×C67C22×D4
hypercenter#417D67
hypocenterC67#3314
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #16864101C2semidirect
3. #16863757C2semidirect
4. #16863730C2semidirect
5. #16863709C2semidirect
6. #16863674C2semidirect
7. #16863642C2semidirect
8. #16863636C2semidirect
9. #16863621C2semidirect
10. #16863558C2semidirect
11. #16863548C2semidirect
12. #16863493C2semidirect
13. #16863465C2semidirect
14. #102×D67C2semidirect
15. #417×C67C2semidirect
17. C2×#11761618C22semidirect
21. #14524042C22semidirect
27. #14524022C22semidirect
33. #14524008C22semidirect
39. #14523977C22semidirect
40. #14523976C22semidirect
41. #14523976C22semidirect
42. Dic67⋊C8C22semidirect
43. #14523968C22semidirect
44. #14523967C22semidirect
49. C22⋊(C67⋊C8)C22semidirect
51. #14523953C22semidirect
53. C2×M16×C67C22semidirect
54. C2×C4⋊C4×C67C22semidirect
55. #103×C67C22semidirect
56. #103×C67C22semidirect
188. C67#3314semidirect
202. #417D67semidirect
209. C2×M16C2×D67semidirect
228. C2×C4⋊C4C2×D67semidirect
236. #103C2×D67semidirect
237. #103C2×D67semidirect