Finite Group Databasebeta

← #22570558

Subgroups — #22570558

Order 8576.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864023C211yesmaximal, characteristic, complemented
2#16863959C211yesmaximal, characteristic
3#16863760C211yesmaximal, characteristic, complemented
4#16863759C211yesmaximal, characteristic, complemented
5#16863758C211yesmaximal, characteristic, complemented
6#16863754C211yesmaximal, characteristic, complemented
7#16863751C211yesmaximal, characteristic, complemented
8#16863749C211yesmaximal, characteristic, complemented
9#16863741C211yesmaximal, characteristic, complemented
10#16863399C211yesmaximal, characteristic, complemented
11#16863394C211yesmaximal, characteristic, complemented
12#16863389C211yesmaximal, characteristic, complemented
13#102×D67C211yesmaximal, characteristic, complemented
14#16863350C211yesmaximal, characteristic, complemented
15#486×C67C211yesmaximal, characteristic, complemented
16C2×#11761627C2211yescharacteristic
17C2×D4×D67C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×C8×D67C2211yescharacteristic
20#14524042C2211yescharacteristic, complemented
21#14524041—21no—
22#14524039C2211yescharacteristic
23#14524035C2211yescharacteristic, complemented
24#14524034C2211yescharacteristic, complemented
25#14524033—21no—
26#14524031C2211yescharacteristic
27#14524030C2211yescharacteristic
28#14524021—21no—
29#14524019—21no—
30#14524015—21no—
31#14524015—21no—
32#14524011—21no—
33#14524011—21no—
34#14524001C2211yescharacteristic
35#14524000C2211yescharacteristic
36C4⋊C4×D67C2211yescharacteristic
37#14523968C2211yescharacteristic, complemented
38#14523968C2211yescharacteristic, complemented
39#14523966C2211yescharacteristic, complemented
40#14523966C2211yescharacteristic, complemented
41#14523965C2211yescharacteristic
42#14523965C2211yescharacteristic
43#14523964C2211yescharacteristic
44#14523964C2211yescharacteristic
45#14523964C2211yescharacteristic, complemented
46#14523964C2211yescharacteristic, complemented
47#14523963C2211yescharacteristic
48#14523963C2211yescharacteristic
49#14523954C2211yescharacteristic, complemented
50#14523953C2211yescharacteristic, complemented
51C8×Dic67C2211yescharacteristic, complemented
52C4⋊Q8×C67C2211yescharacteristic, complemented
53#127×C67C2211yescharacteristic
54#102×C67C2211yescharacteristic
55#102×C67C2211yescharacteristic
56#102×C67C2211yescharacteristic, complemented
57#102×C67C2211yescharacteristic, complemented
58C4×C8×C67C2211yescharacteristic, complemented
59C23×D67—21no—
60#11761627—22no—
61#11761627—22no—
62C2×Dic134—21no—
63C2×Dic134C2311yescharacteristic
64C2×D268C2311yescharacteristic
65C2×C22⋊D67—21no—
66C2×C22⋊D67—21no—
67C2×C22⋊D67—21no—
68C2×C22⋊D67—21no—
69D4×D67—22no—
70D4×D67—22no—
71C22×Dic67—21no—
72C2×C4×D67—21no—
73C2×C4×D67C2311yescharacteristic
74#11761611—22no—
75C2×C67⋊C8C2311yescharacteristic
76C2×C67⋊C8C2311yescharacteristic
77C8×D67—22no—
78C4⋊Dic67—21no—
79C4⋊Dic67C2311yescharacteristic
80C4⋊Dic67C2311yescharacteristic
81Dic67⋊C4—21no—
82Dic67⋊C4—21no—
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86Dic67⋊C4D411yescharacteristic
87Dic67⋊C4D411yescharacteristic
88C22⋊Dic67—21no—
89C22⋊Dic67—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4D411yescharacteristic
94(C2×D67)⋊C4D411yescharacteristic
95C4×Dic67—21no—
96C4×Dic67C2311yescharacteristic
97C2×Q8×C67—21no—
98C2×D4×C67—21no—
99C2×D4×C67—21no—
100C2×D4×C67C2311yescharacteristic
101C2×D4×C67C2311yescharacteristic
102C2×C8×C67C2311yescharacteristic
103C2×C8×C67C2311yescharacteristic
104C4⋊C4×C67—21no—
105C4⋊C4×C67C2311yescharacteristic
106C4⋊C4×C67C2311yescharacteristic
107C42×C67C2311yescharacteristic
108C23×C67—21no—
109C23×C67—21no—
110C22×D67—21no—
111C22×D67—42no—
112C22×D67—81no—
113C22×D67—81no—
114C22×D67C2×D411yescharacteristic
115Q8×C67—81no—
116D4×C67—21no—
117D4×C67—21no—
118D4×C67—21no—
119D4×C67—21no—
120D4×C67—81no—
121D4×C67—81no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2411yescharacteristic
127C22⋊D67—42no—
128C22⋊D67—42no—
129C22⋊D67—81no—
130C22⋊D67—81no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—42no—
135C2×Dic67C2×D411yescharacteristic
136C2×Dic67C2×D411yescharacteristic
137C2×Dic67C2×D411yescharacteristic
138D268—21no—
139D268—21no—
140C4×D67—81no—
141C4×D67#4112yes—
142C4×D67#4112yes—
143Dic134—21no—
144Dic134—21no—
145Dic134—81no—
146C8×C67—21no—
147C8×C67—21no—
148C67⋊C8—21no—
149C67⋊C8—21no—
150C22×C67—41no—
151C22×C67—41no—
152C22×C67—41no—
153C22×C67—41no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67C22×D411yescharacteristic
157C2×D67—21no—
158C2×D67—21no—
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—41no—
162C2×D67—84no—
163C2×D67#12412yes—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67C2×#4111yescharacteristic
169C4×C67C2×#4111yescharacteristic
170Dic67—21no—
171Dic67—21no—
172Dic67—41no—
173Dic67—41no—
174Dic67—41no—
175Dic67#12412yes—
176C2×C67—81no—
177C2×C67—81no—
178C2×C67C2×#12411yescharacteristic
179C2×C67#57611yescharacteristic
180C2×C67#57711yescharacteristic
181D67—22no—
182D67—81no—
183#3136—671nomaximal, Sylow
184C67#313611yescharacteristic, complemented, Sylow
185#544—671no—
186#535—671no—
187#490—671no—
188#490—671no—
189#489—671no—
190#489—671no—
191#488—671no—
192#488—671no—
193#486—671no—
194#486D6711yescharacteristic, complemented
195#418—671no—
196#417—671no—
197#416—671no—
198C2×#102—671no—
199#405—671no—
200C2×#41—671no—
201C22×D4—671no—
202C2×M16—671no—
203C22×C8—671no—
204C4⋊Q8—671no—
205C4⋊Q8—1341no—
206C4⋊Q8C2×D6711yescharacteristic, complemented
207#127—671no—
208#127C2×D6711yescharacteristic
209#125—671no—
210#125—671no—
211#125—1341no—
212#124—671no—
213#124—671no—
214C22⋊Q8—1341no—
215C22⋊Q8—1341no—
216#121—1341no—
217#121—1341no—
218#120—1341no—
219#120—1341no—
220#117—671no—
221#117—671no—
222C2×C4⋊C4—671no—
223#103—671no—
224#103—671no—
225#103—671no—
226#103—671no—
227#102—671no—
228#102—671no—
229#102—671no—
230#102—671no—
231#102—671no—
232#102—671no—
233#102—671no—
234#102—671no—
235#102C2×D6711yescharacteristic
236#102C2×D6711yescharacteristic
237#102C2×D6711yescharacteristic, complemented
238#102C2×D6711yescharacteristic, complemented
239#97—671no—
240#97—671no—
241C4×C8—671no—
242C4×C8C2×D6711yescharacteristic, complemented
243C24—1341no—
244#41—1342no—
245#41—1342no—
246C2×Q8—21no—
247C2×Q8—671no—
248C2×Q8—1341no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—671no—
252C2×D4—1342no—
253C2×D4—1342no—
254C2×D4—1341no—
255C2×D4—1341no—
256C2×D4—1341no—
257C2×D4—1341no—
258C2×D4C22×D6711yescharacteristic
259C2×D4C22×D6711yescharacteristic
260C22×C4—671no—
261C22×C4—1341no—
262C22×C4—1341no—
263M16—1342no—
264C2×C8—671no—
265C2×C8—671no—
266C2×C8—1342no—
267C2×C8C22×D6711yescharacteristic
268C2×C8C22×D6711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—671no—
271C4⋊C4—671no—
272C4⋊C4—671no—
273C4⋊C4—671no—
274C4⋊C4—1341no—
275C4⋊C4—1341no—
276C4⋊C4—1341no—
277C4⋊C4—1341no—
278C4⋊C4—1341no—
279C4⋊C4—1341no—
280C4⋊C4C22×D6711yescharacteristic
281C4⋊C4C22×D6711yescharacteristic
282C22⋊C4—671no—
283C22⋊C4—671no—
284C22⋊C4—1341no—
285C22⋊C4—1341no—
286C22⋊C4—1341no—
287C22⋊C4—1341no—
288C22⋊C4—1341no—
289C42—671no—
290C42—1341no—
291C42C22×D6711yescharacteristic
292C23—21no—
293C23—21no—
294C23—2682no—
295C23—5361no—
296C23—5361no—
297C23—671no—
298C23—1341no—
299Q8—81no—
300Q8—5361no—
301Q8—1341no—
302Q8—1341no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—2682no—
310D4—2682no—
311D4—5361no—
312D4—5361no—
313D4—1341no—
314D4—1341no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—2682no—
318C2×C4—5361no—
319C2×C4—671no—
320C2×C4—672no—
321C2×C4—672no—
322C2×C4—671no—
323C2×C4—671no—
324C2×C4—1341no—
325C2×C4—1341no—
326C2×C4—1341no—
327C2×C4D4×D6711yescharacteristic
328C2×C4D4×D6711yescharacteristic
329C2×C4C23×D6711yescharacteristic
330C8—21no—
331C8—21no—
332C8—1341no—
333C8—1341no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—2681no—
341C22—2681no—
342C22—2681no—
343C22—5364no—
344C22—672no—
345C22—1341no—
346C22—1341no—
347C22C2×D4×D6711yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—2681no—
353C4—2681no—
354C4—2681no—
355C4—672no—
356C4—1341no—
357C4—1341no—
358C4#41×D6711yescharacteristic
359C4#41×D6711yescharacteristic
360C2—81no—
361C2—81no—
362C2—5361no—
363C2—1342no—
364C2#124×D6711yescharacteristic
365C2#1686419711yescharacteristic
366C2#1686420211yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#486×C67C2
socleC22×C67C22×D4
hypercenter#486D67
hypocenterC67#3136
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #16864023C2semidirect
3. #16863760C2semidirect
4. #16863759C2semidirect
5. #16863758C2semidirect
6. #16863754C2semidirect
7. #16863751C2semidirect
8. #16863749C2semidirect
9. #16863741C2semidirect
10. #16863399C2semidirect
11. #16863394C2semidirect
12. #16863389C2semidirect
13. #102×D67C2semidirect
14. #16863350C2semidirect
15. #486×C67C2semidirect
20. #14524042C22semidirect
23. #14524035C22semidirect
24. #14524034C22semidirect
37. #14523968C22semidirect
38. #14523968C22semidirect
39. #14523966C22semidirect
40. #14523966C22semidirect
45. #14523964C22semidirect
46. #14523964C22semidirect
49. #14523954C22semidirect
50. #14523953C22semidirect
51. C8×Dic67C22semidirect
52. C4⋊Q8×C67C22semidirect
56. #102×C67C22semidirect
57. #102×C67C22semidirect
58. C4×C8×C67C22semidirect
184. C67#3136semidirect
194. #486D67semidirect
206. C4⋊Q8C2×D67semidirect
237. #102C2×D67semidirect
238. #102C2×D67semidirect
242. C4×C8C2×D67semidirect