Finite Group Databasebeta

← #22571556

Subgroups — #22571556

Order 8576.

360 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863930C211yesmaximal, characteristic, complemented
2#16863703C211yesmaximal, characteristic, complemented
3#16863700C211yesmaximal, characteristic, complemented
4#16863696C211yesmaximal, characteristic, complemented
5#16863693C211yesmaximal, characteristic, complemented
6#16863635C211yesmaximal, characteristic, complemented
7#16863630C211yesmaximal, characteristic
8#16863627C211yesmaximal, characteristic, complemented
9#16863623C211yesmaximal, characteristic, complemented
10#16863618C211yesmaximal, characteristic, complemented
11#16863615C211yesmaximal, characteristic, complemented
12#16863611C211yesmaximal, characteristic, complemented
13#16863608C211yesmaximal, characteristic, complemented
14C2×C4⋊(C67⋊C8)C211yesmaximal, characteristic, complemented
15#529×C67C211yesmaximal, characteristic, complemented
16C2×#11761618C2211yescharacteristic
17C2×#11761616C2211yescharacteristic
18C2×#11761615C2211yescharacteristic
19C2×#11761613C2211yescharacteristic
20C22×C67⋊C8C2211yescharacteristic
21C22×C67⋊C8C2211yescharacteristic
22#14524037—21no—
23#14524035C2211yescharacteristic
24#14524031C2211yescharacteristic
25#14524026—21no—
26#14524022C2211yescharacteristic
27#14524017C2211yescharacteristic
28C4×Dic134C2211yescharacteristic
29C4×D268C2211yescharacteristic
30C4×C22⋊D67—21no—
31#14524000—21no—
32#14523976C2211yescharacteristic
33#14523976C2211yescharacteristic
34#14523974C2211yescharacteristic
35#14523974C2211yescharacteristic
36C4⋊(C67⋊C8)C2211yescharacteristic
37C4⋊(C67⋊C8)C2211yescharacteristic
38C4⋊(C67⋊C8)C2211yescharacteristic
39C4⋊(C67⋊C8)C2211yescharacteristic
40#14523967C2211yescharacteristic
41#14523967C2211yescharacteristic
42#14523966C2211yescharacteristic
43#14523966C2211yescharacteristic
44#14523964C2211yescharacteristic
45#14523964C2211yescharacteristic
46#14523963C2211yescharacteristic
47#14523963C2211yescharacteristic
48#126×C67—21no—
49#125×C67C2211yescharacteristic
50#124×C67C2211yescharacteristic
51#123×C67—21no—
52C22⋊Q8×C67C2211yescharacteristic
53#121×C67C2211yescharacteristic
54C4×Q8×C67C2211yescharacteristic
55C4×D4×C67—21no—
56C4×D4×C67C2211yescharacteristic
57#117×C67—21no—
58C2×C42×C67C2211yescharacteristic
59C2×Dic134C2311yescharacteristic
60C2×D268C2311yescharacteristic
61C2×C22⋊D67—21no—
62C2×C4×D67—21no—
63#11761618—41no—
64#11761616—41no—
65#11761615—41no—
66#11761613—41no—
67C2×C67⋊C8—41no—
68C2×C67⋊C8—41no—
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C2×C67⋊C8C2311yescharacteristic
72C2×C67⋊C8C2311yescharacteristic
73C4⋊Dic67C2311yescharacteristic
74C4⋊Dic67C2311yescharacteristic
75Dic67⋊C4—21no—
76Dic67⋊C4—21no—
77Dic67⋊C4—21no—
78Dic67⋊C4—21no—
79C22⋊Dic67—21no—
80(C2×D67)⋊C4—21no—
81(C2×D67)⋊C4—21no—
82(C2×D67)⋊C4—21no—
83(C2×D67)⋊C4—21no—
84C4×Dic67—21no—
85C2×Q8×C67C2311yescharacteristic
86C2×D4×C67—21no—
87C2×D4×C67C2311yescharacteristic
88C22×C4×C67—21no—
89C22×C4×C67D411yescharacteristic
90C22×C4×C67D411yescharacteristic
91C22×C4×C67C2311yescharacteristic
92C4⋊C4×C67—21no—
93C4⋊C4×C67—21no—
94C4⋊C4×C67—21no—
95C4⋊C4×C67—21no—
96C4⋊C4×C67C2311yescharacteristic
97C4⋊C4×C67C2311yescharacteristic
98C22⋊C4×C67—21no—
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67—21no—
102C22⋊C4×C67—21no—
103C42×C67—21no—
104C42×C67D411yescharacteristic
105C42×C67D411yescharacteristic
106C42×C67C2311yescharacteristic
107C42×C67C2311yescharacteristic
108C23×C67—21no—
109C23×C67C2×D411yescharacteristic
110C22×D67—21no—
111Q8×C67—21no—
112Q8×C67—21no—
113D4×C67—21no—
114D4×C67—21no—
115D4×C67—42no—
116C2×C4×C67—22no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—41no—
123C2×C4×C67—42no—
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2×D411yescharacteristic
127C2×C4×C67C2×D411yescharacteristic
128C2×C4×C67C2×D411yescharacteristic
129C2×C4×C67C2411yescharacteristic
130C22⋊D67—42no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134D268—21no—
135D268—21no—
136C4×D67—42no—
137Dic134—21no—
138Dic134—21no—
139C67⋊C8—21no—
140C67⋊C8—21no—
141C67⋊C8—21no—
142C67⋊C8—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—41no—
146C22×C67—41no—
147C22×C67—41no—
148C22×C67#13512yes—
149C22×C67C22×D411yescharacteristic
150C2×D67—41no—
151C2×D67—41no—
152C2×D67—41no—
153C4×C67—21no—
154C4×C67—21no—
155C4×C67—21no—
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67H(5,𝔽2)11yescharacteristic
162C4×C672−1+411yescharacteristic
163Dic67—41no—
164Dic67—41no—
165Dic67—41no—
166C2×C67—22no—
167C2×C67—81no—
168C2×C67#53711yescharacteristic
169C2×C67C2×#13511yescharacteristic
170C2×C67#57511yescharacteristic
171D67—81no—
172#3250—671nomaximal, Sylow
173C67#325011yescharacteristic, complemented, Sylow
174#529—671no—
175#529D6711yescharacteristic, complemented
176#479—671no—
177#479—671no—
178#478—671no—
179#478—671no—
180#467—671no—
181#466—671no—
182#465—671no—
183#465—671no—
184#464—671no—
185#464—671no—
186#463—671no—
187#463—671no—
188C2×C4⋊C8—671no—
189C2×Q16—671no—
190C2×SD16—671no—
191C2×SD16—671no—
192C2×D8—671no—
193C22×C8—671no—
194C22×C8—671no—
195#126—21no—
196#126—1341no—
197#125—671no—
198#125C2×D6711yescharacteristic
199#124—671no—
200#124C2×D6711yescharacteristic
201#123—21no—
202#123—1341no—
203C22⋊Q8—671no—
204C22⋊Q8C2×D6711yescharacteristic
205#121—671no—
206#121C2×D6711yescharacteristic
207C4×Q8—671no—
208C4×Q8C2×D6711yescharacteristic
209C4×D4—21no—
210C4×D4—671no—
211C4×D4—1341no—
212C4×D4C2×D6711yescharacteristic
213#117—21no—
214#117—1341no—
215C2×C42C2×D6711yescharacteristic
216#107—671no—
217#107—671no—
218#106—671no—
219#106—671no—
220C4⋊C8—671no—
221C4⋊C8—671no—
222C4⋊C8—671no—
223C4⋊C8—671no—
224#103—671no—
225#103—671no—
226#103—671no—
227#103—671no—
228#102—671no—
229#102—671no—
230#102—671no—
231#102—671no—
232C2×Q8—671no—
233C2×Q8C22×D6711yescharacteristic
234C2×D4—21no—
235C2×D4—671no—
236C2×D4—1341no—
237C2×D4C22×D6711yescharacteristic
238C22×C4—21no—
239C22×C4—1341no—
240C22×C4C22⋊D6711yescharacteristic
241C22×C4C22⋊D6711yescharacteristic
242C22×C4C22×D6711yescharacteristic
243Q16—2681no—
244SD16—2681no—
245SD16—2681no—
246D8—2681no—
247C2×C8—2681no—
248C2×C8—2681no—
249C2×C8—671no—
250C2×C8—671no—
251C2×C8—671no—
252C2×C8—671no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—671no—
258C4⋊C4—671no—
259C4⋊C4—1341no—
260C4⋊C4—1341no—
261C4⋊C4—1341no—
262C4⋊C4—1341no—
263C4⋊C4C22×D6711yescharacteristic
264C4⋊C4C22×D6711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1341no—
271C22⋊C4—1341no—
272C22⋊C4—1341no—
273C22⋊C4—1341no—
274C22⋊C4—1341no—
275C42—21no—
276C42—1341no—
277C42C22⋊D6711yescharacteristic
278C42C22⋊D6711yescharacteristic
279C42C22×D6711yescharacteristic
280C42C22×D6711yescharacteristic
281C23—21no—
282C23—1341no—
283C23C2×C22⋊D6711yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—1341no—
287Q8—1341no—
288D4—21no—
289D4—21no—
290D4—42no—
291D4—2682no—
292D4—1341no—
293D4—1341no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—2682no—
303C2×C4—1341no—
304C2×C4—1341no—
305C2×C4—1341no—
306C2×C4C2×C22⋊D6711yescharacteristic
307C2×C4C2×C22⋊D6711yescharacteristic
308C2×C4C2×C22⋊D6711yescharacteristic
309C2×C4C2×C22⋊D6711yescharacteristic
310C2×C4C2×C22⋊D6711yescharacteristic
311C2×C4C23×D6711yescharacteristic
312C8—1341no—
313C8—1341no—
314C8—1341no—
315C8—1341no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—2681no—
322C22—2681no—
323C22—2681no—
324C22#1452406312yes—
325C22C22×C22⋊D6711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—2681no—
335C4—2681no—
336C4—2681no—
337C4#1452408911yescharacteristic
338C4#1452409011yescharacteristic
339C2—22no—
340C2—81no—
341C2—5361no—
342C2#1686397411yescharacteristic
343C2C2×#1452406311yescharacteristic
344C2#1686419411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#529×C67C2
socleC22×C67C22×D4
hypercenter#529D67
hypocenterC67#3250
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #16863930C2semidirect
2. #16863703C2semidirect
3. #16863700C2semidirect
4. #16863696C2semidirect
5. #16863693C2semidirect
6. #16863635C2semidirect
8. #16863627C2semidirect
9. #16863623C2semidirect
10. #16863618C2semidirect
11. #16863615C2semidirect
12. #16863611C2semidirect
13. #16863608C2semidirect
14. C2×C4⋊(C67⋊C8)C2semidirect
15. #529×C67C2semidirect
173. C67#3250semidirect
175. #529D67semidirect