Finite Group Databasebeta

← #22571991

Subgroups — #22571991

Order 8576.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864210C211yesmaximal, characteristic, complemented
2#16864207C212yesmaximal, complemented
3#16863779C211yesmaximal, characteristic, complemented
4#16863671C211yesmaximal, characteristic, complemented
5#16863663C211yesmaximal, characteristic, complemented
6#16863657C212yesmaximal, complemented
7#16863574C212yesmaximal, complemented
8#16863567C212yesmaximal, complemented
9#16863513C211yesmaximal, characteristic, complemented
10#16863511C211yesmaximal, characteristic, complemented
11#576×C67C211yesmaximal, characteristic, complemented
12#14524090C2212yescomplemented
13#14524073—22no—
14#14524073C2212yes—
15#14524073C2212yescomplemented
16#14524072—22no—
17#14524069—22no—
18#14524069C2212yescomplemented
19#14524063—22no—
20#14524063C2211yescharacteristic
21#14524062—21no—
22C2×Dic268C2211yescharacteristic, complemented
23Q16×D67—21no—
24C2×#11761616—22no—
25C2×#11761616C2212yescomplemented
26#14524050C2212yescomplemented
27#14524049C2211yescharacteristic, complemented
28#14524030C2212yescomplemented
29#14523979C2211yescharacteristic, complemented
30#14523978C2211yescharacteristic, complemented
31#14523970C2211yescharacteristic
32#14523970C2212yescomplemented
33#14523969C2211yescharacteristic, complemented
34#14523961C2212yescomplemented
35#14523960C2212yescomplemented
36C8×Dic67C2211yescharacteristic, complemented
37H(5,𝔽2)×C67C2212yes—
38AGL(1,ℤ/8)×C67—22no—
39AGL(1,ℤ/8)×C67C2212yes—
40#135×C67—21no—
41#135×C67C2211yescharacteristic
42C2×D8×C67—21no—
43C2×D8×C67C2211yescharacteristic
44#131×C67C2211yescharacteristic
45#11761628—22no—
46#11761628C2311yescharacteristic
47#11761627—22no—
48#11761627—42no—
49C2×Dic134—22no—
50C2×Dic134C2312yes—
51Q8×D67—22no—
52Dic268—21no—
53Dic268—21no—
54#11761618—22no—
55#11761618—22no—
56#11761618D411yescharacteristic
57#11761616—22no—
58#11761616—22no—
59#11761616—22no—
60#11761616—22no—
61#11761616—22no—
62#11761616D411yescharacteristic
63#11761615D411yescharacteristic
64#11761613—22no—
65#11761613D411yescharacteristic
66#11761612—22no—
67#11761612C2311yescharacteristic
68#11761612C2312yes—
69#11761611—21no—
70C2×C67⋊C8—22no—
71C2×C67⋊C8C2311yescharacteristic
72C8×D67—21no—
73C22⋊Dic67—42no—
74C4×Dic67C2311yescharacteristic
75#41×C67—22no—
76#41×C67—22no—
77#41×C67C2311yescharacteristic
78#41×C67C2312yes—
79C2×D4×C67—22no—
80C2×D4×C67—22no—
81C2×D4×C67—42no—
82C2×D4×C67C2312yes—
83Q16×C67D411yescharacteristic
84SD16×C67—22no—
85SD16×C67D412yes—
86D8×C67—22no—
87D8×C67—21no—
88D8×C67—21no—
89D8×C67D411yescharacteristic
90M16×C67—21no—
91M16×C67C2311yescharacteristic
92C2×C8×C67—21no—
93C2×C8×C67C2311yescharacteristic
94C23×C67—22no—
95C23×C67—42no—
96Q8×C67C2×D411yescharacteristic
97Q8×C67C2×D412yes—
98D4×C67—21no—
99D4×C67—22no—
100D4×C67—22no—
101D4×C67—22no—
102D4×C67—42no—
103D4×C67—42no—
104D4×C67C2×D411yescharacteristic
105D4×C67C2×D412yes—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—42no—
109C2×C4×C67C2411yescharacteristic
110C22⋊D67—21no—
111C22⋊D67—42no—
112C2×Dic67—21no—
113C2×Dic67—22no—
114C2×Dic67—42no—
115D268C2×D411yescharacteristic
116C4×D67—21no—
117C4×D67—42no—
118Dic134—22no—
119Dic134—22no—
120Dic134—42no—
121Dic134C2×D411yescharacteristic
122C8×C67—21no—
123C8×C67C2×D411yescharacteristic
124C8×C67C2×D411yescharacteristic
125C67⋊C8—21no—
126C67⋊C8—22no—
127C67⋊C8C2×D411yescharacteristic
128C67⋊C8C2×D411yescharacteristic
129C22×C67—22no—
130C22×C67—21no—
131C22×C67—42no—
132C22×C67—82no—
133C22×C67—162no—
134C22×C67H(5,𝔽2)11yescharacteristic
135C2×D67—21no—
136C4×C67—22no—
137C4×C67—21no—
138C4×C67C22×D411yescharacteristic
139C4×C67C22×D411yescharacteristic
140Dic67—21no—
141Dic67—41no—
142Dic67—42no—
143C2×C67—21no—
144C2×C67—42no—
145C2×C67—41no—
146C2×C67—82no—
147C2×C67D4211yescharacteristic
148D67—41no—
149#3290—671nomaximal, Sylow
150C67#329011yescharacteristic, complemented, Sylow
151#578—671no—
152#577—672no—
153#576D6711yescharacteristic, complemented
154#495—671no—
155#473—671no—
156#472—671no—
157#471—672no—
158#455—672no—
159#454—672no—
160#443—671no—
161#443—671no—
1622−1+4—672no—
163H(5,𝔽2)C2×D6712yes—
164#137—672no—
165#137—672no—
166#137—1342no—
167#137—1342no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—672no—
170AGL(1,ℤ/8)—1342no—
171AGL(1,ℤ/8)C2×D6712yes—
172#135—21no—
173#135—671no—
174#135—1342no—
175#135—1341no—
176#135C2×D6711yescharacteristic
177C2×Q16—671no—
178C2×Q16—1341no—
179C2×SD16—672no—
180C2×SD16—1342no—
181C2×D8—21no—
182C2×D8C2×D6711yescharacteristic
183#131—672no—
184#131—671no—
185#131C2×D6711yescharacteristic
186#124—672no—
187#108—671no—
188#108—671no—
189#104—671no—
190#104—672no—
191#104—671no—
192#101—672no—
193#100—672no—
194C4×C8—671no—
195#41—22no—
196#41—22no—
197#41—2682no—
198#41—671no—
199#41—1342no—
200#41—1342no—
201#41C22×D6711yescharacteristic
202#41C22×D6712yes—
203C2×Q8—672no—
204C2×Q8—1342no—
205C2×Q8—1342no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D6712yes—
210Q16—671no—
211Q16—1342no—
212Q16—1342no—
213Q16—1341no—
214Q16—1341no—
215Q16C22⋊D6711yescharacteristic
216SD16—22no—
217SD16—671no—
218SD16—671no—
219SD16—1342no—
220SD16—1342no—
221SD16—1342no—
222SD16—1342no—
223SD16—1342no—
224SD16C22⋊D6712yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—671no—
229D8—1342no—
230D8C22⋊D6711yescharacteristic
231M16—21no—
232M16—671no—
233M16—672no—
234M16—1342no—
235M16—1341no—
236M16C22×D6711yescharacteristic
237C2×C8—21no—
238C2×C8—671no—
239C2×C8—1342no—
240C2×C8—1341no—
241C2×C8C22×D6711yescharacteristic
242C22⋊C4—2682no—
243C42—671no—
244C23—22no—
245C23—42no—
246Q8—2682no—
247Q8—671no—
248Q8—1342no—
249Q8—1342no—
250Q8D4×D6711yescharacteristic
251Q8C2×C22⋊D6712yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—2682no—
259D4—671no—
260D4—1341no—
261D4D4×D6711yescharacteristic
262D4C2×C22⋊D6712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—2682no—
267C2×C4—2682no—
268C2×C4—1341no—
269C2×C4—1342no—
270C2×C4—1341no—
271C2×C4C23×D6711yescharacteristic
272C8—21no—
273C8—671no—
274C8—671no—
275C8—1342no—
276C8—1341no—
277C8C2×C22⋊D6711yescharacteristic
278C8C2×C22⋊D6711yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—1341no—
285C22#1452408811yescharacteristic
286C4—21no—
287C4—22no—
288C4—2681no—
289C4—2682no—
290C4—1341no—
291C4C2×D4×D6711yescharacteristic
292C4C22×C22⋊D6711yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—2681no—
298C2D4×C22⋊D6711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#576×C67C2
socleC2×C67D42
hypercenter#576D67
hypocenterC67#3290
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #16864210C2semidirect
2. #16864207C2semidirect
3. #16863779C2semidirect
4. #16863671C2semidirect
5. #16863663C2semidirect
6. #16863657C2semidirect
7. #16863574C2semidirect
8. #16863567C2semidirect
9. #16863513C2semidirect
10. #16863511C2semidirect
11. #576×C67C2semidirect
12. #14524090C22semidirect
15. #14524073C22semidirect
18. #14524069C22semidirect
22. C2×Dic268C22semidirect
25. C2×#11761616C22semidirect
26. #14524050C22semidirect
27. #14524049C22semidirect
28. #14524030C22semidirect
29. #14523979C22semidirect
30. #14523978C22semidirect
32. #14523970C22semidirect
33. #14523969C22semidirect
34. #14523961C22semidirect
35. #14523960C22semidirect
36. C8×Dic67C22semidirect
150. C67#3290semidirect
153. #576D67semidirect