Finite Group Databasebeta

← #496×D67 #22570639

Subgroups — #496×D67 #22570639

Order 8576.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×SD16×D67C212yesmaximal, complemented
2C2×D8×D67C212yesmaximal, complemented
3#127×D67C211yesmaximal, characteristic, complemented
4#124×D67C211yesmaximal, characteristic, complemented
5#16863786C211yesmaximal, characteristic, complemented
6#16863785C211yesmaximal, characteristic, complemented
7#16863784C212yesmaximal, complemented
8#16863783C211yesmaximal, characteristic, complemented
9#16863782C212yesmaximal, complemented
10#97×D67C211yesmaximal, characteristic, complemented
11#496×C67C211yesmaximal, characteristic, complemented
12C2×Q8×D67C2211yescharacteristic
13C2×D4×D67—22no—
14C2×D4×D67C2211yescharacteristic
15C2×D4×D67C2212yes—
16C2×#11761617C2212yescomplemented
17C2×#11761616C2212yescomplemented
18C2×#11761615C2212yescomplemented
19SD16×D67—28no—
20C2×D536C2212yescomplemented
21C2×#11761613C2212yescomplemented
22C2×#11761613C2212yescomplemented
23D8×D67—28no—
24C2×C8×D67C2212yes—
25#14524040C2211yescharacteristic, complemented
26#14524039—22no—
27#14524039C2212yescomplemented
28#14524032C2211yescharacteristic, complemented
29#14524031C2211yescharacteristic, complemented
30#14524030C2211yescharacteristic, complemented
31#14524029—22no—
32C22⋊C4×D67—22no—
33C42×D67C2211yescharacteristic
34#14523954C2211yescharacteristic, complemented
35#14523953C2212yescomplemented
36C2×SD16×C67C2212yescomplemented
37C2×D8×C67C2212yescomplemented
38#127×C67C2211yescharacteristic, complemented
39#124×C67C2211yescharacteristic, complemented
40#97×C67C2211yescharacteristic, complemented
41C23×D67—21no—
42C23×D67—22no—
43C2×Dic134—21no—
44C2×Dic134C2311yescharacteristic
45Q8×D67—24no—
46C2×D268—22no—
47C2×D268C2311yescharacteristic
48C2×D268C2312yes—
49C2×C22⋊D67—21no—
50C2×C22⋊D67—22no—
51C2×C22⋊D67—24no—
52D4×D67—24no—
53D4×D67—24no—
54D4×D67—24no—
55D4×D67—48no—
56C2×C4×D67—21no—
57C2×C4×D67D412yes—
58C2×C4×D67C2311yescharacteristic
59#11761617—24no—
60#11761616—24no—
61#11761615—24no—
62D536—24no—
63#11761613—24no—
64#11761613—24no—
65C2×C67⋊C8C2312yes—
66C8×D67D418yescomplemented
67C22⋊Dic67—22no—
68(C2×D67)⋊C4—22no—
69(C2×D67)⋊C4—22no—
70C4×Dic67D412yes—
71C4×Dic67C2311yescharacteristic
72C2×Q8×C67C2311yescharacteristic
73C2×D4×C67—22no—
74C2×D4×C67C2311yescharacteristic
75C2×D4×C67C2312yes—
76SD16×C67—24no—
77D8×C67—24no—
78C2×C8×C67C2312yes—
79C22⋊C4×C67—22no—
80C42×C67C2311yescharacteristic
81C23×C67—21no—
82C23×C67—22no—
83C22×D67—21no—
84C22×D67—22no—
85C22×D67—42no—
86C22×D67—44no—
87C22×D67—48no—
88C22×D67—44no—
89C22×D67C2×D411yescharacteristic
90Q8×C67—22no—
91D4×C67—22no—
92D4×C67—22no—
93D4×C67—22no—
94D4×C67—44no—
95C2×C4×C67—21no—
96C2×C4×C67C2×D412yes—
97C2×C4×C67C2411yescharacteristic
98C22⋊D67—48no—
99C22⋊D67—42no—
100C22⋊D67—44no—
101C2×Dic67—21no—
102C2×Dic67C2×D412yes—
103C2×Dic67C2×D411yescharacteristic
104D268—22no—
105D268—22no—
106D268—22no—
107D268—44no—
108C4×D67—24no—
109C4×D67—42no—
110C4×D67C2×D414yes—
111Dic134—22no—
112Dic134—42no—
113C8×C67C2×D414yescomplemented
114C67⋊C8C2×D414yescomplemented
115C22×C67—41no—
116C22×C67—42no—
117C22×C67—44no—
118C22×C67—42no—
119C22×C67C22×D411yescharacteristic
120C2×D67—41no—
121C2×D67—42no—
122C2×D67—44no—
123C2×D67—42no—
124C2×D67—88no—
125C2×D67—84no—
126C2×D67#12712yes—
127C2×D67AGL(1,ℤ/8)14yes—
128C4×C67—22no—
129C4×C67—41no—
130C4×C67C22×D412yes—
131Dic67—22no—
132Dic67—41no—
133Dic67#12712yes—
134C2×C67—82no—
135C2×C67—81no—
136C2×C67C2×#12711yescharacteristic
137C2×C67C2×AGL(1,ℤ/8)12yes—
138D67—82no—
139D67—81no—
140D67#49614yesdirect product factor
141C2×#496—671nomaximal, Sylow
142C67C2×#49611yescharacteristic, complemented, Sylow
143C22×SD16—672no—
144C22×D8—672no—
145C2×#127—671no—
146C2×#124—671no—
147#496—672no—
148#496—672no—
149#496—671no—
150#496—671no—
151#496—671no—
152#496D6711yescharacteristic, direct product factor
153C2×#97—671no—
154C22×Q8—671no—
155C22×D4—671no—
156C22×D4—672no—
157C22×D4—1342no—
158C2×SD16—672no—
159C2×SD16—672no—
160C2×SD16—672no—
161C2×SD16—1348no—
162C2×SD16C2×D6712yescomplemented
163C2×D8—672no—
164C2×D8—672no—
165C2×D8—672no—
166C2×D8—1348no—
167C2×D8C2×D6712yescomplemented
168C22×C8—672no—
169#127—672no—
170#127—671no—
171#127—1342no—
172#127C2×D6711yescharacteristic, complemented
173#124—671no—
174#124—671no—
175#124—671no—
176#124—1342no—
177#124C2×D6711yescharacteristic, complemented
178C2×C22⋊C4—1342no—
179C2×C42—671no—
180#97—672no—
181#97—671no—
182#97C2×D6711yescharacteristic, complemented
183C24—1341no—
184C24—1342no—
185C2×Q8—671no—
186C2×Q8—1341no—
187C2×Q8—1344no—
188C2×Q8C22×D6711yescharacteristic
189C2×D4—22no—
190C2×D4—2688no—
191C2×D4—671no—
192C2×D4—672no—
193C2×D4—1344no—
194C2×D4—1344no—
195C2×D4—1344no—
196C2×D4—1341no—
197C2×D4—1344no—
198C2×D4—1342no—
199C2×D4—1342no—
200C2×D4C22×D6711yescharacteristic
201C2×D4C22×D6712yes—
202C22×C4—671no—
203C22×C4—672no—
204C22×C4—1341no—
205SD16—24no—
206SD16—1344no—
207SD16—1344no—
208SD16—1344no—
209D8—24no—
210D8—1344no—
211D8—1344no—
212D8—1344no—
213C2×C8—672no—
214C2×C8—678no—
215C2×C8C22×D6712yes—
216C22⋊C4—22no—
217C22⋊C4—1342no—
218C22⋊C4—1342no—
219C22⋊C4—1342no—
220C42—672no—
221C42—671no—
222C42C22×D6711yescharacteristic
223C23—21no—
224C23—22no—
225C23—2688no—
226C23—2682no—
227C23—2684no—
228C23—2684no—
229C23—671no—
230C23—1341no—
231C23—1342no—
232Q8—22no—
233Q8—2682no—
234Q8—1342no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—2684no—
240D4—2682no—
241D4—2684no—
242D4—2688no—
243D4—1342no—
244D4—1342no—
245D4—1342no—
246C2×C4—21no—
247C2×C4—2682no—
248C2×C4—671no—
249C2×C4—674no—
250C2×C4—672no—
251C2×C4—1344no—
252C2×C4—1341no—
253C2×C4D4×D6712yes—
254C2×C4C23×D6711yescharacteristic
255C8—674no—
256C8D4×D6714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—2684no—
262C22—2682no—
263C22—2681no—
264C22—2682no—
265C22—5368no—
266C22—5364no—
267C22—674no—
268C22—672no—
269C22C2×D4×D6711yescharacteristic
270C4—22no—
271C4—41no—
272C4—2681no—
273C4—672no—
274C4—1342no—
275C4C2×D4×D6712yes—
276C2—82no—
277C2—81no—
278C2—5362no—
279C2—5361no—
280C2—674no—
281C2#127×D6711yescharacteristic
282C2AGL(1,ℤ/8)×D6712yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#496×C67C2
socleC22×C67C22×D4
hypercenter#496D67
hypocenterC67C2×#496
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D67C2semidirect
2. C2×D8×D67C2semidirect
3. #127×D67C2semidirect
4. #124×D67C2semidirect
5. #16863786C2semidirect
6. #16863785C2semidirect
7. #16863784C2semidirect
8. #16863783C2semidirect
9. #16863782C2semidirect
10. #97×D67C2semidirect
11. #496×C67C2semidirect
16. C2×#11761617C22semidirect
17. C2×#11761616C22semidirect
18. C2×#11761615C22semidirect
20. C2×D536C22semidirect
21. C2×#11761613C22semidirect
22. C2×#11761613C22semidirect
25. #14524040C22semidirect
27. #14524039C22semidirect
28. #14524032C22semidirect
29. #14524031C22semidirect
30. #14524030C22semidirect
34. #14523954C22semidirect
35. #14523953C22semidirect
36. C2×SD16×C67C22semidirect
37. C2×D8×C67C22semidirect
38. #127×C67C22semidirect
39. #124×C67C22semidirect
40. #97×C67C22semidirect
66. C8×D67D4semidirect
113. C8×C67C2×D4semidirect
114. C67⋊C8C2×D4semidirect
140. D67#496direct (N × H)
142. C67C2×#496semidirect
152. #496D67direct (N × H)
162. C2×SD16C2×D67semidirect
167. C2×D8C2×D67semidirect
172. #127C2×D67semidirect
177. #124C2×D67semidirect
182. #97C2×D67semidirect
256. C8D4×D67semidirect