Finite Group Databasebeta

← #22571071

Subgroups — #22571071

Order 8576.

442 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 109 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#16863962C212yesmaximal, complemented
2#16863722C212yesmaximal, complemented
3#16863719C212yesmaximal, complemented
4#16863649C211yesmaximal, characteristic, complemented
5#16863630C212yesmaximal, complemented
6#16863531C212yesmaximal, complemented
7#16863528C212yesmaximal, complemented
8#16863364C211yesmaximal, characteristic, complemented
9#469×C67C211yesmaximal, characteristic, complemented
10C2×#11761628C2211yescharacteristic
11C2×D4×D67C2212yescomplemented
12C2×D536C2211yescharacteristic
13C2×#11761613C2212yes—
14C22×C67⋊C8C2211yescharacteristic, complemented
15#14524039—22no—
16#14524029—22no—
17#14524018C2212yes—
18#14524017C2212yescomplemented
19#14524014C2212yes—
20#14524013—22no—
21#14524011—22no—
22#14524001C2212yescomplemented
23#14523976C2212yescomplemented
24#14523975C2211yescharacteristic, complemented
25#14523965C2212yescomplemented
26#14523964C2212yescomplemented
27#14523963C2212yescomplemented
28(C2×D67)⋊C8C2212yescomplemented
29(C2×D67)⋊C8C2212yescomplemented
30C2×D8×C67C2211yescharacteristic
31C2×M16×C67C2211yescharacteristic, complemented
32#121×C67C2212yescomplemented
33#106×C67C2211yescharacteristic, complemented
34#102×C67C2212yescomplemented
35C23×D67—22no—
36#11761628—41no—
37#11761628—41no—
38C2×Dic134D411yescharacteristic
39C2×D268—22no—
40C2×D268D411yescharacteristic
41C2×D268C2312yes—
42C2×C22⋊D67—22no—
43C2×C22⋊D67—22no—
44C2×C22⋊D67—22no—
45C2×C22⋊D67D412yes—
46D4×D67—42no—
47D4×D67—42no—
48C2×C4×D67C2312yes—
49D536—41no—
50#11761613—42no—
51C2×C67⋊C8—41no—
52C2×C67⋊C8C2312yes—
53C4⋊Dic67C2312yes—
54C22⋊Dic67—22no—
55(C2×D67)⋊C4—22no—
56(C2×D67)⋊C4—22no—
57(C2×D67)⋊C4—22no—
58C4×Dic67—22no—
59C2×D4×C67—22no—
60C2×D4×C67C2312yes—
61C22×C4×C67C2311yescharacteristic
62D8×C67—41no—
63M16×C67—41no—
64C2×C8×C67C2311yescharacteristic
65C2×C8×C67C2311yescharacteristic
66C4⋊C4×C67C2312yes—
67C22⋊C4×C67—22no—
68C23×C67—22no—
69C23×C67C2×D411yescharacteristic
70C22×D67—22no—
71C22×D67—44no—
72C22×D67—82no—
73C22×D67—82no—
74C22×D67C2×D412yes—
75D4×C67—22no—
76D4×C67—22no—
77D4×C67—82no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—22no—
81C2×C4×C67C2×D411yescharacteristic
82C2×C4×C67C2411yescharacteristic
83C22⋊D67—42no—
84C22⋊D67—44no—
85C22⋊D67—82no—
86C22⋊D67—82no—
87C2×Dic67—22no—
88C2×Dic67C2×D412yes—
89D268—22no—
90D268—22no—
91D268—41no—
92D268—82no—
93C4×D67—22no—
94C4×D67—22no—
95Dic134—41no—
96C8×C67—21no—
97C8×C67—21no—
98C67⋊C8—22no—
99C22×C67—21no—
100C22×C67—22no—
101C22×C67—42no—
102C22×C67—42no—
103C22×C67—42no—
104C22×C67C22×D411yescharacteristic
105C2×D67—22no—
106C2×D67—22no—
107C2×D67—22no—
108C2×D67—42no—
109C2×D67—42no—
110C2×D67—42no—
111C2×D67—88no—
112C4×C67—21no—
113C4×C67—42no—
114C4×C67H(5,𝔽2)11yescharacteristic
115C4×C67H(5,𝔽2)11yescharacteristic
116Dic67—22no—
117Dic67—42no—
118C2×C67—41no—
119C2×C67—82no—
120C2×C67#53411yescharacteristic
121C2×C67#57612yes—
122D67—42no—
123D67—82no—
124#3199—671nomaximal, Sylow
125C67#319911yescharacteristic, complemented, Sylow
126#535—672no—
127#482—672no—
128#482—672no—
129#469—671no—
130#469D6711yescharacteristic, complemented
131#466—672no—
132#447—672no—
133#447—672no—
134#408—671no—
135C2×#41—671no—
136C22×D4—672no—
137C2×D8—672no—
138C2×D8—671no—
139C2×D8C2×D6711yescharacteristic
140C2×M16C2×D6711yescharacteristic, complemented
141C22×C8—671no—
142#127—1342no—
143#124—1342no—
144#121—672no—
145#121—672no—
146#121—672no—
147#121C2×D6712yescomplemented
148#120—1342no—
149#120—1342no—
150#117—672no—
151#107—672no—
152#106—671no—
153#106C2×D6711yescharacteristic, complemented
154#102—672no—
155#102—672no—
156#102—672no—
157#102C2×D6712yescomplemented
158C22⋊C8—672no—
159C22⋊C8—672no—
160C24—1342no—
161#41—2681no—
162#41—2681no—
163C2×Q8—671no—
164C2×D4—22no—
165C2×D4—2682no—
166C2×D4—2682no—
167C2×D4—672no—
168C2×D4—671no—
169C2×D4—672no—
170C2×D4—1342no—
171C2×D4—1342no—
172C2×D4—1342no—
173C2×D4—1342no—
174C2×D4C22×D6712yes—
175C22×C4—672no—
176C22×C4C22×D6711yescharacteristic
177D8—41no—
178D8—2681no—
179D8—2682no—
180M16—41no—
181C2×C8—2681no—
182C2×C8—672no—
183C2×C8C22×D6711yescharacteristic
184C2×C8C22×D6711yescharacteristic
185C4⋊C4—672no—
186C4⋊C4C22×D6712yes—
187C22⋊C4—22no—
188C22⋊C4—1342no—
189C22⋊C4—1342no—
190C22⋊C4—1342no—
191C22⋊C4—1342no—
192C42—1342no—
193C23—22no—
194C23—2684no—
195C23—5362no—
196C23—5362no—
197C23—672no—
198C23—1342no—
199C23D4×D6711yescharacteristic
200Q8—2681no—
201D4—22no—
202D4—22no—
203D4—82no—
204D4—2681no—
205D4—2682no—
206D4—2684no—
207D4—5362no—
208D4—5362no—
209D4—5362no—
210D4—1342no—
211D4—1342no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—672no—
216C2×C4—1342no—
217C2×C4—1342no—
218C2×C4—1342no—
219C2×C4D4×D6711yescharacteristic
220C2×C4C23×D6711yescharacteristic
221C8—21no—
222C8—21no—
223C8—1342no—
224C22—22no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—2682no—
230C22—2682no—
231C22—2682no—
232C22—5368no—
233C22—1342no—
234C22—1342no—
235C22—1342no—
236C22C2×D4×D6711yescharacteristic
237C4—21no—
238C4—42no—
239C4—2682no—
240C4—1342no—
241C4#1452408811yescharacteristic
242C4#1452408911yescharacteristic
243C2—41no—
244C2—82no—
245C2—2682no—
246C2—5362no—
247C2#1686395411yescharacteristic
248C2#1686420012yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#469×C67C2
socleC22×C67C22×D4
hypercenter#469D67
hypocenterC67#3199
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #16863962C2semidirect
2. #16863722C2semidirect
3. #16863719C2semidirect
4. #16863649C2semidirect
5. #16863630C2semidirect
6. #16863531C2semidirect
7. #16863528C2semidirect
8. #16863364C2semidirect
9. #469×C67C2semidirect
11. C2×D4×D67C22semidirect
14. C22×C67⋊C8C22semidirect
18. #14524017C22semidirect
22. #14524001C22semidirect
23. #14523976C22semidirect
24. #14523975C22semidirect
25. #14523965C22semidirect
26. #14523964C22semidirect
27. #14523963C22semidirect
28. (C2×D67)⋊C8C22semidirect
29. (C2×D67)⋊C8C22semidirect
31. C2×M16×C67C22semidirect
32. #121×C67C22semidirect
33. #106×C67C22semidirect
34. #102×C67C22semidirect
125. C67#3199semidirect
130. #469D67semidirect
140. C2×M16C2×D67semidirect
147. #121C2×D67semidirect
153. #106C2×D67semidirect
157. #102C2×D67semidirect