Finite Group Databasebeta

← #22571045

Subgroups — #22571045

Order 8576.

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

normal subgroups only

This group has 418 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#16863906C212yesmaximal, complemented
2#16863740C212yesmaximal, complemented
3#16863719C212yesmaximal, complemented
4#16863654C211yesmaximal, characteristic, complemented
5#16863640C212yesmaximal, complemented
6#16863559C212yesmaximal, complemented
7#16863538C212yesmaximal, complemented
8#16863359C211yesmaximal, characteristic, complemented
9#469×C67C211yesmaximal, characteristic, complemented
10C2×#11761627C2212yes—
11C22×C4×D67C2211yescharacteristic
12C2×Dic268C2211yescharacteristic, complemented
13C2×#11761616C2212yescomplemented
14C2×#11761612C2211yescharacteristic
15#14524025—22no—
16#14524023C2212yes—
17#14524022C2212yes—
18#14524014C2212yes—
19#14524011—22no—
20D4×Dic67—22no—
21(C22⋊D67)⋊C4—22no—
22#14524001C2212yes—
23#14523976C2212yescomplemented
24#14523975C2211yescharacteristic, complemented
25#14523968C2212yescomplemented
26#14523966C2212yescomplemented
27#14523964C2212yescomplemented
28(C2×D67)⋊C8C2212yes—
29(C2×D67)⋊C8C2212yes—
30C2×D8×C67C2211yescharacteristic, complemented
31C2×M16×C67C2211yescharacteristic
32#121×C67C2212yes—
33#106×C67C2211yescharacteristic, complemented
34#102×C67C2212yescomplemented
35C23×D67D411yescharacteristic
36#11761627—42no—
37#11761627—42no—
38C2×Dic134C2312yes—
39C2×C22⋊D67—22no—
40C2×C22⋊D67—22no—
41C22×Dic67—22no—
42C22×Dic67D411yescharacteristic
43C2×C4×D67—41no—
44C2×C4×D67—41no—
45C2×C4×D67D412yes—
46C2×C4×D67C2312yes—
47Dic268—41no—
48#11761616—42no—
49#11761612—41no—
50C2×C67⋊C8C2312yes—
51C4⋊Dic67—22no—
52C4⋊Dic67C2312yes—
53Dic67⋊C4—22no—
54C22⋊Dic67—22no—
55C22⋊Dic67—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—41no—
71C22×D67—42no—
72C22×D67C2×D412yes—
73D4×C67—22no—
74D4×C67—22no—
75D4×C67—44no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—22no—
79C2×C4×C67C2×D411yescharacteristic
80C2×C4×C67C2411yescharacteristic
81C22⋊D67—44no—
82C22⋊D67—44no—
83C2×Dic67—22no—
84C2×Dic67—22no—
85C2×Dic67—41no—
86C2×Dic67—44no—
87C2×Dic67C2×D412yes—
88C4×D67—22no—
89C4×D67—22no—
90C4×D67—42no—
91Dic134—22no—
92Dic134—22no—
93C8×C67—21no—
94C8×C67—21no—
95C67⋊C8—22no—
96C22×C67—21no—
97C22×C67—22no—
98C22×C67—42no—
99C22×C67—42no—
100C22×C67—42no—
101C22×C67C22×D411yescharacteristic
102C2×D67—22no—
103C2×D67—22no—
104C2×D67—22no—
105C2×D67—44no—
106C4×C67—21no—
107C4×C67—42no—
108C4×C67H(5,𝔽2)11yescharacteristic
109C4×C67H(5,𝔽2)11yescharacteristic
110Dic67—22no—
111Dic67—42no—
112Dic67—42no—
113C2×C67—41no—
114C2×C67—82no—
115C2×C67#53411yescharacteristic
116C2×C67#57512yes—
117D67—42no—
118#3197—671nomaximal, Sylow
119C67#319711yescharacteristic, complemented, Sylow
120#525—672no—
121#485—672no—
122#482—672no—
123#470—671no—
124#469D6711yescharacteristic, complemented
125#468—672no—
126#452—672no—
127#449—672no—
128#407—671no—
129C2×#41—672no—
130C23×C4—671no—
131C2×Q16—671no—
132C2×SD16—672no—
133C2×D8C2×D6711yescharacteristic, complemented
134C2×M16—671no—
135C2×M16C2×D6711yescharacteristic
136#123—1342no—
137C22⋊Q8—672no—
138C22⋊Q8—672no—
139#121—672no—
140#121C2×D6712yes—
141#120—1342no—
142C4×D4—1342no—
143C4×D4—1342no—
144#117—672no—
145#107—672no—
146#106—671no—
147#106C2×D6711yescharacteristic, complemented
148#103—672no—
149#103—672no—
150#102—672no—
151#102C2×D6712yescomplemented
152C22⋊C8—672no—
153C22⋊C8—672no—
154C24—671no—
155#41—2682no—
156#41—2682no—
157C2×Q8—672no—
158C2×D4—22no—
159C2×D4—1342no—
160C2×D4—1342no—
161C2×D4C22×D6712yes—
162C22×C4—2681no—
163C22×C4—2681no—
164C22×C4—671no—
165C22×C4—672no—
166C22×C4—672no—
167C22×C4—1342no—
168C22×C4C22×D6711yescharacteristic
169Q16—2681no—
170SD16—2682no—
171D8—41no—
172M16—41no—
173M16—2681no—
174C2×C8—672no—
175C2×C8C22×D6711yescharacteristic
176C2×C8C22×D6711yescharacteristic
177C4⋊C4—672no—
178C4⋊C4—1342no—
179C4⋊C4—1342no—
180C4⋊C4C22×D6712yes—
181C22⋊C4—22no—
182C22⋊C4—1342no—
183C22⋊C4—1342no—
184C22⋊C4—1342no—
185C22⋊C4—1342no—
186C42—1342no—
187C23—22no—
188C23—2681no—
189C23—2682no—
190C23—672no—
191C23D4×D6711yescharacteristic
192Q8—1342no—
193Q8—1342no—
194D4—22no—
195D4—22no—
196D4—44no—
197D4—2684no—
198D4—2684no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—2682no—
203C2×C4—2684no—
204C2×C4—2681no—
205C2×C4—672no—
206C2×C4—1342no—
207C2×C4—1342no—
208C2×C4—1342no—
209C2×C4—1342no—
210C2×C4D4×D6711yescharacteristic
211C2×C4C23×D6711yescharacteristic
212C8—21no—
213C8—21no—
214C8—1342no—
215C22—22no—
216C22—21no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—2684no—
221C22—1342no—
222C22—1342no—
223C22—1342no—
224C22C2×D4×D6711yescharacteristic
225C4—21no—
226C4—42no—
227C4—2682no—
228C4—2682no—
229C4—1342no—
230C4#1452408811yescharacteristic
231C4#1452408911yescharacteristic
232C2—41no—
233C2—82no—
234C2—2682no—
235C2#1686395411yescharacteristic
236C2#1686419212yes—
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#469×C67C2
socleC22×C67C22×D4
hypercenter#469D67
hypocenterC67#3197
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. #16863906C2semidirect
2. #16863740C2semidirect
3. #16863719C2semidirect
4. #16863654C2semidirect
5. #16863640C2semidirect
6. #16863559C2semidirect
7. #16863538C2semidirect
8. #16863359C2semidirect
9. #469×C67C2semidirect
12. C2×Dic268C22semidirect
13. C2×#11761616C22semidirect
23. #14523976C22semidirect
24. #14523975C22semidirect
25. #14523968C22semidirect
26. #14523966C22semidirect
27. #14523964C22semidirect
30. C2×D8×C67C22semidirect
33. #106×C67C22semidirect
34. #102×C67C22semidirect
119. C67#3197semidirect
124. #469D67semidirect
133. C2×D8C2×D67semidirect
147. #106C2×D67semidirect
151. #102C2×D67semidirect