Finite Group Databasebeta

← #22570158

Subgroups — #22570158

Order 8576.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14524022C211yesmaximal, characteristic
2C2×#14524002C211yesmaximal, characteristic
3#16863738C214yesmaximal, complemented
4#16863732C212yesmaximal, complemented
5#16863727C212yesmaximal, complemented
6#16863438C212yesmaximal, complemented
7#16863406C212yesmaximal, complemented
8#407×C67C211yesmaximal, characteristic
9C22×Dic134C2211yescharacteristic
10#14524022—21no—
11#14524022—21no—
12#14524022C2212yes—
13#14524022C2212yes—
14#14524002—21no—
15#14524002—21no—
16#14524002C2212yes—
17#14524002C2212yes—
18C2×C4⋊Dic67C2211yescharacteristic
19C2×C4⋊Dic67C2211yescharacteristic
20C2×C4⋊Dic67C2211yescharacteristic
21C2×Dic67⋊C4—21no—
22C2×C22⋊Dic67—21no—
23C2×C22⋊Dic67—21no—
24C2×C4×Dic67—21no—
25#14523977C2214yescomplemented
26#14523975C2214yescomplemented
27#14523968C2214yescomplemented
28#14523968C2214yescomplemented
29C23×C4×C67C2211yescharacteristic
30C2×M16×C67C2212yes—
31C22⋊C8×C67C2212yes—
32C22⋊C8×C67C2212yes—
33C2×Dic134—24no—
34C2×Dic134—21no—
35C2×Dic134C2312yes—
36C22×Dic67—21no—
37C22×Dic67—21no—
38C22×Dic67—21no—
39C22×Dic67—21no—
40C4⋊Dic67—21no—
41C4⋊Dic67—21no—
42C4⋊Dic67—21no—
43C4⋊Dic67C2312yes—
44C4⋊Dic67C2312yes—
45C4⋊Dic67C2312yes—
46Dic67⋊C4—22no—
47Dic67⋊C4—22no—
48C22⋊Dic67—22no—
49C22⋊Dic67—22no—
50C22⋊Dic67—22no—
51C22⋊Dic67—22no—
52C4×Dic67—22no—
53C4×Dic67—22no—
54C24×C67D411yescharacteristic
55C22×C4×C67—24no—
56C22×C4×C67D411yescharacteristic
57C22×C4×C67D411yescharacteristic
58C22×C4×C67D411yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C22×C4×C67C2311yescharacteristic
61C22×C4×C67C2311yescharacteristic
62M16×C67—24no—
63C2×C8×C67C2314yes—
64C23×C67—22no—
65C23×C67—22no—
66C23×C67—41no—
67C23×C67C2×D411yescharacteristic
68C23×C67C2×D411yescharacteristic
69C23×C67C2×D411yescharacteristic
70C2×C4×C67—22no—
71C2×C4×C67—22no—
72C2×C4×C67—22no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67C2×D411yescharacteristic
76C2×C4×C67C2×D411yescharacteristic
77C2×C4×C67C2×D411yescharacteristic
78C2×C4×C67#4114yes—
79C2×C4×C67C2411yescharacteristic
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—22no—
83C2×Dic67—22no—
84C2×Dic67—42no—
85C2×Dic67—42no—
86C2×Dic67—42no—
87C2×Dic67—42no—
88Dic134—24no—
89Dic134—42no—
90C8×C67—24no—
91C22×C67—21no—
92C22×C67—21no—
93C22×C67—21no—
94C22×C67—21no—
95C22×C67—21no—
96C22×C67—21no—
97C22×C67—21no—
98C22×C67—44no—
99C22×C67#12312yes—
100C22×C67#13712yes—
101C22×C67C22×D411yescharacteristic
102C4×C67—21no—
103C4×C67—21no—
104C4×C67#12312yes—
105C4×C67C2×#4112yes—
106Dic67—42no—
107Dic67—42no—
108Dic67—42no—
109Dic67—42no—
110C2×C67—22no—
111C2×C67—22no—
112C2×C67—41no—
113C2×C67C2×#12311yescharacteristic
114C2×C67C2×#13711yescharacteristic
115C2×C67#57511yescharacteristic
116#3095—671nomaximal, Sylow
117C67#309511yescharacteristic, complemented, Sylow
118C2×C22⋊Q8—671no—
119C2×#117—671no—
120#485—674no—
121#484—672no—
122#483—672no—
123#428—672no—
124#419—672no—
125#407D6711yescharacteristic
126C22×Q8—671no—
127C23×C4C2×D6711yescharacteristic
128C2×M16C2×D6712yes—
129C22⋊Q8—672no—
130C22⋊Q8—672no—
131C22⋊Q8—1341no—
132C22⋊Q8—1341no—
133#117—672no—
134#117—672no—
135#117—1341no—
136#117—1341no—
137C2×C4⋊C4—671no—
138C2×C4⋊C4—671no—
139C2×C4⋊C4—671no—
140C2×C4⋊C4—1341no—
141C2×C22⋊C4—1341no—
142C2×C22⋊C4—1341no—
143C2×C42—1341no—
144#107—674no—
145#106—674no—
146#103—674no—
147#103—674no—
148C22⋊C8C2×D6712yes—
149C22⋊C8C2×D6712yes—
150C24D26811yescharacteristic
151C2×Q8—672no—
152C2×Q8—1344no—
153C2×Q8—1341no—
154C22×C4—24no—
155C22×C4—1341no—
156C22×C4—1341no—
157C22×C4—1341no—
158C22×C4—1341no—
159C22×C4D26811yescharacteristic
160C22×C4D26811yescharacteristic
161C22×C4D26811yescharacteristic
162C22×C4C22×D6711yescharacteristic
163C22×C4C22×D6711yescharacteristic
164C22×C4C22×D6711yescharacteristic
165M16—24no—
166C2×C8C22×D6714yes—
167C4⋊C4—672no—
168C4⋊C4—672no—
169C4⋊C4—672no—
170C4⋊C4—1341no—
171C4⋊C4—1341no—
172C4⋊C4—1341no—
173C4⋊C4—1342no—
174C4⋊C4—1342no—
175C22⋊C4—1342no—
176C22⋊C4—1342no—
177C22⋊C4—1342no—
178C22⋊C4—1342no—
179C42—1342no—
180C42—1342no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×D26811yescharacteristic
185C23C2×D26811yescharacteristic
186C23C2×D26811yescharacteristic
187Q8—2682no—
188Q8—1344no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—2682no—
195C2×C4—2682no—
196C2×C4—2682no—
197C2×C4—2682no—
198C2×C4—1342no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4—1342no—
202C2×C4C2×D26811yescharacteristic
203C2×C4C2×D26811yescharacteristic
204C2×C4C2×D26811yescharacteristic
205C2×C4#1176162714yes—
206C2×C4C23×D6711yescharacteristic
207C8—24no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#1452402712yes—
217C22#1452407612yes—
218C22C22×D26811yescharacteristic
219C4—21no—
220C4—21no—
221C4—2682no—
222C4—2682no—
223C4—2682no—
224C4—2682no—
225C4#1452402712yes—
226C4C2×#1176162712yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#1452402711yescharacteristic
231C2C2×#1452407611yescharacteristic
232C2#1686419611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#407×C67C2
socleC22×C67C22×D4
hypercenter#407D67
hypocenterC67#3095
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #16863738C2semidirect
4. #16863732C2semidirect
5. #16863727C2semidirect
6. #16863438C2semidirect
7. #16863406C2semidirect
25. #14523977C22semidirect
26. #14523975C22semidirect
27. #14523968C22semidirect
28. #14523968C22semidirect
117. C67#3095semidirect