Finite Group Databasebeta

← #22568493

Subgroups — #22568493

Order 8576.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D67C211yesmaximal, characteristic
2C2×#14524019C212yesmaximal, complemented
3#16863268C212yesmaximal, complemented
4#16863267C214yesmaximal, complemented
5#16863256C211yesmaximal, characteristic, complemented
6#16863253C212yesmaximal, complemented
7#16863251C212yesmaximal
8C2×C22⋊Q8×C67C211yesmaximal, characteristic, complemented
9C24×D67C2211yescharacteristic
10C23×Dic67C2211yescharacteristic
11C22×C4×D67—28no—
12C22×C4×D67—24no—
13C22×C4×D67C2212yes—
14C22×C4×D67C2212yes—
15#14524019—28no—
16C2×C4⋊Dic67C2212yescomplemented
17C2×Dic67⋊C4C2214yescomplemented
18C2×C22⋊Dic67C2212yes—
19C2×(C2×D67)⋊C4C2214yes—
20C2×(C2×D67)⋊C4C2214yes—
21C67⋊#95C2214yescomplemented
22C67⋊#95C2212yescomplemented
23C22×Q8×C67C2211yescharacteristic, complemented
24C23×C4×C67C2211yescharacteristic
25C22⋊Q8×C67—24no—
26C2×C4⋊C4×C67C2211yescharacteristic, complemented
27C2×C4⋊C4×C67C2212yescomplemented
28C2×C22⋊C4×C67C2212yes—
29C23×D67—22no—
30C23×D67—22no—
31C23×D67—24no—
32C23×D67—24no—
33C23×D67Q814yescomplemented
34C23×D67C2312yes—
35C22×Dic67—22no—
36C22×Dic67—24no—
37C22×Dic67C2314yes—
38C22×Dic67C2312yes—
39C2×C4×D67—232no—
40C2×C4×D67—24no—
41C2×C4×D67—24no—
42C2×C4×D67—24no—
43C2×C4×D67—24no—
44C2×C4×D67—24no—
45C2×C4×D67D418yes—
46C4⋊Dic67—24no—
47Dic67⋊C4—28no—
48C22⋊Dic67—24no—
49(C2×D67)⋊C4—28no—
50(C2×D67)⋊C4—28no—
51C24×C67C2311yescharacteristic
52C2×Q8×C67—22no—
53C2×Q8×C67—24no—
54C22×C4×C67—24no—
55C22×C4×C67D414yes—
56C22×C4×C67C2312yes—
57C22×C4×C67C2312yes—
58C22×C4×C67C2311yescharacteristic
59C22×C4×C67C2311yescharacteristic
60C4⋊C4×C67—22no—
61C4⋊C4×C67—24no—
62C22⋊C4×C67—24no—
63C23×C67—22no—
64C23×C67—22no—
65C23×C67C2×D412yes—
66C23×C67C2×Q812yes—
67C23×C67#4112yes—
68C23×C67C2411yescharacteristic
69C22×D67—28no—
70C22×D67—216no—
71C22×D67—28no—
72C22×D67—28no—
73C22×D67—22no—
74C22×D67—22no—
75C22×D67—22no—
76C22×D67—22no—
77C22×D67—216no—
78C22×D67C2×D414yes—
79C22×D67C2×Q814yes—
80C22×D67#4114yes—
81Q8×C67—44no—
82C2×C4×C67—22no—
83C2×C4×C67—28no—
84C2×C4×C67—22no—
85C2×C4×C67—22no—
86C2×C4×C67—24no—
87C2×C4×C67—24no—
88C2×C4×C67C2×D414yes—
89C2×C4×C67C2×D412yes—
90C2×C4×C67C2×D412yes—
91C2×Dic67—28no—
92C2×Dic67—22no—
93C2×Dic67—22no—
94C2×Dic67—24no—
95C2×Dic67—24no—
96C2×Dic67—24no—
97C2×Dic67C2×D414yes—
98C4×D67—216no—
99C4×D67—216no—
100C22×C67—22no—
101C22×C67—24no—
102C22×C67—22no—
103C22×C67—24no—
104C22×C67C22⋊Q814yescomplemented
105C22×C67C22×D412yes—
106C22×C67C22×D411yescharacteristic
107C22×C67C22×Q811yescharacteristic
108C22×C67C2×#4112yes—
109C22×C67C2×#4111yescharacteristic
110C2×D67—232no—
111C2×D67—24no—
112C2×D67—24no—
113C2×D67—24no—
114C2×D67—24no—
115C2×D67—24no—
116C2×D67—24no—
117C2×D67C22⋊Q818yescomplemented
118C4×C67—22no—
119C4×C67—42no—
120C4×C67—42no—
121C4×C67#12014yes—
122Dic67—24no—
123Dic67—44no—
124C2×C67—24no—
125C2×C67C2×#12011yescharacteristic
126C2×C67C2×C22⋊Q812yes—
127C2×C67C2×C22⋊Q811yescharacteristic
128C2×C67#52511yescharacteristic
129C2×C67#52512yes—
130D67—28no—
131#2848—671nomaximal, Sylow
132C67#284811yescharacteristic, complemented, Sylow
133C24×C4—671no—
134C2×C22⋊Q8—672no—
135C2×C22⋊Q8D6711yescharacteristic, complemented
136#385—674no—
137#385—672no—
138#380—671no—
139#380—672no—
140#379—672no—
141C25—671no—
142C22×Q8C2×D6711yescharacteristic, complemented
143C23×C4—672no—
144C23×C4—671no—
145C23×C4—672no—
146C23×C4—1348no—
147C23×C4—1344no—
148C23×C4C2×D6711yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—1348no—
151C2×C4⋊C4—674no—
152C2×C4⋊C4—672no—
153C2×C4⋊C4C2×D6711yescharacteristic, complemented
154C2×C4⋊C4C2×D6712yescomplemented
155C2×C22⋊C4—672no—
156C2×C22⋊C4—674no—
157C2×C22⋊C4—674no—
158C2×C22⋊C4C2×D6712yes—
159#95—674no—
160#95—672no—
161C24—672no—
162C24—674no—
163C24—1344no—
164C24—1344no—
165C24—1342no—
166C24—1342no—
167C24C22×D6711yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—672no—
172C22×C4—678no—
173C22×C4—674no—
174C22×C4—1344no—
175C22×C4—1344no—
176C22×C4—13432no—
177C22×C4—1344no—
178C22×C4—1342no—
179C22×C4—1344no—
180C22×C4—1344no—
181C22×C4—1344no—
182C22×C4C22⋊D6714yes—
183C22×C4C22×D6712yes—
184C22×C4C22×D6712yes—
185C22×C4C22×D6711yescharacteristic
186C22×C4C22×D6711yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—1348no—
190C4⋊C4—1344no—
191C22⋊C4—24no—
192C22⋊C4—1348no—
193C22⋊C4—1348no—
194C22⋊C4—1344no—
195C23—22no—
196C23—22no—
197C23—674no—
198C23—674no—
199C23—674no—
200C23—13416no—
201C23—1348no—
202C23—1348no—
203C23—13416no—
204C23—1348no—
205C23—1342no—
206C23—1342no—
207C23—1342no—
208C23—1342no—
209C23C2×C22⋊D6712yes—
210C23Q8×D6712yes—
211C23#1176162912yes—
212C23C23×D6711yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—674no—
221C2×C4—13416no—
222C2×C4—1344no—
223C2×C4—1344no—
224C2×C4—1344no—
225C2×C4—1348no—
226C2×C4—1342no—
227C2×C4—1342no—
228C2×C4—13416no—
229C2×C4D4×D6714yes—
230C2×C4C2×C22⋊D6712yes—
231C2×C4C2×C22⋊D6712yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—678no—
237C22—13432no—
238C22—1344no—
239C22—1344no—
240C22—1344no—
241C22—1344no—
242C22—1344no—
243C22—1344no—
244C22#1452401914yescomplemented
245C22C2×D4×D6712yes—
246C22C22×C22⋊D6711yescharacteristic
247C22C2×Q8×D6711yescharacteristic
248C22#41×D6712yes—
249C22C2×#1176162911yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—2684no—
254C4—1344no—
255C4#1452401114yes—
256C2—24no—
257C2—1348no—
258C2C2×#1452401111yescharacteristic
259C2C22⋊Q8×D6712yes—
260C2C2×#1452401911yescharacteristic
261C2#1686390311yescharacteristic
262C2#1686390912yes—
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
FittingC2×C22⋊Q8×C67C2
hypercenterC2×C22⋊Q8D67
hypocenterC67#2848
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14524019C2semidirect
3. #16863268C2semidirect
4. #16863267C2semidirect
5. #16863256C2semidirect
6. #16863253C2semidirect
8. C2×C22⋊Q8×C67C2semidirect
16. C2×C4⋊Dic67C22semidirect
17. C2×Dic67⋊C4C22semidirect
21. C67⋊#95C22semidirect
22. C67⋊#95C22semidirect
23. C22×Q8×C67C22semidirect
26. C2×C4⋊C4×C67C22semidirect
27. C2×C4⋊C4×C67C22semidirect
33. C23×D67Q8semidirect
104. C22×C67C22⋊Q8semidirect
117. C2×D67C22⋊Q8semidirect
132. C67#2848semidirect
135. C2×C22⋊Q8D67semidirect
142. C22×Q8C2×D67semidirect
153. C2×C4⋊C4C2×D67semidirect
154. C2×C4⋊C4C2×D67semidirect
244. C22#14524019semidirect