Finite Group Databasebeta

← #22570134

Subgroups — #22570134

Order 8576.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#16863907C211yesmaximal, characteristic
2C22×C4⋊Dic67C211yesmaximal, characteristic, complemented
3#16863732C212yesmaximal, complemented
4#16863727C212yesmaximal, complemented
5#16863715C212yesmaximal, complemented
6#16863710C212yesmaximal, complemented
7#16863438C212yesmaximal, complemented
8#16863398C212yesmaximal, complemented
9#407×C67C211yesmaximal, characteristic, complemented
10C2×#11761628C2211yescharacteristic
11C23×Dic67—21no—
12#14524026—21no—
13#14524022C2211yescharacteristic
14#14524022C2211yescharacteristic
15#14524017C2211yescharacteristic
16#14524017C2211yescharacteristic
17#14524012—21no—
18C4×C22⋊D67—21no—
19C4×C22⋊D67—21no—
20#14524002C2211yescharacteristic
21C2×C4⋊Dic67—21no—
22C2×C4⋊Dic67—21no—
23C2×C4⋊Dic67—21no—
24C2×C4⋊Dic67C2212yes—
25C2×C4⋊Dic67C2212yes—
26C2×C4⋊Dic67C2212yes—
27#14523977C2214yescomplemented
28#14523975C2214yescomplemented
29#14523968C2214yescomplemented
30#14523965C2214yescomplemented
31C23×C4×C67C2211yescharacteristic
32C2×M16×C67C2212yes—
33C22⋊C8×C67C2212yes—
34C22⋊C8×C67C2212yes—
35#11761628—24no—
36C2×Dic134C2311yescharacteristic
37C2×D268C2311yescharacteristic
38C2×C22⋊D67—21no—
39C2×C22⋊D67—21no—
40C2×C22⋊D67—21no—
41C22×Dic67—22no—
42C22×Dic67—22no—
43C22×Dic67—22no—
44C22×Dic67—44no—
45C2×C4×D67—21no—
46C4⋊Dic67—22no—
47C4⋊Dic67—22no—
48C4⋊Dic67—22no—
49C4⋊Dic67C2314yes—
50C4⋊Dic67C2311yescharacteristic
51C4⋊Dic67C2311yescharacteristic
52Dic67⋊C4—21no—
53Dic67⋊C4—21no—
54C22⋊Dic67—21no—
55C22⋊Dic67—21no—
56C22⋊Dic67—21no—
57(C2×D67)⋊C4—21no—
58(C2×D67)⋊C4—21no—
59C4×Dic67—21no—
60C24×C67D411yescharacteristic
61C22×C4×C67—24no—
62C22×C4×C67D411yescharacteristic
63C22×C4×C67D411yescharacteristic
64C22×C4×C67D411yescharacteristic
65C22×C4×C67C2311yescharacteristic
66C22×C4×C67C2311yescharacteristic
67C22×C4×C67C2311yescharacteristic
68M16×C67—24no—
69C2×C8×C67C2314yes—
70C23×C67—22no—
71C23×C67—22no—
72C23×C67—22no—
73C23×C67C2×D411yescharacteristic
74C23×C67C2×D411yescharacteristic
75C23×C67C2×D411yescharacteristic
76C22×D67—21no—
77C2×C4×C67—22no—
78C2×C4×C67—22no—
79C2×C4×C67—22no—
80C2×C4×C67—22no—
81C2×C4×C67—22no—
82C2×C4×C67C2×D411yescharacteristic
83C2×C4×C67C2×D411yescharacteristic
84C2×C4×C67C2×D411yescharacteristic
85C2×C4×C67#4114yes—
86C2×C4×C67C2411yescharacteristic
87C22⋊D67—42no—
88C22⋊D67—42no—
89C22⋊D67—42no—
90C2×Dic67—21no—
91C2×Dic67—21no—
92C2×Dic67—21no—
93C2×Dic67—24no—
94C2×Dic67—44no—
95C2×Dic67—44no—
96C2×Dic67—44no—
97D268—22no—
98C4×D67—42no—
99Dic134—22no—
100C8×C67—24no—
101C22×C67—28no—
102C22×C67—21no—
103C22×C67—21no—
104C22×C67—21no—
105C22×C67—21no—
106C22×C67—21no—
107C22×C67—21no—
108C22×C67#12312yes—
109C22×C67AGL(1,ℤ/8)12yes—
110C22×C67#13712yes—
111C22×C67C22×D411yescharacteristic
112C2×D67—41no—
113C2×D67—41no—
114C2×D67—41no—
115C4×C67—21no—
116C4×C67—21no—
117C4×C67#12312yes—
118C4×C67C2×#4112yes—
119Dic67—41no—
120Dic67—44no—
121Dic67—41no—
122Dic67—41no—
123C2×C67—22no—
124C2×C67—22no—
125C2×C67—22no—
126C2×C67C2×#12311yescharacteristic
127C2×C67C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C67C2×#13711yescharacteristic
129D67—81no—
130#3092—671nomaximal, Sylow
131C67#309211yescharacteristic, complemented, Sylow
132#525—671no—
133C22×C4⋊C4—671no—
134#484—672no—
135#483—672no—
136#481—672no—
137#480—672no—
138#428—672no—
139#417—672no—
140#407D6711yescharacteristic, complemented
141C2×#41—671no—
142C23×C4—1341no—
143C23×C4C2×D6711yescharacteristic
144C2×M16C2×D6712yes—
145#123—1341no—
146C22⋊Q8—671no—
147C22⋊Q8—671no—
148#121—671no—
149#121—671no—
150#120—1341no—
151C4×D4—1341no—
152C4×D4—1341no—
153#117—671no—
154C2×C4⋊C4—672no—
155C2×C4⋊C4—672no—
156C2×C4⋊C4—672no—
157C2×C4⋊C4—1341no—
158C2×C4⋊C4—1341no—
159C2×C4⋊C4—1341no—
160#107—674no—
161#106—674no—
162#103—674no—
163#102—674no—
164C22⋊C8C2×D6712yes—
165C22⋊C8C2×D6712yes—
166C24D26811yescharacteristic
167#41—1344no—
168C2×Q8—671no—
169C2×D4—671no—
170C2×D4—1341no—
171C2×D4—1341no—
172C2×D4—1341no—
173C22×C4—24no—
174C22×C4—2684no—
175C22×C4—1342no—
176C22×C4—1342no—
177C22×C4—1342no—
178C22×C4—1341no—
179C22×C4D26811yescharacteristic
180C22×C4D26811yescharacteristic
181C22×C4D26811yescharacteristic
182C22×C4C22×D6711yescharacteristic
183C22×C4C22×D6711yescharacteristic
184C22×C4C22×D6711yescharacteristic
185M16—24no—
186C2×C8C22×D6714yes—
187C4⋊C4—674no—
188C4⋊C4—671no—
189C4⋊C4—671no—
190C4⋊C4—1342no—
191C4⋊C4—1342no—
192C4⋊C4—1342no—
193C4⋊C4—1341no—
194C4⋊C4—1341no—
195C22⋊C4—1341no—
196C22⋊C4—1341no—
197C22⋊C4—1341no—
198C22⋊C4—1341no—
199C22⋊C4—1341no—
200C42—1341no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—1341no—
205C23C2×D26811yescharacteristic
206C23C2×D26811yescharacteristic
207C23C2×D26811yescharacteristic
208Q8—1342no—
209D4—2682no—
210D4—2682no—
211D4—2682no—
212D4—1342no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—2684no—
219C2×C4—2684no—
220C2×C4—2684no—
221C2×C4—2682no—
222C2×C4—1341no—
223C2×C4—1341no—
224C2×C4—1341no—
225C2×C4—1344no—
226C2×C4C2×D26811yescharacteristic
227C2×C4C2×D26811yescharacteristic
228C2×C4C2×D26811yescharacteristic
229C2×C4#1176162714yes—
230C2×C4C23×D6711yescharacteristic
231C8—24no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—2681no—
240C22—2681no—
241C22—2681no—
242C22#1452402712yes—
243C22#1452407112yes—
244C22#1452407612yes—
245C22C22×D26811yescharacteristic
246C4—21no—
247C4—21no—
248C4—2681no—
249C4—2684no—
250C4—2681no—
251C4—2681no—
252C4#1452402712yes—
253C4C2×#1176162712yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—5361no—
258C2C2×#1452402711yescharacteristic
259C2C2×#1452407111yescharacteristic
260C2C2×#1452407611yescharacteristic
subgroupG/H
centerC22C22×D268
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#407×C67C2
socleC22×C67C22×D4
hypercenter#407D67
hypocenterC67#3092
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
2. C22×C4⋊Dic67C2semidirect
3. #16863732C2semidirect
4. #16863727C2semidirect
5. #16863715C2semidirect
6. #16863710C2semidirect
7. #16863438C2semidirect
8. #16863398C2semidirect
9. #407×C67C2semidirect
27. #14523977C22semidirect
28. #14523975C22semidirect
29. #14523968C22semidirect
30. #14523965C22semidirect
131. C67#3092semidirect
140. #407D67semidirect