Finite Group Databasebeta

← #21456600

Subgroups — #21456600

Order 7808.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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#16840273C211yesmaximal, characteristic
2#16840100C212yesmaximal, complemented
3#16840092C211yesmaximal, characteristic, complemented
4#16840075C211yesmaximal, characteristic, complemented
5#16840001C212yesmaximal, complemented
6#16839993C212yesmaximal, complemented
7#16839922C212yesmaximal, complemented
8#16839918C211yesmaximal, characteristic, complemented
9#16839901C211yesmaximal, characteristic, complemented
10C2×C22⋊#92418C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C61C211yesmaximal, characteristic, complemented
12C2×#11758222C2211yescharacteristic
13C2×#11758202C2212yescomplemented
14C2×#11758201C2212yescomplemented
15C22×#92418C2212yes—
16#14519760—21no—
17#14519756C2211yescharacteristic
18#14519756C2211yescharacteristic
19#14519751C2211yescharacteristic
20#14519751C2211yescharacteristic
21#14519746—21no—
22C4×C22⋊D61—21no—
23C4×C22⋊D61—21no—
24#14519736C2211yescharacteristic
25#14519710C2212yescomplemented
26#14519708C2212yescomplemented
27#14519701C2212yescomplemented
28#14519701C2212yescomplemented
29#14519700C2212yescomplemented
30#14519697C2212yescomplemented
31C22⋊#92418C2211yescharacteristic
32C22⋊#92418C2211yescharacteristic
33C22⋊#92418C2212yes—
34C22×Q8×C61C2211yescharacteristic
35C23×C4×C61C2211yescharacteristic
36C22⋊Q8×C61—21no—
37C22⋊Q8×C61—21no—
38C22⋊Q8×C61C2212yes—
39C22⋊Q8×C61C2212yes—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61C2211yescharacteristic
42C2×C22⋊C4×C61—21no—
43C24×C61D411yescharacteristic
44C2×Q8×C61—21no—
45C2×Q8×C61—42no—
46C2×Q8×C61C2312yes—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—42no—
50C22×C4×C61D411yescharacteristic
51C22×C4×C61D411yescharacteristic
52C22×C4×C61D411yescharacteristic
53C22×C4×C61C2311yescharacteristic
54C22×C4×C61C2311yescharacteristic
55C22×C4×C61C2311yescharacteristic
56C2×C22⋊D61—21no—
57C2×C22⋊D61—21no—
58C2×C22⋊D61—21no—
59#11758222—42no—
60C2×D244C2311yescharacteristic
61C2×C4×D61—21no—
62C2×Dic122C2311yescharacteristic
63C4⋊C4×C61—22no—
64C4⋊C4×C61—22no—
65C4⋊C4×C61—21no—
66C4⋊C4×C61C2312yes—
67C22⋊C4×C61—22no—
68C22⋊C4×C61—22no—
69C22⋊Dic61—21no—
70C22⋊Dic61—21no—
71C22⋊Dic61—21no—
72#11758202—42no—
73#11758201—42no—
74#11758198—21no—
75#11758198—21no—
76C4⋊Dic61C2311yescharacteristic
77C4⋊Dic61C2311yescharacteristic
78#11758196—21no—
79#11758196—21no—
80C4×Dic61—21no—
81C2×#92418—42no—
82C2×#92418C2312yes—
83C2×#92418C2312yes—
84C23×C61—22no—
85C23×C61—22no—
86C23×C61—41no—
87C23×C61C2×D411yescharacteristic
88C23×C61C2×D411yescharacteristic
89C23×C61C2×D411yescharacteristic
90C22×D61—21no—
91Q8×C61—22no—
92Q8×C61—22no—
93Q8×C61—42no—
94C2×C4×C61—22no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61—42no—
103C2×C4×C61—42no—
104C2×C4×C61C2×D411yescharacteristic
105C2×C4×C61C2×D411yescharacteristic
106C2×C4×C61C2×D411yescharacteristic
107C2×C4×C61C2411yescharacteristic
108C22⋊D61—42no—
109C22⋊D61—42no—
110C22⋊D61—42no—
111C2×Dic61—21no—
112C2×Dic61—21no—
113C2×Dic61—21no—
114D244—22no—
115C4×D61—42no—
116Dic122—22no—
117#92418—22no—
118#92418—22no—
119C22×C61—21no—
120C22×C61—21no—
121C22×C61—21no—
122C22×C61—21no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—21no—
126C22×C61—44no—
127C22×C61#13512yes—
128C22×C61#13712yes—
129C22×C61C22×D411yescharacteristic
130C2×D61—41no—
131C2×D61—41no—
132C2×D61—41no—
133C4×C61—21no—
134C4×C61—21no—
135C4×C61—21no—
136C4×C61—42no—
137C4×C61—42no—
138C4×C61H(5,𝔽2)12yes—
139Dic61—41no—
140Dic61—41no—
141Dic61—41no—
142#3190—611nomaximal, Sylow
143C2×C61—22no—
144C2×C61—22no—
145C2×C61—41no—
146C2×C61#53411yescharacteristic
147C2×C61C2×#13511yescharacteristic
148C2×C61C2×#13711yescharacteristic
149D61—81no—
150#525—611no—
151C2×C22⋊Q8D6111yescharacteristic, complemented
152#485—612no—
153#483—611no—
154#480—611no—
155#467—612no—
156#465—612no—
157#452—612no—
158#451—611no—
159#448—611no—
160C2×C22⋊C8—611no—
161C61#319011yescharacteristic, complemented, Sylow
162C2×#41—611no—
163C22×Q8C2×D6111yescharacteristic
164C23×C4C2×D6111yescharacteristic
165C2×Q16—612no—
166C2×SD16—612no—
167C22×C8—612no—
168#123—1221no—
169C22⋊Q8—21no—
170C22⋊Q8—21no—
171C22⋊Q8—611no—
172C22⋊Q8—611no—
173C22⋊Q8C2×D6112yes—
174C22⋊Q8C2×D6112yes—
175#121—611no—
176#121—611no—
177#120—1221no—
178C4×D4—1221no—
179C4×D4—1221no—
180#117—611no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×D6111yescharacteristic
183C2×C22⋊C4—21no—
184#107—612no—
185#106—612no—
186#103—612no—
187#103—612no—
188#103—612no—
189#102—612no—
190C22⋊C8—611no—
191C22⋊C8—611no—
192C22⋊C8—612no—
193C24C22⋊D6111yescharacteristic
194#41—2442no—
195C2×Q8—21no—
196C2×Q8—42no—
197C2×Q8—611no—
198C2×Q8C22×D6112yes—
199C2×D4—611no—
200C2×D4—1221no—
201C2×D4—1221no—
202C2×D4—1221no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—1221no—
207C22×C4C22⋊D6111yescharacteristic
208C22×C4C22⋊D6111yescharacteristic
209C22×C4C22⋊D6111yescharacteristic
210C22×C4C22×D6111yescharacteristic
211C22×C4C22×D6111yescharacteristic
212C22×C4C22×D6111yescharacteristic
213Q16—2442no—
214SD16—2442no—
215C2×C8—612no—
216C2×C8—612no—
217C2×C8—2442no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—21no—
221C4⋊C4—611no—
222C4⋊C4—611no—
223C4⋊C4—1221no—
224C4⋊C4—1221no—
225C4⋊C4C22×D6112yes—
226C22⋊C4—22no—
227C22⋊C4—22no—
228C22⋊C4—1221no—
229C22⋊C4—1221no—
230C22⋊C4—1221no—
231C22⋊C4—1221no—
232C22⋊C4—1221no—
233C42—1221no—
234C23—22no—
235C23—22no—
236C23—41no—
237C23—1221no—
238C23C2×C22⋊D6111yescharacteristic
239C23C2×C22⋊D6111yescharacteristic
240C23C2×C22⋊D6111yescharacteristic
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—1222no—
245D4—1222no—
246D4—2442no—
247D4—2442no—
248D4—2442no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—1221no—
260C2×C4—1221no—
261C2×C4—1221no—
262C2×C4—2442no—
263C2×C4C2×C22⋊D6111yescharacteristic
264C2×C4C2×C22⋊D6111yescharacteristic
265C2×C4C2×C22⋊D6111yescharacteristic
266C2×C4C23×D6111yescharacteristic
267C8—1222no—
268C8—1222no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—2441no—
278C22—2441no—
279C22—2441no—
280C22#1451979712yes—
281C22#1451980812yes—
282C22C22×C22⋊D6111yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—2441no—
289C4—2441no—
290C4—2441no—
291C4#1451982212yes—
292C2—22no—
293C2—22no—
294C2—41no—
295C2—4881no—
296C2#1684032111yescharacteristic
297C2C2×#1451979711yescharacteristic
298C2C2×#1451980811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC2×C22⋊Q8×C61C2
socleC22×C61C22×D4
hypercenterC2×C22⋊Q8D61
hypocenterC61#3190
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #16840100C2semidirect
3. #16840092C2semidirect
4. #16840075C2semidirect
5. #16840001C2semidirect
6. #16839993C2semidirect
7. #16839922C2semidirect
8. #16839918C2semidirect
9. #16839901C2semidirect
10. C2×C22⋊#92418C2semidirect
11. C2×C22⋊Q8×C61C2semidirect
13. C2×#11758202C22semidirect
14. C2×#11758201C22semidirect
25. #14519710C22semidirect
26. #14519708C22semidirect
27. #14519701C22semidirect
28. #14519701C22semidirect
29. #14519700C22semidirect
30. #14519697C22semidirect
151. C2×C22⋊Q8D61semidirect
161. C61#3190semidirect