Finite Group Databasebeta

← #21455909

Subgroups — #21455909

Order 7808.

488 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 121 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#16840310C211yesmaximal, characteristic, complemented
2C4×D4×D61C211yesmaximal, characteristic, complemented
3#16840079C212yesmaximal, complemented
4#16840053C211yesmaximal, characteristic, complemented
5#16840052C211yesmaximal, characteristic, complemented
6#16839954C211yesmaximal, characteristic, complemented
7#16839894C212yesmaximal, complemented
8#16839874C212yesmaximal, complemented
9#16839871C212yesmaximal, complemented
10#16839818C211yesmaximal, characteristic, complemented
11C61×#459C211yesmaximal, characteristic, complemented
12C2×#11758224—21no—
13C2×D4×D61C2211yescharacteristic
14C22×C4×D61—21no—
15C2×#11758199C2212yes—
16#14519777C2211yescharacteristic, complemented
17#14519764—21no—
18#14519748—21no—
19#14519748C2212yes—
20C4×D244C2211yescharacteristic
21D4×Dic61—21no—
22D4×Dic61C2211yescharacteristic
23D4×Dic61C2212yescomplemented
24#14519739C2211yescharacteristic
25C4×C22⋊D61—21no—
26#14519737—21no—
27C4⋊C4×D61C2211yescharacteristic
28C22⋊C4×D61—21no—
29C42×D61C2211yescharacteristic
30#14519709C2212yescomplemented
31#14519708C2212yescomplemented
32C4⋊#92418C2211yescharacteristic, complemented
33#14519699C2212yes—
34#14519698C2212yes—
35#14519698C2212yescomplemented
36#14519689C2212yes—
37#14519687C2212yescomplemented
38C2×D8×C61C2212yes—
39#127×C61C2211yescharacteristic, complemented
40C4×D4×C61C2211yescharacteristic
41C4⋊C8×C61C2211yescharacteristic, complemented
42#102×C61C2212yescomplemented
43C23×D61—21no—
44C2×D4×C61—21no—
45C2×D4×C61—21no—
46C2×D4×C61C2312yes—
47C2×D4×C61C2311yescharacteristic
48C22×C4×C61—21no—
49C2×C22⋊D61—22no—
50C2×C22⋊D61—21no—
51C22×Dic61—21no—
52C22×Dic61—22no—
53#11758224—44no—
54D4×D61—22no—
55D4×D61—22no—
56C2×D244C2311yescharacteristic
57C2×C4×D61—21no—
58C2×C4×D61—21no—
59C2×C4×D61—21no—
60C2×C4×D61—44no—
61C2×C4×D61D411yescharacteristic
62C2×C4×D61D411yescharacteristic
63C2×C4×D61C2311yescharacteristic
64C2×Dic122—21no—
65D8×C61—24no—
66C2×C8×C61C2312yes—
67C4⋊C4×C61C2311yescharacteristic
68C22⋊C4×C61—21no—
69C42×C61C2311yescharacteristic
70C22⋊Dic61—21no—
71C22⋊Dic61—22no—
72C22⋊Dic61—22no—
73#11758199—24no—
74#11758198—21no—
75#11758198—21no—
76C4⋊Dic61—21no—
77C4⋊Dic61C2312yes—
78C4⋊Dic61C2311yescharacteristic
79#11758196—21no—
80C4×Dic61D411yescharacteristic
81C4×Dic61D411yescharacteristic
82C4×Dic61C2311yescharacteristic
83C2×#92418C2312yes—
84C23×C61—21no—
85C23×C61—22no—
86C22×D61—24no—
87C22×D61—21no—
88C22×D61—42no—
89C22×D61—42no—
90C22×D61C2×D411yescharacteristic
91D4×C61—21no—
92D4×C61—22no—
93D4×C61—22no—
94D4×C61—42no—
95D4×C61—42no—
96D4×C61#4112yescomplemented
97C2×C4×C61—21no—
98C2×C4×C61—42no—
99C2×C4×C61C2×D411yescharacteristic
100C2×C4×C61C2×D411yescharacteristic
101C2×C4×C61C2411yescharacteristic
102C22⋊D61—24no—
103C22⋊D61—44no—
104C2×Dic61—21no—
105C2×Dic61—22no—
106C2×Dic61—42no—
107C2×Dic61—44no—
108C2×Dic61C2×D411yescharacteristic
109C2×Dic61C2×D411yescharacteristic
110C2×Dic61C2×D411yescharacteristic
111D244—21no—
112D244#4112yescomplemented
113C4×D61—22no—
114C4×D61—22no—
115C4×D61—21no—
116C4×D61—21no—
117C4×D61—42no—
118C4×D61—42no—
119C4×D61—42no—
120Dic122—42no—
121C8×C61—22no—
122#92418—22no—
123C22×C61—22no—
124C22×C61—41no—
125C22×C61—41no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—42no—
129C22×C61C22×D411yescharacteristic
130C2×D61—21no—
131C2×D61—21no—
132C2×D61—22no—
133C2×D61—48no—
134C2×D61—41no—
135C2×D61—41no—
136C2×D61AGL(1,ℤ/8)12yes—
137C4×C61—21no—
138C4×C61—41no—
139C4×C61AGL(1,ℤ/8)12yes—
140C4×C61C2×#4111yescharacteristic
141C4×C61C2×#4111yescharacteristic
142Dic61—21no—
143Dic61—21no—
144Dic61—21no—
145Dic61—42no—
146Dic61—41no—
147#3109—611nomaximal, Sylow
148C2×C61—42no—
149C2×C61—82no—
150C2×C61#52511yescharacteristic
151C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
153D61—22no—
154D61—42no—
155#532—611no—
156C2×C4×D4—611no—
157#481—612no—
158#476—611no—
159#476—611no—
160#459—611no—
161#459D6111yescharacteristic, complemented
162#447—612no—
163#442—612no—
164#442—612no—
165#432—611no—
166C61#310911yescharacteristic, complemented, Sylow
167C2×#41—1221no—
168C22×D4—611no—
169C23×C4—1221no—
170C2×D8—612no—
171C2×D8C2×D6112yes—
172C4⋊Q8—611no—
173#127C2×D6111yescharacteristic, complemented
174#124—1221no—
175#121—612no—
176#121—1221no—
177C4×D4—611no—
178C4×D4—612no—
179C4×D4—611no—
180C4×D4—611no—
181C4×D4—1221no—
182C4×D4—1221no—
183C4×D4—1221no—
184C4×D4C2×D6111yescharacteristic
185C2×C4⋊C4—611no—
186C2×C22⋊C4—1221no—
187C2×C42—611no—
188#106—612no—
189#106—612no—
190C4⋊C8—611no—
191C4⋊C8C2×D6111yescharacteristic, complemented
192#102—612no—
193#102—612no—
194#102—612no—
195#102C2×D6112yescomplemented
196C22⋊C8—612no—
197#97—612no—
198C24—1221no—
199#41—2444no—
200C2×Q8—1221no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—611no—
204C2×D4—1221no—
205C2×D4—1222no—
206C2×D4—1222no—
207C2×D4—1222no—
208C2×D4C22×D6112yes—
209C2×D4C22×D6111yescharacteristic
210C22×C4—21no—
211C22×C4—611no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—1221no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—1222no—
218C22×C4—1221no—
219C22×C4—2444no—
220D8—24no—
221D8—1224no—
222C2×C8—612no—
223C2×C8C22×D6112yes—
224C4⋊C4—611no—
225C4⋊C4—612no—
226C4⋊C4—1221no—
227C4⋊C4—1221no—
228C4⋊C4C22×D6111yescharacteristic
229C22⋊C4—21no—
230C22⋊C4—1221no—
231C22⋊C4—1221no—
232C22⋊C4—1222no—
233C22⋊C4—1222no—
234C22⋊C4—1221no—
235C42—611no—
236C42—611no—
237C42—611no—
238C42C22×D6111yescharacteristic
239C23—21no—
240C23—22no—
241C23—611no—
242C23—1224no—
243C23—1221no—
244C23—2442no—
245C23—2442no—
246Q8—2442no—
247D4—21no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—612no—
253D4—1224no—
254D4—1221no—
255D4—2444no—
256D4#1175822412yescomplemented
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—611no—
260C2×C4—611no—
261C2×C4—611no—
262C2×C4—1222no—
263C2×C4—1221no—
264C2×C4—1221no—
265C2×C4—1222no—
266C2×C4—1221no—
267C2×C4—1222no—
268C2×C4—2442no—
269C2×C4—2442no—
270C2×C4—2442no—
271C2×C4—2444no—
272C2×C4—2442no—
273C2×C4D4×D6111yescharacteristic
274C2×C4D4×D6111yescharacteristic
275C2×C4C23×D6111yescharacteristic
276C8—22no—
277C8—1222no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—612no—
285C22—1221no—
286C22—1221no—
287C22—1222no—
288C22—2448no—
289C22—2441no—
290C22—2441no—
291C22C2×D4×D6111yescharacteristic
292C4—21no—
293C4—41no—
294C4—1221no—
295C4—1221no—
296C4—1221no—
297C4—2442no—
298C4—2441no—
299C4#1451980112yes—
300C4#41×D6111yescharacteristic
301C4C2×#1175822411yescharacteristic
302C2—42no—
303C2—82no—
304C2—1222no—
305C2—2442no—
306C2#1684027211yescharacteristic
307C2AGL(1,ℤ/8)×D6111yescharacteristic
308C2C2×#1451980111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#459C2
socleC22×C61C22×D4
hypercenter#459D61
hypocenterC61#3109
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #16840310C2semidirect
2. C4×D4×D61C2semidirect
3. #16840079C2semidirect
4. #16840053C2semidirect
5. #16840052C2semidirect
6. #16839954C2semidirect
7. #16839894C2semidirect
8. #16839874C2semidirect
9. #16839871C2semidirect
10. #16839818C2semidirect
11. C61×#459C2semidirect
16. #14519777C22semidirect
23. D4×Dic61C22semidirect
30. #14519709C22semidirect
31. #14519708C22semidirect
32. C4⋊#92418C22semidirect
35. #14519698C22semidirect
37. #14519687C22semidirect
39. #127×C61C22semidirect
41. C4⋊C8×C61C22semidirect
42. #102×C61C22semidirect
96. D4×C61#41semidirect
112. D244#41semidirect
161. #459D61semidirect
166. C61#3109semidirect
173. #127C2×D61semidirect
191. C4⋊C8C2×D61semidirect
195. #102C2×D61semidirect
256. D4#11758224semidirect