Finite Group Databasebeta

← #21456622

Subgroups — #21456622

Order 7808.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22⋊Q8×D61C211yesmaximal, characteristic, complemented
2C2×#14519748C211yesmaximal, characteristic
3#16840098C211yesmaximal, characteristic, complemented
4#16840095C211yesmaximal, characteristic, complemented
5#16840074C212yesmaximal, complemented
6#16840007C212yesmaximal, complemented
7#16840005C212yesmaximal, complemented
8#16839913C211yesmaximal, characteristic, complemented
9#16839899C212yesmaximal, complemented
10#16839724C211yesmaximal, characteristic, complemented
11C61×#450C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2211yescharacteristic, complemented
13C22×C22⋊D61—21no—
14C22×C4×D61C2211yescharacteristic
15C2×#11758200C2212yescomplemented
16C2×#11758189C2212yescomplemented
17#14519757C2211yescharacteristic, complemented
18#14519756C2211yescharacteristic
19#14519755—21no—
20#14519754—21no—
21#14519753C2211yescharacteristic, complemented
22#14519748—21no—
23#14519748—21no—
24#14519748C2212yes—
25#14519748C2212yes—
26C2×C4⋊Dic61C2211yescharacteristic
27C4⋊C4×D61—21no—
28C4⋊C4×D61C2211yescharacteristic, complemented
29C2×C22⋊Dic61—21no—
30C22⋊C4×D61—21no—
31#14519711C2212yescomplemented
32#14519710C2212yescomplemented
33#14519702C2212yescomplemented
34#14519701C2212yescomplemented
35#14519698C2212yescomplemented
36C22⋊#92418C2211yescharacteristic
37#14519689C2212yescomplemented
38C22×D4×C61C2211yescharacteristic
39C2×SD16×C61C2212yescomplemented
40C22⋊Q8×C61C2211yescharacteristic
41#102×C61C2212yescomplemented
42C22⋊C8×C61C2211yescharacteristic
43C24×C61—21no—
44C23×D61D411yescharacteristic
45C2×Q8×C61C2311yescharacteristic
46C2×D4×C61—21no—
47C2×D4×C61—42no—
48C2×D4×C61C2312yes—
49C22×C4×C61C2311yescharacteristic
50C2×C22⋊D61—22no—
51C2×C22⋊D61—22no—
52C2×C22⋊D61—44no—
53C22×Dic61—21no—
54C22×Dic61D411yescharacteristic
55Q8×D61—42no—
56C2×C4×D61—21no—
57C2×C4×D61—21no—
58C2×C4×D61—42no—
59C2×C4×D61D411yescharacteristic
60C2×C4×D61D411yescharacteristic
61C2×C4×D61C2311yescharacteristic
62C2×C4×D61C2311yescharacteristic
63C2×Dic122—21no—
64C2×Dic122C2311yescharacteristic
65SD16×C61—42no—
66C2×C8×C61C2312yes—
67C4⋊C4×C61—21no—
68C4⋊C4×C61C2311yescharacteristic
69C22⋊C4×C61—21no—
70C22⋊Dic61—21no—
71C22⋊Dic61—22no—
72C22⋊Dic61—22no—
73#11758200—42no—
74#11758198—21no—
75#11758198—21no—
76C4⋊Dic61—21no—
77C4⋊Dic61—21no—
78C4⋊Dic61C2311yescharacteristic
79C4⋊Dic61C2312yes—
80#11758196—21no—
81#11758196—21no—
82#11758196—21no—
83C2×#92418C2312yes—
84#11758189—42no—
85C23×C61—22no—
86C23×C61—42no—
87C23×C61—42no—
88C23×C61—42no—
89C23×C61C2×D411yescharacteristic
90C22×D61—22no—
91C22×D61—22no—
92C22×D61—41no—
93C22×D61C2×D411yescharacteristic
94C22×D61C2×D411yescharacteristic
95Q8×C61—22no—
96D4×C61—22no—
97D4×C61—22no—
98D4×C61—42no—
99C2×C4×C61—22no—
100C2×C4×C61—21no—
101C2×C4×C61—21no—
102C2×C4×C61C2×D411yescharacteristic
103C2×C4×C61C2411yescharacteristic
104C22⋊D61—44no—
105C22⋊D61—44no—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—22no—
109C2×Dic61—41no—
110C2×Dic61—42no—
111C2×Dic61C2×D411yescharacteristic
112C2×Dic61C2×D411yescharacteristic
113C4×D61—22no—
114C4×D61—22no—
115C4×D61—41no—
116C4×D61—41no—
117C4×D61—42no—
118C4×D61—42no—
119Dic122—22no—
120Dic122—42no—
121C8×C61—22no—
122#92418—22no—
123C22×C61—21no—
124C22×C61—21no—
125C22×C61—42no—
126C22×C61—42no—
127C22×C61—42no—
128C22×C61—84no—
129C22×C61#13712yes—
130C22×C61C22×D411yescharacteristic
131C2×D61—21no—
132C2×D61—21no—
133C2×D61—21no—
134C2×D61—21no—
135C2×D61—21no—
136C2×D61—44no—
137C2×D61AGL(1,ℤ/8)12yes—
138C4×C61—21no—
139C4×C61—41no—
140C4×C61—41no—
141C4×C61H(5,𝔽2)12yes—
142Dic61—21no—
143Dic61—21no—
144Dic61—41no—
145Dic61—41no—
146Dic61—42no—
147#3193—611nomaximal, Sylow
148C2×C61—22no—
149C2×C61—82no—
150C2×C61#53411yescharacteristic
151C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C61C2×#13711yescharacteristic
153D61—22no—
154D61—41no—
155C2×C22⋊Q8—611no—
156C2×#121—611no—
157#484—611no—
158#484—611no—
159#480—612no—
160#468—612no—
161#468—612no—
162#450—611no—
163#450D6111yescharacteristic, complemented
164#448—612no—
165#407—611no—
166C61#319311yescharacteristic, complemented, Sylow
167C22×Q8—611no—
168C22×D4—1221no—
169C22×D4C2×D6111yescharacteristic
170C23×C4—611no—
171C2×SD16—612no—
172C2×SD16C2×D6112yescomplemented
173C2×M16—612no—
174C22⋊Q8—611no—
175C22⋊Q8—611no—
176C22⋊Q8—611no—
177C22⋊Q8—1221no—
178C22⋊Q8—1221no—
179C22⋊Q8C2×D6111yescharacteristic
180#121—612no—
181#121—612no—
182#121—1221no—
183#121—1221no—
184C2×C4⋊C4—611no—
185C2×C4⋊C4—611no—
186C2×C4⋊C4—1221no—
187C2×C22⋊C4—1221no—
188C2×C22⋊C4—1221no—
189#107—612no—
190#107—612no—
191#103—612no—
192#103—612no—
193#102—612no—
194#102C2×D6112yescomplemented
195C22⋊C8—611no—
196C22⋊C8—612no—
197C22⋊C8C2×D6111yescharacteristic
198C24—21no—
199C24—611no—
200C2×Q8—611no—
201C2×Q8—1221no—
202C2×Q8—2442no—
203C2×Q8C22×D6111yescharacteristic
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—1222no—
207C2×D4—1222no—
208C2×D4—2444no—
209C2×D4C22×D6112yes—
210C22×C4—611no—
211C22×C4—611no—
212C22×C4—611no—
213C22×C4—611no—
214C22×C4—611no—
215C22×C4—1221no—
216C22×C4—1221no—
217C22×C4—1221no—
218C22×C4—2442no—
219C22×C4C22×D6111yescharacteristic
220SD16—42no—
221SD16—2442no—
222M16—2442no—
223C2×C8—612no—
224C2×C8C22×D6112yes—
225C4⋊C4—21no—
226C4⋊C4—611no—
227C4⋊C4—612no—
228C4⋊C4—1221no—
229C4⋊C4—1221no—
230C4⋊C4—1221no—
231C4⋊C4—1221no—
232C4⋊C4—1221no—
233C4⋊C4C22×D6111yescharacteristic
234C22⋊C4—21no—
235C22⋊C4—1221no—
236C22⋊C4—1221no—
237C22⋊C4—1222no—
238C22⋊C4—1222no—
239C22⋊C4—1221no—
240C23—22no—
241C23—42no—
242C23—42no—
243C23—42no—
244C23—611no—
245C23—611no—
246C23—1222no—
247C23—1222no—
248C23—2441no—
249C23D4×D6111yescharacteristic
250Q8—22no—
251Q8—1222no—
252Q8—2442no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—2444no—
257D4—2444no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—611no—
262C2×C4—611no—
263C2×C4—1222no—
264C2×C4—1222no—
265C2×C4—1221no—
266C2×C4—1222no—
267C2×C4—1221no—
268C2×C4—2441no—
269C2×C4—2441no—
270C2×C4—2442no—
271C2×C4—2442no—
272C2×C4—2441no—
273C2×C4—2442no—
274C2×C4D4×D6111yescharacteristic
275C2×C4C23×D6111yescharacteristic
276C8—22no—
277C8—1222no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—42no—
283C22—84no—
284C22—612no—
285C22—1221no—
286C22—1221no—
287C22—1221no—
288C22—1221no—
289C22—1221no—
290C22—2444no—
291C22#1451980612yes—
292C22C2×D4×D6111yescharacteristic
293C4—21no—
294C4—41no—
295C4—41no—
296C4—1221no—
297C4—1221no—
298C4—2441no—
299C4—2442no—
300C4—2441no—
301C4#1451982212yes—
302C2—22no—
303C2—82no—
304C2—1222no—
305C2—2441no—
306C2#1684032211yescharacteristic
307C2AGL(1,ℤ/8)×D6111yescharacteristic
308C2C2×#1451980611yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#450C2
socleC22×C61C22×D4
hypercenter#450D61
hypocenterC61#3193
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D61C2semidirect
3. #16840098C2semidirect
4. #16840095C2semidirect
5. #16840074C2semidirect
6. #16840007C2semidirect
7. #16840005C2semidirect
8. #16839913C2semidirect
9. #16839899C2semidirect
10. #16839724C2semidirect
11. C61×#450C2semidirect
12. C2×Q8×D61C22semidirect
15. C2×#11758200C22semidirect
16. C2×#11758189C22semidirect
17. #14519757C22semidirect
21. #14519753C22semidirect
28. C4⋊C4×D61C22semidirect
31. #14519711C22semidirect
32. #14519710C22semidirect
33. #14519702C22semidirect
34. #14519701C22semidirect
35. #14519698C22semidirect
37. #14519689C22semidirect
39. C2×SD16×C61C22semidirect
41. #102×C61C22semidirect
163. #450D61semidirect
166. C61#3193semidirect
172. C2×SD16C2×D61semidirect
194. #102C2×D61semidirect