Finite Group Databasebeta

← #496×D61 #21456249

Subgroups — #496×D61 #21456249

Order 7808.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×SD16×D61C212yesmaximal, complemented
2C2×D8×D61C212yesmaximal, complemented
3#127×D61C211yesmaximal, characteristic, complemented
4#124×D61C211yesmaximal, characteristic, complemented
5#16840152C211yesmaximal, characteristic, complemented
6#16840151C211yesmaximal, characteristic, complemented
7#16840150C212yesmaximal, complemented
8#16840149C211yesmaximal, characteristic, complemented
9#16840148C212yesmaximal, complemented
10#97×D61C211yesmaximal, characteristic, complemented
11C61×#496C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2211yescharacteristic
13C2×D4×D61—22no—
14C2×D4×D61C2211yescharacteristic
15C2×D4×D61C2212yes—
16C2×#11758190C2212yescomplemented
17C2×#11758200C2212yescomplemented
18C2×#11758201C2212yescomplemented
19SD16×D61—28no—
20C2×D488C2212yescomplemented
21C2×#11758199C2212yescomplemented
22C2×#11758199C2212yescomplemented
23D8×D61—28no—
24C2×C8×D61C2212yes—
25#14519774C2211yescharacteristic, complemented
26#14519773—22no—
27#14519773C2212yescomplemented
28#14519766C2211yescharacteristic, complemented
29#14519765C2211yescharacteristic, complemented
30#14519764C2211yescharacteristic, complemented
31#14519763—22no—
32C22⋊C4×D61—22no—
33C42×D61C2211yescharacteristic
34#14519688C2211yescharacteristic, complemented
35#14519687C2212yescomplemented
36C2×SD16×C61C2212yescomplemented
37C2×D8×C61C2212yescomplemented
38#127×C61C2211yescharacteristic, complemented
39#124×C61C2211yescharacteristic, complemented
40#97×C61C2211yescharacteristic, complemented
41C23×D61—22no—
42C23×D61—21no—
43C2×Q8×C61C2311yescharacteristic
44C2×D4×C61—22no—
45C2×D4×C61C2311yescharacteristic
46C2×D4×C61C2312yes—
47C2×C22⋊D61—24no—
48C2×C22⋊D61—21no—
49C2×C22⋊D61—22no—
50Q8×D61—24no—
51D4×D61—24no—
52D4×D61—24no—
53D4×D61—24no—
54D4×D61—48no—
55C2×D244—22no—
56C2×D244C2311yescharacteristic
57C2×D244C2312yes—
58C2×C4×D61—21no—
59C2×C4×D61D412yes—
60C2×C4×D61C2311yescharacteristic
61C2×Dic122—21no—
62C2×Dic122C2311yescharacteristic
63SD16×C61—24no—
64D8×C61—24no—
65C2×C8×C61C2312yes—
66C22⋊C4×C61—22no—
67C42×C61C2311yescharacteristic
68C22⋊Dic61—22no—
69#11758201—24no—
70#11758200—24no—
71#11758199—24no—
72#11758199—24no—
73#11758198—22no—
74#11758198—22no—
75C4×Dic61D412yes—
76C4×Dic61C2311yescharacteristic
77C2×#92418C2312yes—
78D488—24no—
79#11758190—24no—
80C8×D61D418yescomplemented
81C23×C61—21no—
82C23×C61—22no—
83C22×D61—21no—
84C22×D61—22no—
85C22×D61—42no—
86C22×D61—44no—
87C22×D61—48no—
88C22×D61—44no—
89C22×D61C2×D411yescharacteristic
90Q8×C61—22no—
91D4×C61—22no—
92D4×C61—22no—
93D4×C61—22no—
94D4×C61—44no—
95C2×C4×C61—21no—
96C2×C4×C61C2×D412yes—
97C2×C4×C61C2411yescharacteristic
98C22⋊D61—48no—
99C22⋊D61—42no—
100C22⋊D61—44no—
101C2×Dic61—21no—
102C2×Dic61C2×D412yes—
103C2×Dic61C2×D411yescharacteristic
104D244—22no—
105D244—22no—
106D244—22no—
107D244—44no—
108C4×D61—24no—
109C4×D61—42no—
110C4×D61C2×D414yes—
111Dic122—22no—
112Dic122—42no—
113C8×C61C2×D414yescomplemented
114#92418C2×D414yescomplemented
115C22×C61—41no—
116C22×C61—42no—
117C22×C61—44no—
118C22×C61—42no—
119C22×C61C22×D411yescharacteristic
120C2×D61—42no—
121C2×D61—41no—
122C2×D61—44no—
123C2×D61—42no—
124C2×D61—88no—
125C2×D61—84no—
126C2×D61#12712yes—
127C2×D61AGL(1,ℤ/8)14yes—
128C4×C61—22no—
129C4×C61—41no—
130C4×C61C22×D412yes—
131Dic61—22no—
132Dic61—41no—
133Dic61#12712yes—
134C2×#496—611nomaximal, Sylow
135C2×C61—82no—
136C2×C61—81no—
137C2×C61C2×#12711yescharacteristic
138C2×C61C2×AGL(1,ℤ/8)12yes—
139D61—82no—
140D61—81no—
141D61#49614yesdirect product factor
142C22×SD16—612no—
143C22×D8—612no—
144C2×#127—611no—
145C2×#124—611no—
146#496—612no—
147#496—612no—
148#496—611no—
149#496—611no—
150#496—611no—
151#496D6111yescharacteristic, direct product factor
152C2×#97—611no—
153C61C2×#49611yescharacteristic, complemented, Sylow
154C22×Q8—611no—
155C22×D4—611no—
156C22×D4—612no—
157C22×D4—1222no—
158C2×SD16—612no—
159C2×SD16—612no—
160C2×SD16—612no—
161C2×SD16—1228no—
162C2×SD16C2×D6112yescomplemented
163C2×D8—612no—
164C2×D8—612no—
165C2×D8—612no—
166C2×D8—1228no—
167C2×D8C2×D6112yescomplemented
168C22×C8—612no—
169#127—612no—
170#127—611no—
171#127—1222no—
172#127C2×D6111yescharacteristic, complemented
173#124—611no—
174#124—611no—
175#124—611no—
176#124—1222no—
177#124C2×D6111yescharacteristic, complemented
178C2×C22⋊C4—1222no—
179C2×C42—611no—
180#97—612no—
181#97—611no—
182#97C2×D6111yescharacteristic, complemented
183C24—1221no—
184C24—1222no—
185C2×Q8—611no—
186C2×Q8—1221no—
187C2×Q8—1224no—
188C2×Q8C22×D6111yescharacteristic
189C2×D4—22no—
190C2×D4—611no—
191C2×D4—612no—
192C2×D4—1224no—
193C2×D4—1224no—
194C2×D4—1224no—
195C2×D4—1221no—
196C2×D4—1224no—
197C2×D4—1222no—
198C2×D4—1222no—
199C2×D4—2448no—
200C2×D4C22×D6111yescharacteristic
201C2×D4C22×D6112yes—
202C22×C4—611no—
203C22×C4—612no—
204C22×C4—1221no—
205SD16—24no—
206SD16—1224no—
207SD16—1224no—
208SD16—1224no—
209D8—24no—
210D8—1224no—
211D8—1224no—
212D8—1224no—
213C2×C8—612no—
214C2×C8—618no—
215C2×C8C22×D6112yes—
216C22⋊C4—22no—
217C22⋊C4—1222no—
218C22⋊C4—1222no—
219C22⋊C4—1222no—
220C42—612no—
221C42—611no—
222C42C22×D6111yescharacteristic
223C23—21no—
224C23—22no—
225C23—611no—
226C23—1221no—
227C23—1222no—
228C23—2442no—
229C23—2444no—
230C23—2448no—
231C23—2444no—
232Q8—22no—
233Q8—1222no—
234Q8—2442no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—1222no—
240D4—1222no—
241D4—1222no—
242D4—2444no—
243D4—2442no—
244D4—2444no—
245D4—2448no—
246C2×C4—21no—
247C2×C4—611no—
248C2×C4—614no—
249C2×C4—612no—
250C2×C4—1224no—
251C2×C4—1221no—
252C2×C4—2442no—
253C2×C4D4×D6112yes—
254C2×C4C23×D6111yescharacteristic
255C8—614no—
256C8D4×D6114yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—614no—
262C22—612no—
263C22—4888no—
264C22—4884no—
265C22—2444no—
266C22—2442no—
267C22—2441no—
268C22—2442no—
269C22C2×D4×D6111yescharacteristic
270C4—22no—
271C4—41no—
272C4—612no—
273C4—1222no—
274C4—2441no—
275C4C2×D4×D6112yes—
276C2—82no—
277C2—81no—
278C2—614no—
279C2—4882no—
280C2—4881no—
281C2#127×D6111yescharacteristic
282C2AGL(1,ℤ/8)×D6112yes—
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#496C2
socleC22×C61C22×D4
hypercenter#496D61
hypocenterC61C2×#496
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D61C2semidirect
2. C2×D8×D61C2semidirect
3. #127×D61C2semidirect
4. #124×D61C2semidirect
5. #16840152C2semidirect
6. #16840151C2semidirect
7. #16840150C2semidirect
8. #16840149C2semidirect
9. #16840148C2semidirect
10. #97×D61C2semidirect
11. C61×#496C2semidirect
16. C2×#11758190C22semidirect
17. C2×#11758200C22semidirect
18. C2×#11758201C22semidirect
20. C2×D488C22semidirect
21. C2×#11758199C22semidirect
22. C2×#11758199C22semidirect
25. #14519774C22semidirect
27. #14519773C22semidirect
28. #14519766C22semidirect
29. #14519765C22semidirect
30. #14519764C22semidirect
34. #14519688C22semidirect
35. #14519687C22semidirect
36. C2×SD16×C61C22semidirect
37. C2×D8×C61C22semidirect
38. #127×C61C22semidirect
39. #124×C61C22semidirect
40. #97×C61C22semidirect
80. C8×D61D4semidirect
113. C8×C61C2×D4semidirect
114. #92418C2×D4semidirect
141. D61#496direct (N × H)
151. #496D61direct (N × H)
153. C61C2×#496semidirect
162. C2×SD16C2×D61semidirect
167. C2×D8C2×D61semidirect
172. #127C2×D61semidirect
177. #124C2×D61semidirect
182. #97C2×D61semidirect
256. C8D4×D61semidirect