Finite Group Databasebeta

← #21457510

Subgroups — #21457510

Order 7808.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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
2Q8×D244C211yesmaximal, characteristic, complemented
3#16840397C211yesmaximal, characteristic
4#16840134C211yesmaximal, characteristic, complemented
5#16839990C212yesmaximal, complemented
6#16839968C211yesmaximal, characteristic, complemented
7#16839914C212yesmaximal, complemented
8#16839899C212yesmaximal, complemented
9C4×#11758201C211yesmaximal, characteristic, complemented
10#16839827C211yesmaximal, characteristic, complemented
11C61×#460C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2212yescomplemented
13C2×D4×D61—21no—
14C2×D4×D61C2212yes—
15C2×#11758190C2212yes—
16C2×#11758200C2212yescomplemented
17C2×#11758200C2212yescomplemented
18C2×#11758201C2211yescharacteristic
19SD16×D61—28no—
20C2×C8×D61C2212yescomplemented
21#14519777—21no—
22#14519777C2211yescharacteristic, complemented
23#14519757—22no—
24#14519757C2212yescomplemented
25#14519748—21no—
26#14519748C2212yes—
27#14519745—22no—
28C4×D244—21no—
29C4×D244C2211yescharacteristic
30C4×D244C2211yescharacteristic
31#14519708C2211yescharacteristic, complemented
32#14519702C2212yescomplemented
33#14519701C2211yescharacteristic, complemented
34#14519697C2211yescharacteristic
35#14519689C2212yescomplemented
36C4×#92418C2211yescharacteristic, complemented
37C2×SD16×C61C2212yescomplemented
38#127×C61C2211yescharacteristic
39C4×Q8×C61C2211yescharacteristic, complemented
40C4⋊C8×C61C2211yescharacteristic, complemented
41#102×C61C2212yescomplemented
42C23×D61—22no—
43C2×Q8×C61C2311yescharacteristic
44C2×D4×C61—21no—
45C2×D4×C61—21no—
46C2×D4×C61C2312yes—
47C2×C22⋊D61—22no—
48C2×C22⋊D61—22no—
49Q8×D61—24no—
50Q8×D61—42no—
51D4×D61—24no—
52D4×D61—42no—
53D4×D61—44no—
54C2×D244D411yescharacteristic
55C2×D244C2311yescharacteristic
56C2×C4×D61—22no—
57C2×C4×D61—21no—
58C2×C4×D61C2312yes—
59C2×Dic122—22no—
60C2×Dic122C2312yes—
61SD16×C61—24no—
62C2×C8×C61C2312yes—
63C4⋊C4×C61—21no—
64C4⋊C4×C61C2311yescharacteristic
65C42×C61—21no—
66C42×C61C2311yescharacteristic
67C22⋊Dic61—22no—
68#11758201D414yescomplemented
69#11758200—24no—
70#11758200—42no—
71#11758198—22no—
72#11758198—21no—
73#11758198D412yes—
74C4⋊Dic61—21no—
75C4⋊Dic61—21no—
76C4⋊Dic61—22no—
77C4⋊Dic61D411yescharacteristic
78C4⋊Dic61C2311yescharacteristic
79C2×#92418C2311yescharacteristic
80C2×#92418C2311yescharacteristic
81#11758190—24no—
82C8×D61—24no—
83C23×C61—22no—
84C22×D61—21no—
85C22×D61—44no—
86C22×D61—44no—
87C22×D61—82no—
88C22×D61C2×D412yes—
89Q8×C61—21no—
90Q8×C61C2×D412yescomplemented
91D4×C61—22no—
92D4×C61—22no—
93D4×C61—42no—
94D4×C61—81no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61C2×D411yescharacteristic
98C2×C4×C61C2×D411yescharacteristic
99C2×C4×C61C2411yescharacteristic
100C22⋊D61—44no—
101C22⋊D61—44no—
102C2×Dic61—22no—
103C2×Dic61—21no—
104C2×Dic61C2×D412yes—
105D244—21no—
106D244SD1614yescomplemented
107D244C2×D412yes—
108C4×D61—22no—
109C4×D61—22no—
110C4×D61—44no—
111C4×D61—42no—
112Dic122—22no—
113Dic122—22no—
114Dic122—44no—
115C8×C61—22no—
116#92418—21no—
117#92418C2×D412yescomplemented
118C22×C61—42no—
119C22×C61—42no—
120C22×C61—42no—
121C22×C61C22×D411yescharacteristic
122C2×D61—22no—
123C2×D61—22no—
124C2×D61—22no—
125C2×D61—41no—
126C2×D61—41no—
127C2×D61—88no—
128C2×D61C2×SD1614yescomplemented
129C4×C61—22no—
130C4×C61—21no—
131C4×C61—41no—
132C4×C61C2×SD1612yescomplemented
133C4×C61C22×D411yescharacteristic
134C4×C61H(5,𝔽2)11yescharacteristic
135Dic61—22no—
136Dic61—42no—
137Dic61—41no—
138D4×SD16—611nomaximal, Sylow
139C2×C61—82no—
140C2×C61D4211yescharacteristic
141C2×C61C22×SD1611yescharacteristic
142C2×C61#57711yescharacteristic
143D61—24no—
144D61—42no—
145C22×SD16—612no—
146D4×Q8—611no—
147D42—611no—
148#492—611no—
149#465—612no—
150#461—611no—
151#460D6111yescharacteristic, complemented
152#450—612no—
153#448—612no—
154C4×SD16—611no—
155C8×D4—611no—
156C61D4×SD1611yescharacteristic, complemented, Sylow
157C22×Q8—612no—
158C22×D4—612no—
159C22×D4—1221no—
160C2×SD16—611no—
161C2×SD16—612no—
162C2×SD16—612no—
163C2×SD16—612no—
164C2×SD16—1228no—
165C2×SD16C2×D6112yescomplemented
166C22×C8—612no—
167C4⋊Q8—611no—
168C4⋊Q8—1221no—
169#127C2×D6111yescharacteristic
170C22⋊Q8—612no—
171C22⋊Q8—1222no—
172#121—612no—
173#121—1221no—
174#120—1222no—
175C4×Q8C2×D6111yescharacteristic, complemented
176C4×D4—611no—
177C4×D4—611no—
178C4×D4—1221no—
179#106—611no—
180C4⋊C8C2×D6111yescharacteristic, complemented
181#103—611no—
182#103—612no—
183#102—611no—
184#102C2×D6112yescomplemented
185C22⋊C8—612no—
186C4×C8—611no—
187C24—1222no—
188C2×Q8—612no—
189C2×Q8—1222no—
190C2×Q8—1224no—
191C2×Q8—2442no—
192C2×Q8C22×D6111yescharacteristic
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—611no—
196C2×D4—611no—
197C2×D4—1222no—
198C2×D4—1224no—
199C2×D4—1222no—
200C2×D4—2442no—
201C2×D4—2444no—
202C2×D4C22×D6112yes—
203C22×C4—612no—
204C22×C4—1222no—
205C22×C4—1221no—
206SD16—24no—
207SD16—614no—
208SD16—1224no—
209SD16—1224no—
210SD16—2442no—
211C2×C8—611no—
212C2×C8—611no—
213C2×C8—1224no—
214C2×C8C22×D6112yes—
215C4⋊C4—21no—
216C4⋊C4—611no—
217C4⋊C4—611no—
218C4⋊C4—1222no—
219C4⋊C4—1221no—
220C4⋊C4—1221no—
221C4⋊C4C22×D6111yescharacteristic
222C22⋊C4—612no—
223C22⋊C4—1222no—
224C22⋊C4—1222no—
225C22⋊C4—1221no—
226C42—21no—
227C42C22×D6111yescharacteristic
228C23—22no—
229C23—612no—
230C23—1221no—
231C23—4882no—
232C23—2444no—
233C23—2444no—
234Q8—21no—
235Q8—1222no—
236Q8—1222no—
237Q8—2444no—
238Q8D4×D6112yescomplemented
239D4—22no—
240D4—22no—
241D4—42no—
242D4—81no—
243D4—612no—
244D4—614no—
245D4—1221no—
246D4—2444no—
247D4—2444no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—612no—
251C2×C4—1222no—
252C2×C4—1222no—
253C2×C4—1221no—
254C2×C4—1222no—
255C2×C4—2444no—
256C2×C4—2442no—
257C2×C4D4×D6111yescharacteristic
258C2×C4D4×D6111yescharacteristic
259C2×C4C23×D6111yescharacteristic
260C8—22no—
261C8—612no—
262C8—1221no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—614no—
267C22—1222no—
268C22—1222no—
269C22—1222no—
270C22—4888no—
271C22—2441no—
272C22—2441no—
273C22C2×D4×D6111yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—1222no—
278C4—2441no—
279C4—2442no—
280C4SD16×D6112yescomplemented
281C4C2×D4×D6111yescharacteristic
282C4#1451982311yescharacteristic
283C2—82no—
284C2—1224no—
285C2—2442no—
286C2#1684039611yescharacteristic
287C2C2×SD16×D6111yescharacteristic
288C2#1684057211yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#460C2
socleC22×C61C22×D4
hypercenter#460D61
hypocenterC61D4×SD16
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D61C2semidirect
2. Q8×D244C2semidirect
4. #16840134C2semidirect
5. #16839990C2semidirect
6. #16839968C2semidirect
7. #16839914C2semidirect
8. #16839899C2semidirect
9. C4×#11758201C2semidirect
10. #16839827C2semidirect
11. C61×#460C2semidirect
12. C2×Q8×D61C22semidirect
16. C2×#11758200C22semidirect
17. C2×#11758200C22semidirect
20. C2×C8×D61C22semidirect
22. #14519777C22semidirect
24. #14519757C22semidirect
31. #14519708C22semidirect
32. #14519702C22semidirect
33. #14519701C22semidirect
35. #14519689C22semidirect
36. C4×#92418C22semidirect
37. C2×SD16×C61C22semidirect
39. C4×Q8×C61C22semidirect
40. C4⋊C8×C61C22semidirect
41. #102×C61C22semidirect
68. #11758201D4semidirect
90. Q8×C61C2×D4semidirect
106. D244SD16semidirect
117. #92418C2×D4semidirect
128. C2×D61C2×SD16semidirect
132. C4×C61C2×SD16semidirect
151. #460D61semidirect
156. C61D4×SD16semidirect
165. C2×SD16C2×D61semidirect
175. C4×Q8C2×D61semidirect
180. C4⋊C8C2×D61semidirect
184. #102C2×D61semidirect
238. Q8D4×D61semidirect
280. C4SD16×D61semidirect