Finite Group Databasebeta

← #21456204

Subgroups — #21456204

Order 7808.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#16840317C211yesmaximal, characteristic, complemented
2#124×D61C211yesmaximal, characteristic
3#16840124C211yesmaximal, characteristic, complemented
4#16840122C211yesmaximal, characteristic, complemented
5#16840121C211yesmaximal, characteristic, complemented
6#16840118C212yesmaximal, complemented
7#16840115C212yesmaximal, complemented
8#16839769C212yesmaximal, complemented
9#16839765C212yesmaximal
10#16839714C211yesmaximal, characteristic, complemented
11C61×#489C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2211yescharacteristic
13C2×D4×D61C2211yescharacteristic
14C2×#11758189C2212yes—
15#14519777C2211yescharacteristic, complemented
16#14519768C2212yescomplemented
17#14519766C2211yescharacteristic
18#14519765C2211yescharacteristic
19#14519764C2211yescharacteristic
20#14519763—21no—
21#14519763—21no—
22#14519757—21no—
23#14519753—21no—
24Q8×Dic61—21no—
25#14519742—21no—
26#14519735C2212yes—
27C22⋊C4×D61—21no—
28C22⋊C4×D61—21no—
29C42×D61C2211yescharacteristic
30#14519702C2212yescomplemented
31#14519701C2212yescomplemented
32#14519700C2212yescomplemented
33#14519699C2212yes—
34#14519698C2212yes—
35#14519697C2212yes—
36#14519688C2211yescharacteristic, complemented
37C8×Dic61C2212yescomplemented
38C4⋊Q8×C61C2211yescharacteristic, complemented
39#124×C61C2211yescharacteristic
40#103×C61C2212yescomplemented
41#102×C61C2212yes—
42#97×C61C2211yescharacteristic, complemented
43C23×D61—21no—
44C2×Q8×C61—21no—
45C2×Q8×C61C2311yescharacteristic
46C2×D4×C61C2311yescharacteristic
47C2×C22⋊D61—21no—
48Q8×D61—24no—
49D4×D61—24no—
50C2×D244C2311yescharacteristic
51C2×C4×D61—21no—
52C2×C4×D61D411yescharacteristic
53C2×C4×D61D411yescharacteristic
54C2×C4×D61C2311yescharacteristic
55C2×Dic122—21no—
56C2×Dic122—21no—
57C2×Dic122C2311yescharacteristic
58C2×C8×C61C2312yes—
59C4⋊C4×C61—21no—
60C4⋊C4×C61C2312yes—
61C22⋊C4×C61—21no—
62C22⋊C4×C61—21no—
63C42×C61C2311yescharacteristic
64C22⋊Dic61—21no—
65C22⋊Dic61—21no—
66#11758198—22no—
67#11758198—21no—
68#11758198—21no—
69#11758198—21no—
70#11758198—21no—
71C4⋊Dic61—21no—
72C4⋊Dic61—22no—
73C4⋊Dic61C2312yes—
74#11758196—22no—
75C4×Dic61—22no—
76C4×Dic61D411yescharacteristic
77C4×Dic61D411yescharacteristic
78C4×Dic61C2311yescharacteristic
79C2×#92418C2312yes—
80#11758189—24no—
81C23×C61—21no—
82C22×D61—21no—
83C22×D61—42no—
84C22×D61—42no—
85C22×D61—42no—
86C22×D61C2×D411yescharacteristic
87Q8×C61—22no—
88Q8×C61—42no—
89D4×C61—22no—
90C2×C4×C61—22no—
91C2×C4×C61—21no—
92C2×C4×C61C2×D411yescharacteristic
93C2×C4×C61C2×D411yescharacteristic
94C2×C4×C61C2411yescharacteristic
95C22⋊D61—42no—
96C2×Dic61—22no—
97C2×Dic61—21no—
98C2×Dic61C2×D411yescharacteristic
99C2×Dic61C2×D411yescharacteristic
100C2×Dic61C2×D411yescharacteristic
101D244—22no—
102C4×D61—41no—
103C4×D61—41no—
104C4×D61—42no—
105C4×D61#4114yes—
106Dic122—22no—
107Dic122—42no—
108Dic122—42no—
109C8×C61—22no—
110#92418—22no—
111C22×C61—41no—
112C22×C61—41no—
113C22×C61—41no—
114C22×C61C22×D411yescharacteristic
115C2×D61—21no—
116C2×D61—21no—
117C2×D61—41no—
118C2×D61—41no—
119C2×D61—41no—
120C2×D61—84no—
121C2×D61#12412yes—
122C4×C61—21no—
123C4×C61—21no—
124C4×C61—41no—
125C4×C61—42no—
126C4×C61C2×#4112yes—
127Dic61—21no—
128Dic61—21no—
129Dic61—41no—
130Dic61—42no—
131Dic61#12412yes—
132#3140—611nomaximal, Sylow
133C2×C61—81no—
134C2×C61C2×#12411yescharacteristic
135C2×C61#57511yescharacteristic
136C2×C61#57511yescharacteristic
137D61—22no—
138D61—81no—
139#533—611no—
140C2×#124—611no—
141#489—611no—
142#489—611no—
143#489—611no—
144#489D6111yescharacteristic, complemented
145#488—612no—
146#488—612no—
147#419—612no—
148#418—612no—
149C4×M16—611no—
150C61#314011yescharacteristic, complemented, Sylow
151C22×Q8—611no—
152C22×D4—611no—
153C2×M16—612no—
154C4⋊Q8—611no—
155C4⋊Q8C2×D6111yescharacteristic, complemented
156#125—612no—
157#124—611no—
158#124—611no—
159#124—611no—
160#124—1221no—
161#124—1221no—
162#124C2×D6111yescharacteristic
163C22⋊Q8—1221no—
164C22⋊Q8—1221no—
165C4×Q8—1221no—
166C4×Q8—1221no—
167#117—612no—
168C2×C22⋊C4—1221no—
169C2×C22⋊C4—1221no—
170C2×C42—611no—
171#103—612no—
172#103—612no—
173#103—612no—
174#103C2×D6112yescomplemented
175#102—612no—
176#102—612no—
177#102—612no—
178#102C2×D6112yes—
179#97—611no—
180#97C2×D6111yescharacteristic, complemented
181C4×C8—612no—
182C24—1221no—
183C2×Q8—21no—
184C2×Q8—611no—
185C2×Q8—1221no—
186C2×Q8—1224no—
187C2×Q8—1221no—
188C2×Q8C22×D6111yescharacteristic
189C2×D4—611no—
190C2×D4—1224no—
191C2×D4—1221no—
192C2×D4C22×D6111yescharacteristic
193C22×C4—611no—
194C22×C4—611no—
195C22×C4—611no—
196C22×C4—1221no—
197M16—1224no—
198C2×C8—612no—
199C2×C8C22×D6112yes—
200C4⋊C4—21no—
201C4⋊C4—612no—
202C4⋊C4—1222no—
203C4⋊C4—1222no—
204C4⋊C4—1221no—
205C4⋊C4C22×D6112yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—1221no—
209C22⋊C4—1221no—
210C22⋊C4—1222no—
211C22⋊C4—1221no—
212C22⋊C4—1221no—
213C22⋊C4—1221no—
214C22⋊C4—1221no—
215C42—611no—
216C42—611no—
217C42—611no—
218C42—1222no—
219C42C22×D6111yescharacteristic
220C23—21no—
221C23—611no—
222C23—1221no—
223C23—2442no—
224C23—2442no—
225C23—2442no—
226Q8—22no—
227Q8—42no—
228Q8—1222no—
229Q8—2442no—
230Q8—2442no—
231D4—22no—
232D4—1222no—
233D4—2442no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—611no—
237C2×C4—614no—
238C2×C4—611no—
239C2×C4—611no—
240C2×C4—1222no—
241C2×C4—1221no—
242C2×C4—2441no—
243C2×C4—2441no—
244C2×C4—2442no—
245C2×C4D4×D6111yescharacteristic
246C2×C4D4×D6111yescharacteristic
247C2×C4C23×D6111yescharacteristic
248C8—22no—
249C8—1222no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—612no—
254C22—1221no—
255C22—1221no—
256C22—4884no—
257C22—2441no—
258C22—2441no—
259C22—2441no—
260C22C2×D4×D6111yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—612no—
266C4—1221no—
267C4—1221no—
268C4—2441no—
269C4—2442no—
270C4#41×D6112yes—
271C2—81no—
272C2—1222no—
273C2—4881no—
274C2#124×D6111yescharacteristic
275C2#1684055811yescharacteristic
276C2#1684056111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#489C2
socleC22×C61C22×D4
hypercenter#489D61
hypocenterC61#3140
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16840317C2semidirect
3. #16840124C2semidirect
4. #16840122C2semidirect
5. #16840121C2semidirect
6. #16840118C2semidirect
7. #16840115C2semidirect
8. #16839769C2semidirect
10. #16839714C2semidirect
11. C61×#489C2semidirect
15. #14519777C22semidirect
16. #14519768C22semidirect
30. #14519702C22semidirect
31. #14519701C22semidirect
32. #14519700C22semidirect
36. #14519688C22semidirect
37. C8×Dic61C22semidirect
38. C4⋊Q8×C61C22semidirect
40. #103×C61C22semidirect
42. #97×C61C22semidirect
144. #489D61semidirect
150. C61#3140semidirect
155. C4⋊Q8C2×D61semidirect
174. #103C2×D61semidirect
180. #97C2×D61semidirect