Finite Group Databasebeta

← #21456789

Subgroups — #21456789

Order 7808.

410 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 113 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#16840298C211yesmaximal, characteristic, complemented
2C2×#14519766C211yesmaximal, characteristic, complemented
3#16840147C212yesmaximal, complemented
4#16840118C212yesmaximal, complemented
5#16839986C212yesmaximal, complemented
6#16839980C212yesmaximal, complemented
7#16839912C211yesmaximal, characteristic, complemented
8#16839901C211yesmaximal, characteristic, complemented
9#16839898C211yesmaximal, characteristic
10#16839812C211yesmaximal, characteristic, complemented
11C61×#451C211yesmaximal, characteristic, complemented
12C22×D244C2211yescharacteristic
13C2×#11758201C2212yes—
14C2×#11758201C2212yescomplemented
15C2×D488C2212yes—
16#14519772—21no—
17#14519768C2211yescharacteristic, complemented
18#14519766—21no—
19#14519766—21no—
20#14519766C2212yes—
21#14519766C2212yes—
22#14519766C2211yescharacteristic, complemented
23#14519761—21no—
24#14519751C2211yescharacteristic
25Q8×Dic61C2211yescharacteristic, complemented
26#14519739C2211yescharacteristic, complemented
27#14519737—21no—
28#14519735—21no—
29C2×#11758198—21no—
30C2×#11758198—21no—
31C2×C4×Dic61C2211yescharacteristic
32#14519706C2212yescomplemented
33#14519701C2212yescomplemented
34#14519699C2212yes—
35#14519697C2212yes—
36C22⋊#92418C2211yescharacteristic
37C8×Dic61C2212yescomplemented
38C22×Q8×C61C2211yescharacteristic
39C2×Q16×C61C2212yescomplemented
40C22⋊Q8×C61C2211yescharacteristic
41#103×C61C2212yescomplemented
42C22⋊C8×C61C2211yescharacteristic
43C23×D61—21no—
44C2×Q8×C61—21no—
45C2×Q8×C61—42no—
46C2×Q8×C61C2312yes—
47C2×Q8×C61C2311yescharacteristic
48C22×C4×C61—21no—
49C22×C4×C61C2311yescharacteristic
50C2×C22⋊D61—21no—
51C22×Dic61D411yescharacteristic
52C22×Dic61D411yescharacteristic
53C2×D244—21no—
54C2×D244—42no—
55C2×D244C2312yes—
56C2×D244C2311yescharacteristic
57C2×C4×D61—21no—
58Q16×C61—42no—
59C2×C8×C61C2312yes—
60C4⋊C4×C61—21no—
61C4⋊C4×C61C2311yescharacteristic
62C22⋊C4×C61—21no—
63#11758201—42no—
64#11758201—42no—
65#11758198—21no—
66#11758198—22no—
67#11758198—22no—
68#11758198—22no—
69#11758198—22no—
70#11758198—21no—
71#11758198—21no—
72#11758198—21no—
73C4⋊Dic61—21no—
74C4⋊Dic61—21no—
75C4⋊Dic61C2311yescharacteristic
76#11758196—21no—
77C4×Dic61—21no—
78C4×Dic61D411yescharacteristic
79C4×Dic61D411yescharacteristic
80C4×Dic61C2311yescharacteristic
81C4×Dic61C2311yescharacteristic
82C2×#92418C2312yes—
83D488—42no—
84C23×C61C2×D411yescharacteristic
85C22×D61—21no—
86C22×D61—22no—
87C22×D61—42no—
88C22×D61—42no—
89C22×D61—42no—
90Q8×C61—22no—
91Q8×C61—22no—
92Q8×C61—22no—
93Q8×C61—42no—
94C2×C4×C61—22no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—22no—
98C2×C4×C61—42no—
99C2×C4×C61C2×D411yescharacteristic
100C2×C4×C61C2411yescharacteristic
101C22⋊D61—42no—
102C2×Dic61—22no—
103C2×Dic61—21no—
104C2×Dic61—41no—
105C2×Dic61C2×D411yescharacteristic
106C2×Dic61C2×D411yescharacteristic
107C2×Dic61C2×D411yescharacteristic
108C2×Dic61C2×D411yescharacteristic
109D244—22no—
110D244—22no—
111D244—22no—
112D244—42no—
113C4×D61—42no—
114C8×C61—22no—
115#92418—22no—
116C22×C61—21no—
117C22×C61—21no—
118C22×C61#13512yes—
119C22×C61C22×D411yescharacteristic
120C2×D61—42no—
121C2×D61—41no—
122C2×D61—41no—
123C2×D61—42no—
124C2×D61—41no—
125C2×D61—42no—
126C2×D61—84no—
127C4×C61—21no—
128C4×C61—42no—
129C4×C61—41no—
130C4×C61—41no—
131C4×C61H(5,𝔽2)12yes—
132Dic61—21no—
133Dic61—21no—
134Dic61—21no—
135Dic61—21no—
136Dic61—41no—
137#3211—611nomaximal, Sylow
138C2×C61—22no—
139C2×C61#53511yescharacteristic
140C2×C61C2×#13511yescharacteristic
141C2×C61#57511yescharacteristic
142D61—81no—
143D61—82no—
144#529—611no—
145C2×#124—611no—
146#495—612no—
147#488—612no—
148#464—612no—
149#463—612no—
150#451D6111yescharacteristic, complemented
151#450—611no—
152#448—611no—
153#447—611no—
154#431—611no—
155C61#321111yescharacteristic, complemented, Sylow
156C22×Q8C2×D6111yescharacteristic
157C22×D4—611no—
158C2×Q16C2×D6112yescomplemented
159C2×SD16—612no—
160C2×SD16—612no—
161C2×D8—612no—
162#126—1221no—
163#125—611no—
164#124—612no—
165#124—612no—
166#124—611no—
167#124—1221no—
168#124—1221no—
169#123—1221no—
170C22⋊Q8C2×D6111yescharacteristic
171#121—611no—
172C4×Q8—611no—
173C4×D4—611no—
174C4×D4—1221no—
175#117—1221no—
176C2×C22⋊C4—1221no—
177C2×C22⋊C4—1221no—
178C2×C42—611no—
179C4⋊C8—612no—
180#103—612no—
181#103C2×D6112yescomplemented
182#102—612no—
183#102—612no—
184C22⋊C8—611no—
185C22⋊C8C2×D6111yescharacteristic
186C4×C8—612no—
187C24—1221no—
188C2×Q8—21no—
189C2×Q8—42no—
190C2×Q8C22×D6112yes—
191C2×Q8C22×D6111yescharacteristic
192C2×D4—612no—
193C2×D4—611no—
194C2×D4—1221no—
195C2×D4—1221no—
196C2×D4—2442no—
197C22×C4—21no—
198C22×C4—611no—
199C22×C4—611no—
200C22×C4—1221no—
201C22×C4C22×D6111yescharacteristic
202Q16—42no—
203SD16—2442no—
204SD16—2442no—
205D8—2442no—
206C2×C8—612no—
207C2×C8C22×D6112yes—
208C4⋊C4—21no—
209C4⋊C4—611no—
210C4⋊C4—1221no—
211C4⋊C4—1221no—
212C4⋊C4—1221no—
213C4⋊C4C22×D6111yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—1222no—
216C22⋊C4—1222no—
217C22⋊C4—1222no—
218C22⋊C4—1222no—
219C22⋊C4—1221no—
220C22⋊C4—1221no—
221C22⋊C4—1221no—
222C22⋊C4—1221no—
223C42—611no—
224C42—611no—
225C42—611no—
226C42—611no—
227C42—1221no—
228C23—1221no—
229C23—1222no—
230C23—2442no—
231C23—2442no—
232C23—2442no—
233C23D4×D6111yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—22no—
237Q8—42no—
238D4—1222no—
239D4—1222no—
240D4—1222no—
241D4—2442no—
242D4—2442no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—611no—
249C2×C4—611no—
250C2×C4—611no—
251C2×C4—611no—
252C2×C4—1222no—
253C2×C4—1221no—
254C2×C4—2442no—
255C2×C4—2441no—
256C2×C4D4×D6111yescharacteristic
257C2×C4C23×D6111yescharacteristic
258C8—22no—
259C8—1222no—
260C22—21no—
261C22—21no—
262C22—4884no—
263C22—2441no—
264C22—2442no—
265C22—2441no—
266C22—2442no—
267C22—2442no—
268C22—2441no—
269C22#1451980012yes—
270C22C2×D4×D6111yescharacteristic
271C4—21no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—1221no—
276C4—1221no—
277C4—1221no—
278C4—1221no—
279C4—2441no—
280C4#1451982212yes—
281C2—22no—
282C2—4882no—
283C2—4881no—
284C2#1684032711yescharacteristic
285C2C2×#1451980011yescharacteristic
286C2#1684056111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#451C2
socleC22×C61C22×D4
hypercenter#451D61
hypocenterC61#3211
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #16840298C2semidirect
2. C2×#14519766C2semidirect
3. #16840147C2semidirect
4. #16840118C2semidirect
5. #16839986C2semidirect
6. #16839980C2semidirect
7. #16839912C2semidirect
8. #16839901C2semidirect
10. #16839812C2semidirect
11. C61×#451C2semidirect
14. C2×#11758201C22semidirect
17. #14519768C22semidirect
22. #14519766C22semidirect
25. Q8×Dic61C22semidirect
26. #14519739C22semidirect
32. #14519706C22semidirect
33. #14519701C22semidirect
37. C8×Dic61C22semidirect
39. C2×Q16×C61C22semidirect
41. #103×C61C22semidirect
150. #451D61semidirect
155. C61#3211semidirect
158. C2×Q16C2×D61semidirect
181. #103C2×D61semidirect