Finite Group Databasebeta

← #21456743

Subgroups — #21456743

Order 7808.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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#16840309C211yesmaximal, characteristic
2C4⋊Q8×D61C211yesmaximal, characteristic, complemented
3#16840133C212yesmaximal, complemented
4#16840111C212yesmaximal, complemented
5#16839968C211yesmaximal, characteristic, complemented
6#16839967C211yesmaximal, characteristic, complemented
7#16839964C211yesmaximal, characteristic, complemented
8#16839902C212yesmaximal, complemented
9#16839899C212yesmaximal, complemented
10#16839814C211yesmaximal, characteristic, complemented
11C61×#461C211yesmaximal, characteristic, complemented
12C2×#11758222—21no—
13C2×Q8×D61C2212yes—
14C2×#11758190C2212yes—
15C2×#11758200C2212yes—
16C2×#11758201C2212yes—
17#14519777C2211yescharacteristic
18#14519776—21no—
19#14519776—21no—
20#14519776C2211yescharacteristic
21#14519775C2212yes—
22#14519773C2211yescharacteristic
23#14519763—21no—
24#14519752C2211yescharacteristic
25#14519750—21no—
26#14519748C2211yescharacteristic
27C4×D244C2211yescharacteristic
28C4×C22⋊D61—21no—
29C4⋊C4×D61—21no—
30C4⋊C4×D61—21no—
31C42×D61C2211yescharacteristic
32C4⋊#92418C2211yescharacteristic
33#14519702C2212yes—
34#14519701C2212yes—
35#14519700C2212yes—
36#14519689C2212yes—
37C8×Dic61C2212yes—
38C2×SD16×C61C2212yes—
39C4⋊Q8×C61C2211yescharacteristic
40C4×D4×C61C2211yescharacteristic
41C4⋊C8×C61C2211yescharacteristic
42#103×C61C2212yes—
43C2×Q8×C61C2312yes—
44C2×D4×C61C2311yescharacteristic
45C22×C4×C61—21no—
46C2×C22⋊D61—21no—
47C2×C22⋊D61—21no—
48C2×C22⋊D61—21no—
49Q8×D61—42no—
50Q8×D61—42no—
51#11758222—44no—
52C2×D244—21no—
53C2×D244C2311yescharacteristic
54C2×C4×D61—22no—
55C2×C4×D61—21no—
56C2×C4×D61D411yescharacteristic
57C2×C4×D61D411yescharacteristic
58C2×C4×D61C2311yescharacteristic
59C2×Dic122—22no—
60C2×Dic122—21no—
61C2×Dic122C2312yes—
62SD16×C61—42no—
63C2×C8×C61C2312yes—
64C4⋊C4×C61—21no—
65C4⋊C4×C61—21no—
66C4⋊C4×C61C2311yescharacteristic
67C22⋊C4×C61—21no—
68C42×C61C2311yescharacteristic
69C22⋊Dic61—21no—
70#11758201—42no—
71#11758200—42no—
72#11758198—21no—
73#11758198—21no—
74C4⋊Dic61—21no—
75C4⋊Dic61—21no—
76C4⋊Dic61C2311yescharacteristic
77#11758196—22no—
78#11758196—22no—
79#11758196—21no—
80C4×Dic61D411yescharacteristic
81C4×Dic61D411yescharacteristic
82C4×Dic61C2311yescharacteristic
83C2×#92418C2312yes—
84#11758190—42no—
85C23×C61—21no—
86C22×D61—21no—
87C22×D61C2×D411yescharacteristic
88Q8×C61—22no—
89Q8×C61—22no—
90D4×C61—21no—
91D4×C61—21no—
92C2×C4×C61—22no—
93C2×C4×C61—21no—
94C2×C4×C61—42no—
95C2×C4×C61C2×D411yescharacteristic
96C2×C4×C61C2×D411yescharacteristic
97C2×C4×C61C2411yescharacteristic
98C22⋊D61—42no—
99C22⋊D61—42no—
100C22⋊D61—42no—
101C2×Dic61—22no—
102C2×Dic61—21no—
103C2×Dic61C2×D411yescharacteristic
104C2×Dic61C2×D411yescharacteristic
105C2×Dic61C2×D411yescharacteristic
106D244—21no—
107D244—21no—
108D244—42no—
109C4×D61—21no—
110C4×D61—21no—
111C4×D61—22no—
112C4×D61—44no—
113C4×D61—42no—
114C4×D61SD1614yescomplemented
115Dic122—22no—
116Dic122—22no—
117Dic122—44no—
118Dic122—42no—
119C8×C61—22no—
120#92418—22no—
121C22×C61—41no—
122C22×C61—41no—
123C22×C61—41no—
124C22×C61C22×D411yescharacteristic
125C2×D61—21no—
126C2×D61—21no—
127C2×D61—41no—
128C2×D61—41no—
129C2×D61—41no—
130C2×D61C2×SD1612yes—
131C4×C61—21no—
132C4×C61—42no—
133C4×C61—41no—
134C4×C61C2×SD1612yes—
135C4×C61H(5,𝔽2)11yescharacteristic
136C4×C61H(5,𝔽2)11yescharacteristic
137Dic61—21no—
138Dic61—42no—
139Dic61—41no—
140Dic61C2×SD1612yes—
141Dic61#13712yes—
142#3206—611nomaximal, Sylow
143C2×C61—81no—
144C2×C61#53511yescharacteristic
145C2×C61C22×SD1611yescharacteristic
146C2×C61C2×#13711yescharacteristic
147D61—22no—
148D61—81no—
149#532—611no—
150C2×C4⋊Q8—611no—
151#492—612no—
152#487—612no—
153#461—611no—
154#461—611no—
155#461—611no—
156#461D6111yescharacteristic, complemented
157#448—612no—
158#448—612no—
159#431—611no—
160C61#320611yescharacteristic, complemented, Sylow
161C2×#41—1221no—
162C22×Q8—612no—
163C2×SD16—612no—
164C2×SD16—612no—
165C2×SD16—612no—
166C2×SD16C2×D6112yes—
167C4⋊Q8—611no—
168C4⋊Q8—612no—
169C4⋊Q8—611no—
170C4⋊Q8—1221no—
171C4⋊Q8—1221no—
172C4⋊Q8C2×D6111yescharacteristic
173#127—611no—
174#124—1221no—
175#121—611no—
176#121—611no—
177#121—1221no—
178C4×D4—611no—
179C4×D4—1221no—
180C4×D4C2×D6111yescharacteristic
181C2×C4⋊C4—1221no—
182C2×C4⋊C4—1221no—
183C2×C42—611no—
184C4⋊C8—611no—
185C4⋊C8C2×D6111yescharacteristic
186#103—612no—
187#103—612no—
188#103—612no—
189#103C2×D6112yes—
190C22⋊C8—612no—
191C4×C8—612no—
192#41—2444no—
193C2×Q8—612no—
194C2×Q8—1221no—
195C2×Q8—1222no—
196C2×Q8—2442no—
197C2×Q8—2442no—
198C2×Q8C22×D6112yes—
199C2×D4—611no—
200C2×D4—1221no—
201C2×D4—1221no—
202C2×D4—1221no—
203C2×D4—1221no—
204C2×D4C22×D6111yescharacteristic
205C22×C4—21no—
206C22×C4—611no—
207C22×C4—611no—
208C22×C4—611no—
209C22×C4—1222no—
210C22×C4—1221no—
211SD16—42no—
212SD16—2442no—
213SD16—2442no—
214SD16—2442no—
215C2×C8—612no—
216C2×C8C22×D6112yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—611no—
220C4⋊C4—1222no—
221C4⋊C4—1222no—
222C4⋊C4—1221no—
223C4⋊C4—1221no—
224C4⋊C4—1221no—
225C4⋊C4C22×D6111yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—1221no—
228C22⋊C4—1221no—
229C22⋊C4—1221no—
230C42—611no—
231C42—611no—
232C42—611no—
233C42C22×D6111yescharacteristic
234C23—21no—
235C23—611no—
236C23—1221no—
237Q8—22no—
238Q8—22no—
239Q8—1222no—
240Q8—1222no—
241Q8—2442no—
242Q8—2444no—
243D4—21no—
244D4—21no—
245D4—1221no—
246D4—1221no—
247D4—2442no—
248D4—2442no—
249D4—2442no—
250D4—2442no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—611no—
255C2×C4—611no—
256C2×C4—614no—
257C2×C4—611no—
258C2×C4—1222no—
259C2×C4—1221no—
260C2×C4—1221no—
261C2×C4—1221no—
262C2×C4—1222no—
263C2×C4—2444no—
264C2×C4—2442no—
265C2×C4D4×D6111yescharacteristic
266C2×C4D4×D6111yescharacteristic
267C2×C4C23×D6111yescharacteristic
268C8—22no—
269C8—1222no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—612no—
274C22—1221no—
275C22—1221no—
276C22—2441no—
277C22—2441no—
278C22—2441no—
279C22C2×D4×D6111yescharacteristic
280C4—21no—
281C4—41no—
282C4—42no—
283C4—612no—
284C4—612no—
285C4—1221no—
286C4—2441no—
287C4—2442no—
288C4SD16×D6112yes—
289C4#1451982211yescharacteristic
290C4#1451982311yescharacteristic
291C2—81no—
292C2—1222no—
293C2—4881no—
294C2#1684032811yescharacteristic
295C2C2×SD16×D6111yescharacteristic
296C2#137×D6111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#461C2
socleC22×C61C22×D4
hypercenter#461D61
hypocenterC61#3206
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D61C2semidirect
3. #16840133C2semidirect
4. #16840111C2semidirect
5. #16839968C2semidirect
6. #16839967C2semidirect
7. #16839964C2semidirect
8. #16839902C2semidirect
9. #16839899C2semidirect
10. #16839814C2semidirect
11. C61×#461C2semidirect
114. C4×D61SD16semidirect
156. #461D61semidirect
160. C61#3206semidirect