Finite Group Databasebeta

← #21456842

Subgroups — #21456842

Order 7808.

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

normal subgroups only

This group has 386 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
1C4⋊Q8×D61C211yesmaximal, characteristic, complemented
2#16840291C211yesmaximal, characteristic
3#16840157C211yesmaximal, characteristic, complemented
4#16840131C211yesmaximal, characteristic, complemented
5#16840130C211yesmaximal, characteristic, complemented
6#16839970C212yesmaximal, complemented
7#16839965C212yesmaximal, complemented
8#16839916C212yesmaximal, complemented
9#16839899C212yesmaximal
10#16839817C211yesmaximal, characteristic, complemented
11C61×#497C211yesmaximal, characteristic, complemented
12C2×Q8×D61C2212yes—
13C2×#11758202C2212yes—
14C2×#11758200C2212yes—
15#14519777C2211yescharacteristic
16#14519776—21no—
17#14519776—21no—
18#14519775C2212yes—
19#14519770—21no—
20#14519769C2211yescharacteristic
21#14519759—21no—
22#14519753C2211yescharacteristic
23#14519748C2211yescharacteristic
24Q8×Dic61C2211yescharacteristic
25D4×Dic61C2211yescharacteristic
26#14519737—21no—
27#14519733—21no—
28C4⋊C4×D61—21no—
29C4⋊C4×D61—21no—
30C42×D61C2211yescharacteristic
31#14519706C2212yes—
32#14519702C2212yes—
33#14519702C2212yes—
34#14519701C2212yes—
35#14519701C2212yes—
36#14519689C2212yes—
37#14519688C2211yescharacteristic
38C2×Q16×C61C2212yes—
39C2×SD16×C61C2212yes—
40C4⋊Q8×C61C2211yescharacteristic
41#124×C61C2211yescharacteristic
42#97×C61C2211yescharacteristic
43C2×Q8×C61C2312yes—
44C2×Q8×C61C2311yescharacteristic
45C2×D4×C61C2311yescharacteristic
46C2×C22⋊D61—21no—
47C22×Dic61—21no—
48Q8×D61—42no—
49Q8×D61—42no—
50C2×C4×D61—22no—
51C2×C4×D61D411yescharacteristic
52C2×C4×D61D411yescharacteristic
53C2×C4×D61C2311yescharacteristic
54C2×Dic122—22no—
55C2×Dic122C2312yes—
56Q16×C61—42no—
57SD16×C61—42no—
58C2×C8×C61C2312yes—
59C4⋊C4×C61—21no—
60C4⋊C4×C61—21no—
61C22⋊C4×C61—21no—
62C22⋊C4×C61—21no—
63C42×C61C2311yescharacteristic
64C22⋊Dic61—21no—
65C22⋊Dic61—21no—
66#11758202—42no—
67#11758200—42no—
68#11758198—21no—
69C4⋊Dic61—21no—
70C4⋊Dic61—21no—
71C4⋊Dic61—21no—
72C4⋊Dic61C2311yescharacteristic
73C4⋊Dic61C2311yescharacteristic
74#11758196—21no—
75#11758196—21no—
76#11758196—22no—
77#11758196—22no—
78#11758196—21no—
79C4×Dic61—21no—
80C4×Dic61D411yescharacteristic
81C4×Dic61D411yescharacteristic
82C4×Dic61C2311yescharacteristic
83C2×#92418C2312yes—
84C23×C61—21no—
85C22×D61C2×D411yescharacteristic
86Q8×C61—22no—
87Q8×C61—22no—
88Q8×C61—22no—
89D4×C61—22no—
90C2×C4×C61—21no—
91C2×C4×C61—22no—
92C2×C4×C61C2×D411yescharacteristic
93C2×C4×C61C2×D411yescharacteristic
94C2×C4×C61C2411yescharacteristic
95C22⋊D61—42no—
96C2×Dic61—21no—
97C2×Dic61—21no—
98C2×Dic61—22no—
99C2×Dic61—42no—
100C2×Dic61C2×D411yescharacteristic
101C2×Dic61C2×D411yescharacteristic
102C2×Dic61C2×D411yescharacteristic
103C4×D61—22no—
104C4×D61—22no—
105C4×D61—22no—
106C4×D61—44no—
107Dic122—22no—
108Dic122—22no—
109Dic122—44no—
110C8×C61—22no—
111#92418—22no—
112C22×C61—41no—
113C22×C61—41no—
114C22×C61—41no—
115C22×C61C22×D411yescharacteristic
116C2×D61—21no—
117C2×D61—21no—
118C2×D61#13712yes—
119C4×C61—21no—
120C4×C61—21no—
121C4×C61—42no—
122C4×C61—41no—
123C4×C61H(5,𝔽2)12yes—
124Dic61—21no—
125Dic61—21no—
126Dic61—42no—
127Dic61—41no—
128Dic61—41no—
129Dic61#13712yes—
130#3218—611nomaximal, Sylow
131C2×C61—81no—
132C2×C61#53511yescharacteristic
133C2×C61C2×#13711yescharacteristic
134C2×C61C2×#13711yescharacteristic
135D61—22no—
136C2×C4⋊Q8—611no—
137#529—611no—
138#497—611no—
139#497D6111yescharacteristic, complemented
140#491—611no—
141#491—611no—
142#462—612no—
143#461—612no—
144#451—612no—
145#448—612no—
146#432—611no—
147C61#321811yescharacteristic, complemented, Sylow
148C22×Q8—612no—
149C2×Q16—612no—
150C2×Q16C2×D6112yes—
151C2×SD16—612no—
152C2×SD16C2×D6112yes—
153C4⋊Q8—612no—
154C4⋊Q8—611no—
155C4⋊Q8—1221no—
156C4⋊Q8—1221no—
157C4⋊Q8C2×D6111yescharacteristic
158#126—1221no—
159#125—611no—
160#124C2×D6111yescharacteristic
161#123—1221no—
162C22⋊Q8—611no—
163#121—611no—
164C4×Q8—611no—
165C4×D4—611no—
166C4×D4—1221no—
167#117—1221no—
168C2×C4⋊C4—1221no—
169C2×C4⋊C4—1221no—
170C2×C42—611no—
171C4⋊C8—612no—
172#103—612no—
173#103—612no—
174#103—612no—
175#103—612no—
176C22⋊C8—612no—
177#97—611no—
178#97C2×D6111yescharacteristic
179C2×Q8—612no—
180C2×Q8—1222no—
181C2×Q8—2442no—
182C2×Q8—2442no—
183C2×Q8C22×D6112yes—
184C2×Q8C22×D6111yescharacteristic
185C2×D4—1221no—
186C2×D4C22×D6111yescharacteristic
187C22×C4—611no—
188C22×C4—611no—
189C22×C4—611no—
190C22×C4—1222no—
191C22×C4—1221no—
192Q16—42no—
193Q16—2442no—
194SD16—42no—
195SD16—2442no—
196C2×C8—612no—
197C2×C8C22×D6112yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—611no—
201C4⋊C4—611no—
202C4⋊C4—1222no—
203C4⋊C4—1222no—
204C4⋊C4—1221no—
205C4⋊C4—1221no—
206C4⋊C4—1221no—
207C4⋊C4—1221no—
208C4⋊C4—1221no—
209C4⋊C4—1221no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—1221no—
213C22⋊C4—1221no—
214C22⋊C4—1221no—
215C42—611no—
216C42—611no—
217C42—611no—
218C42—1221no—
219C42C22×D6111yescharacteristic
220C23—21no—
221C23—611no—
222Q8—22no—
223Q8—22no—
224Q8—22no—
225Q8—1222no—
226Q8—1222no—
227Q8—2444no—
228D4—22no—
229D4—2442no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—611no—
233C2×C4—611no—
234C2×C4—611no—
235C2×C4—1222no—
236C2×C4—1222no—
237C2×C4—1222no—
238C2×C4—1221no—
239C2×C4—1221no—
240C2×C4—1222no—
241C2×C4—2444no—
242C2×C4—2442no—
243C2×C4D4×D6111yescharacteristic
244C2×C4D4×D6111yescharacteristic
245C2×C4C23×D6111yescharacteristic
246C8—22no—
247C8—1222no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—612no—
252C22—1221no—
253C22—1221no—
254C22C2×D4×D6111yescharacteristic
255C4—21no—
256C4—21no—
257C4—42no—
258C4—41no—
259C4—612no—
260C4—1221no—
261C4—1221no—
262C4—2441no—
263C4—2441no—
264C4—2442no—
265C4#1451982212yes—
266C2—81no—
267C2—1222no—
268C2#1684032511yescharacteristic
269C2#137×D6111yescharacteristic
270C2#137×D6111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#497C2
socleC22×C61C22×D4
hypercenter#497D61
hypocenterC61#3218
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D61C2semidirect
3. #16840157C2semidirect
4. #16840131C2semidirect
5. #16840130C2semidirect
6. #16839970C2semidirect
7. #16839965C2semidirect
8. #16839916C2semidirect
10. #16839817C2semidirect
11. C61×#497C2semidirect
139. #497D61semidirect
147. C61#3218semidirect