Finite Group Databasebeta

← #21456809

Subgroups — #21456809

Order 7808.

434 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 111 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#16840311C211yesmaximal, characteristic, complemented
2#16840310C211yesmaximal, characteristic, complemented
3#16840149C211yesmaximal, characteristic, complemented
4#16840122C211yesmaximal, characteristic, complemented
5#16840121C211yesmaximal, characteristic, complemented
6#16839965C212yesmaximal, complemented
7#16839955C212yesmaximal, complemented
8#16839906C212yesmaximal, complemented
9#16839904C212yesmaximal, complemented
10#16839817C211yesmaximal, characteristic, complemented
11C61×#496C211yesmaximal, characteristic, complemented
12C2×#11758226C2211yescharacteristic
13C2×#11758224—21no—
14C2×#11758224C2211yescharacteristic
15C2×#11758200C2212yes—
16C2×#11758199C2212yes—
17#14519777C2211yescharacteristic
18#14519775C2211yescharacteristic
19#14519773C2211yescharacteristic
20#14519765C2211yescharacteristic
21#14519764—21no—
22#14519750—21no—
23#14519750—21no—
24#14519748—21no—
25#14519748C2212yes—
26D4×Dic61—21no—
27D4×Dic61C2212yes—
28#14519737—21no—
29#14519737—21no—
30C42×D61C2211yescharacteristic
31#14519706C2212yes—
32#14519702C2212yes—
33#14519701C2212yes—
34#14519699C2212yes—
35#14519698C2212yes—
36#14519689C2212yes—
37#14519688C2211yescharacteristic
38C2×SD16×C61C2212yes—
39C2×D8×C61C2212yes—
40#127×C61C2211yescharacteristic
41#124×C61C2211yescharacteristic
42#97×C61C2211yescharacteristic
43C2×Q8×C61C2311yescharacteristic
44C2×D4×C61—21no—
45C2×D4×C61—21no—
46C2×D4×C61C2311yescharacteristic
47C2×D4×C61C2312yes—
48C2×C22⋊D61—22no—
49C2×C22⋊D61—21no—
50C2×C22⋊D61—21no—
51C2×C22⋊D61—21no—
52C22×Dic61—22no—
53C22×Dic61—21no—
54#11758226—42no—
55#11758224—42no—
56#11758224—44no—
57C2×D244—21no—
58C2×D244C2311yescharacteristic
59C2×C4×D61—21no—
60C2×C4×D61D411yescharacteristic
61C2×C4×D61D411yescharacteristic
62C2×C4×D61C2311yescharacteristic
63C2×Dic122—21no—
64C2×Dic122C2311yescharacteristic
65SD16×C61—42no—
66D8×C61—42no—
67C2×C8×C61C2312yes—
68C22⋊C4×C61—21no—
69C22⋊C4×C61—21no—
70C42×C61C2311yescharacteristic
71C22⋊Dic61—22no—
72C22⋊Dic61—22no—
73#11758200—42no—
74#11758199—42no—
75#11758198—21no—
76#11758198—21no—
77C4⋊Dic61—21no—
78C4⋊Dic61C2312yes—
79#11758196—21no—
80#11758196—21no—
81C4×Dic61D411yescharacteristic
82C4×Dic61D411yescharacteristic
83C4×Dic61C2311yescharacteristic
84C2×#92418C2312yes—
85C23×C61—22no—
86C23×C61—21no—
87C22×D61—21no—
88C22×D61C2×D411yescharacteristic
89Q8×C61—22no—
90D4×C61—22no—
91D4×C61—22no—
92D4×C61—22no—
93D4×C61—42no—
94D4×C61—42no—
95C2×C4×C61—21no—
96C2×C4×C61C2×D411yescharacteristic
97C2×C4×C61C2×D411yescharacteristic
98C2×C4×C61C2411yescharacteristic
99C22⋊D61—42no—
100C22⋊D61—42no—
101C22⋊D61—44no—
102C22⋊D61—42no—
103C2×Dic61—21no—
104C2×Dic61—22no—
105C2×Dic61—42no—
106C2×Dic61—44no—
107C2×Dic61C2×D411yescharacteristic
108C2×Dic61C2×D411yescharacteristic
109C2×Dic61C2×D411yescharacteristic
110D244—22no—
111D244—42no—
112C4×D61—22no—
113C4×D61—22no—
114C4×D61—41no—
115C4×D61—42no—
116Dic122—22no—
117Dic122—42no—
118C8×C61—22no—
119#92418—22no—
120C22×C61—41no—
121C22×C61—42no—
122C22×C61—42no—
123C22×C61—41no—
124C22×C61—41no—
125C22×C61—42no—
126C22×C61C22×D411yescharacteristic
127C2×D61—21no—
128C2×D61—21no—
129C2×D61—21no—
130C2×D61—41no—
131C2×D61—41no—
132C2×D61—41no—
133C4×C61—21no—
134C4×C61—21no—
135C4×C61—41no—
136C4×C61H(5,𝔽2)12yes—
137Dic61—21no—
138Dic61—21no—
139Dic61—41no—
140Dic61—42no—
141Dic61AGL(1,ℤ/8)12yes—
142#3214—611nomaximal, Sylow
143C2×C61—81no—
144C2×C61—82no—
145C2×C61#53511yescharacteristic
146C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C61#57511yescharacteristic
148D61—41no—
149D61—81no—
150#532—611no—
151#532—611no—
152#496—611no—
153#496D6111yescharacteristic, complemented
154#489—611no—
155#489—611no—
156#461—612no—
157#459—612no—
158#449—612no—
159#449—612no—
160#432—611no—
161C61#321411yescharacteristic, complemented, Sylow
162C2×#41—611no—
163C2×#41—611no—
164C2×#41—1221no—
165C2×SD16—612no—
166C2×SD16C2×D6112yes—
167C2×D8—612no—
168C2×D8C2×D6112yes—
169C4⋊Q8—611no—
170C4⋊Q8—611no—
171#127—611no—
172#127C2×D6111yescharacteristic
173#124—611no—
174#124—1221no—
175#124C2×D6111yescharacteristic
176#121—612no—
177#121—1221no—
178#121—1221no—
179#121—1221no—
180C4×D4—612no—
181C4×D4—1221no—
182C4×D4—1221no—
183C4×D4—1221no—
184C2×C42—611no—
185C4⋊C8—612no—
186#103—612no—
187#103—612no—
188#102—612no—
189#102—612no—
190C22⋊C8—612no—
191#97—611no—
192#97C2×D6111yescharacteristic
193#41—2442no—
194#41—2444no—
195#41—2442no—
196C2×Q8—611no—
197C2×Q8—1221no—
198C2×Q8C22×D6111yescharacteristic
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—611no—
202C2×D4—1221no—
203C2×D4—1221no—
204C2×D4—1221no—
205C2×D4—1222no—
206C2×D4—1221no—
207C2×D4C22×D6111yescharacteristic
208C2×D4C22×D6112yes—
209C22×C4—611no—
210C22×C4—611no—
211C22×C4—611no—
212C22×C4—1221no—
213C22×C4—1222no—
214C22×C4—1221no—
215SD16—42no—
216SD16—2442no—
217D8—42no—
218D8—2442no—
219C2×C8—612no—
220C2×C8C22×D6112yes—
221C4⋊C4—612no—
222C4⋊C4—1221no—
223C4⋊C4—1221no—
224C4⋊C4—1221no—
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—1222no—
228C22⋊C4—1222no—
229C22⋊C4—1221no—
230C22⋊C4—1221no—
231C42—611no—
232C42—611no—
233C42—611no—
234C42C22×D6111yescharacteristic
235C23—21no—
236C23—22no—
237C23—611no—
238C23—1221no—
239Q8—22no—
240Q8—1222no—
241Q8—2442no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4—42no—
246D4—42no—
247D4—1222no—
248D4—2442no—
249D4—2442no—
250D4—2442no—
251D4—2444no—
252D4—2442no—
253C2×C4—21no—
254C2×C4—611no—
255C2×C4—611no—
256C2×C4—611no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—1221no—
261C2×C4—2441no—
262C2×C4—2442no—
263C2×C4—2444no—
264C2×C4—2442no—
265C2×C4D4×D6111yescharacteristic
266C2×C4D4×D6111yescharacteristic
267C2×C4C23×D6111yescharacteristic
268C8—22no—
269C8—1222no—
270C22—42no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—41no—
275C22—42no—
276C22—1221no—
277C22—1221no—
278C22—1221no—
279C22—2441no—
280C22—2441no—
281C22—2441no—
282C22C2×D4×D6111yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—612no—
287C4—1221no—
288C4—1221no—
289C4—2442no—
290C4—2441no—
291C4#1451982212yes—
292C2—82no—
293C2—81no—
294C2—4881no—
295C2—2441no—
296C2#1684032511yescharacteristic
297C2AGL(1,ℤ/8)×D6111yescharacteristic
298C2#1684055811yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#496C2
socleC22×C61C22×D4
hypercenter#496D61
hypocenterC61#3214
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #16840311C2semidirect
2. #16840310C2semidirect
3. #16840149C2semidirect
4. #16840122C2semidirect
5. #16840121C2semidirect
6. #16839965C2semidirect
7. #16839955C2semidirect
8. #16839906C2semidirect
9. #16839904C2semidirect
10. #16839817C2semidirect
11. C61×#496C2semidirect
153. #496D61semidirect
161. C61#3214semidirect