Finite Group Databasebeta

← #21456735

Subgroups — #21456735

Order 7808.

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

normal subgroups only

This group has 532 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#16840307C211yesmaximal, characteristic, complemented
2C2×#14519773C211yesmaximal, characteristic, complemented
3#16840137C212yesmaximal, complemented
4#16840107C212yesmaximal, complemented
5#16839956C212yesmaximal, complemented
6#16839955C212yesmaximal, complemented
7#16839898C211yesmaximal, characteristic, complemented
8#16839896C211yesmaximal, characteristic, complemented
9#16839895C211yesmaximal, characteristic, complemented
10#16839812C211yesmaximal, characteristic, complemented
11C61×#447C211yesmaximal, characteristic, complemented
12C2×#11758224—21no—
13C22×D244C2211yescharacteristic
14C22×C22⋊D61—21no—
15C22×C22⋊D61—21no—
16C2×D488C2212yescomplemented
17C2×#11758199C2212yescomplemented
18C2×#11758199C2212yescomplemented
19#14519776C2211yescharacteristic, complemented
20#14519773—21no—
21#14519773—21no—
22#14519773C2212yes—
23#14519773C2212yes—
24#14519773C2211yescharacteristic, complemented
25#14519763—21no—
26#14519751C2211yescharacteristic
27#14519749—21no—
28D4×Dic61C2211yescharacteristic, complemented
29#14519739C2211yescharacteristic, complemented
30#14519737—21no—
31C2×C4×Dic61C2211yescharacteristic
32#14519706C2212yescomplemented
33#14519699C2212yescomplemented
34#14519698C2212yescomplemented
35#14519697C2212yescomplemented
36C22⋊#92418C2211yescharacteristic
37C8×Dic61C2212yescomplemented
38C22×D4×C61C2211yescharacteristic
39C2×D8×C61C2212yescomplemented
40#121×C61C2211yescharacteristic
41#102×C61C2212yescomplemented
42C22⋊C8×C61C2211yescharacteristic
43C24×C61—21no—
44C23×D61—21no—
45C2×D4×C61—21no—
46C2×D4×C61—21no—
47C2×D4×C61—42no—
48C2×D4×C61C2312yes—
49C2×D4×C61C2311yescharacteristic
50C22×C4×C61C2311yescharacteristic
51C2×C22⋊D61—21no—
52C2×C22⋊D61—22no—
53C2×C22⋊D61—22no—
54C2×C22⋊D61—22no—
55C2×C22⋊D61—22no—
56C2×C22⋊D61—21no—
57C2×C22⋊D61—21no—
58C2×C22⋊D61—44no—
59C2×C22⋊D61—44no—
60C22×Dic61—21no—
61C22×Dic61D411yescharacteristic
62C22×Dic61D411yescharacteristic
63#11758224—44no—
64C2×D244—21no—
65C2×D244—42no—
66C2×D244C2312yes—
67C2×D244C2311yescharacteristic
68C2×C4×D61—21no—
69C2×Dic122—21no—
70D8×C61—42no—
71C2×C8×C61C2312yes—
72C4⋊C4×C61C2311yescharacteristic
73C22⋊C4×C61—21no—
74C22⋊Dic61—21no—
75#11758199—42no—
76#11758199—42no—
77#11758198—21no—
78#11758198—21no—
79C4⋊Dic61C2311yescharacteristic
80#11758196—21no—
81C4×Dic61D411yescharacteristic
82C4×Dic61D411yescharacteristic
83C4×Dic61C2311yescharacteristic
84C4×Dic61C2311yescharacteristic
85C2×#92418C2312yes—
86D488—42no—
87C23×C61—21no—
88C23×C61—22no—
89C23×C61—42no—
90C23×C61—42no—
91C23×C61—42no—
92C23×C61C2×D411yescharacteristic
93C22×D61—21no—
94C22×D61—22no—
95C22×D61—42no—
96C22×D61—42no—
97C22×D61—42no—
98D4×C61—22no—
99D4×C61—22no—
100D4×C61—22no—
101D4×C61—42no—
102D4×C61—42no—
103C2×C4×C61—22no—
104C2×C4×C61—21no—
105C2×C4×C61C2×D411yescharacteristic
106C2×C4×C61C2411yescharacteristic
107C22⋊D61—44no—
108C22⋊D61—42no—
109C22⋊D61—42no—
110C22⋊D61—44no—
111C22⋊D61—44no—
112C22⋊D61—42no—
113C22⋊D61—44no—
114C2×Dic61—22no—
115C2×Dic61—21no—
116C2×Dic61—42no—
117C2×Dic61D814yescomplemented
118C2×Dic61C2×D411yescharacteristic
119C2×Dic61C2×D411yescharacteristic
120C2×Dic61C2×D411yescharacteristic
121C2×Dic61C2×D411yescharacteristic
122D244—22no—
123D244—22no—
124D244—22no—
125D244—42no—
126C4×D61—42no—
127Dic122—42no—
128C8×C61—22no—
129#92418—22no—
130C22×C61—21no—
131C22×C61—21no—
132C22×C61—42no—
133C22×C61—41no—
134C22×C61—41no—
135C22×C61—42no—
136C22×C61—42no—
137C22×C61—41no—
138C22×C61—84no—
139C22×C61C2×D812yes—
140C22×C61C22×D411yescharacteristic
141C2×D61—42no—
142C2×D61—41no—
143C2×D61—41no—
144C2×D61—42no—
145C2×D61—42no—
146C2×D61—41no—
147C2×D61—84no—
148C4×C61—21no—
149C4×C61—41no—
150C4×C61H(5,𝔽2)12yes—
151Dic61—21no—
152Dic61—41no—
153Dic61C2×D812yes—
154Dic61C2×D812yes—
155Dic61AGL(1,ℤ/8)12yes—
156#3205—611nomaximal, Sylow
157C2×C61—22no—
158C2×C61—81no—
159C2×C61—82no—
160C2×C61#53511yescharacteristic
161C2×C61C22×D811yescharacteristic
162C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
163D61—81no—
164D61—82no—
165#532—611no—
166C2×#127—611no—
167#493—612no—
168#486—612no—
169#459—612no—
170#459—612no—
171#447—611no—
172#447—611no—
173#447—611no—
174#447D6111yescharacteristic, complemented
175#431—611no—
176C61#320511yescharacteristic, complemented, Sylow
177C2×#41—1221no—
178C22×D4—611no—
179C22×D4—1221no—
180C22×D4—1221no—
181C22×D4C2×D6111yescharacteristic
182C2×D8—612no—
183C2×D8—612no—
184C2×D8—612no—
185C2×D8C2×D6112yescomplemented
186C4⋊Q8—611no—
187#127—612no—
188#127—612no—
189#127—611no—
190#127—1221no—
191#127—1221no—
192#124—1221no—
193#121—611no—
194#121—1221no—
195#121C2×D6111yescharacteristic
196C4×D4—611no—
197C4×D4—611no—
198C4×D4—1221no—
199C2×C42—611no—
200C4⋊C8—612no—
201#102—612no—
202#102—612no—
203#102—612no—
204#102C2×D6112yescomplemented
205C22⋊C8—611no—
206C22⋊C8C2×D6111yescharacteristic
207C4×C8—612no—
208C24—21no—
209C24—1221no—
210#41—2444no—
211C2×Q8—1221no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—42no—
215C2×D4—612no—
216C2×D4—611no—
217C2×D4—1221no—
218C2×D4—1221no—
219C2×D4—1222no—
220C2×D4—1222no—
221C2×D4—1222no—
222C2×D4—1222no—
223C2×D4—1221no—
224C2×D4—1221no—
225C2×D4—2442no—
226C2×D4—2444no—
227C2×D4—2444no—
228C2×D4C22×D6112yes—
229C2×D4C22×D6111yescharacteristic
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—1221no—
233C22×C4—1221no—
234C22×C4C22×D6111yescharacteristic
235D8—42no—
236D8—2442no—
237D8—2442no—
238D8—2442no—
239C2×C8—612no—
240C2×C8C22×D6112yes—
241C4⋊C4—611no—
242C4⋊C4—1221no—
243C4⋊C4C22×D6111yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—1221no—
246C22⋊C4—1221no—
247C22⋊C4—1221no—
248C42—611no—
249C42—611no—
250C42—611no—
251C42—611no—
252C23—21no—
253C23—22no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—1221no—
258C23—1222no—
259C23—2442no—
260C23—2442no—
261C23—2442no—
262C23D4×D6111yescharacteristic
263Q8—2442no—
264D4—22no—
265D4—22no—
266D4—22no—
267D4—42no—
268D4—42no—
269D4—1222no—
270D4—1222no—
271D4—1222no—
272D4—2442no—
273D4—2442no—
274D4—2444no—
275D4—2444no—
276D4—2444no—
277D4—2442no—
278D4—2444no—
279D4—2442no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—611no—
283C2×C4—611no—
284C2×C4—611no—
285C2×C4—614no—
286C2×C4—611no—
287C2×C4—1222no—
288C2×C4—1221no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4D4×D6111yescharacteristic
292C2×C4C23×D6111yescharacteristic
293C8—22no—
294C8—1222no—
295C22—21no—
296C22—21no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—41no—
303C22—84no—
304C22—4884no—
305C22—2441no—
306C22—2442no—
307C22—2442no—
308C22—2441no—
309C22—2442no—
310C22—2441no—
311C22D8×D6112yes—
312C22C2×D4×D6111yescharacteristic
313C4—21no—
314C4—41no—
315C4—612no—
316C4—612no—
317C4—612no—
318C4—1221no—
319C4—2441no—
320C4#1451982212yes—
321C2—22no—
322C2—82no—
323C2—81no—
324C2—4882no—
325C2—4881no—
326C2#1684032711yescharacteristic
327C2C2×D8×D6111yescharacteristic
328C2AGL(1,ℤ/8)×D6111yescharacteristic
subgroupG/H
centerC22C2×D4×D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#447C2
socleC22×C61C22×D4
hypercenter#447D61
hypocenterC61#3205
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16840307C2semidirect
2. C2×#14519773C2semidirect
3. #16840137C2semidirect
4. #16840107C2semidirect
5. #16839956C2semidirect
6. #16839955C2semidirect
7. #16839898C2semidirect
8. #16839896C2semidirect
9. #16839895C2semidirect
10. #16839812C2semidirect
11. C61×#447C2semidirect
16. C2×D488C22semidirect
17. C2×#11758199C22semidirect
18. C2×#11758199C22semidirect
19. #14519776C22semidirect
24. #14519773C22semidirect
28. D4×Dic61C22semidirect
29. #14519739C22semidirect
32. #14519706C22semidirect
33. #14519699C22semidirect
34. #14519698C22semidirect
35. #14519697C22semidirect
37. C8×Dic61C22semidirect
39. C2×D8×C61C22semidirect
41. #102×C61C22semidirect
117. C2×Dic61D8semidirect
174. #447D61semidirect
176. C61#3205semidirect
185. C2×D8C2×D61semidirect
204. #102C2×D61semidirect