Finite Group Databasebeta

← #27181267

Subgroups — #27181267

Order 12928.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265708C211yesmaximal, characteristic, complemented
2C2×#15692217C211yesmaximal, characteristic, complemented
3#20265507C211yesmaximal, characteristic, complemented
4#20265480C212yesmaximal, complemented
5#20265391C212yesmaximal, complemented
6#20265330C211yesmaximal, characteristic, complemented
7#20265310C211yesmaximal, characteristic
8#20265286C212yesmaximal, complemented
9#20265276C212yesmaximal, complemented
10#20265231C211yesmaximal, characteristic, complemented
11#480×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C22×C4×D101—21no—
14C2×#13108063C2212yes—
15C2×#13108059C2212yes—
16#15692248—21no—
17#15692246C2211yescharacteristic, complemented
18#15692242C2211yescharacteristic, complemented
19#15692236—21no—
20#15692234C2211yescharacteristic
21#15692220C2211yescharacteristic, complemented
22D4×Dic101—21no—
23D4×Dic101C2211yescharacteristic, complemented
24#15692217—21no—
25#15692217—21no—
26#15692217C2212yes—
27#15692217C2212yes—
28#15692211—21no—
29C2×#13108051—21no—
30C2×C4×Dic101C2211yescharacteristic
31#15692186C2212yescomplemented
32#15692184C2212yescomplemented
33#15692178C2212yescomplemented
34#15692177C2212yes—
35#15692175C2212yes—
36C22⋊#11717579C2211yescharacteristic
37#15692165C2212yescomplemented
38#121×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic
40#107×C101C2212yescomplemented
41#102×C101C2212yescomplemented
42C22⋊C8×C101C2211yescharacteristic
43C23×D101—21no—
44C2×Dic202C2311yescharacteristic
45C2×D404—21no—
46C2×D404—24no—
47C2×D404C2312yes—
48C22×Dic101—21no—
49C22×Dic101D411yescharacteristic
50C22×Dic101D411yescharacteristic
51C2×C4×D101—22no—
52C2×C4×D101—22no—
53C2×C4×D101—44no—
54#13108063—24no—
55#13108059—24no—
56C2×#11717579C2312yes—
57C4⋊Dic101—21no—
58C4⋊Dic101C2311yescharacteristic
59#13108053—21no—
60#13108053—21no—
61#13108053—21no—
62C22⋊Dic101—21no—
63C22⋊Dic101—21no—
64C22⋊Dic101—21no—
65C22⋊Dic101—21no—
66#13108051—22no—
67#13108051—22no—
68C4×Dic101—21no—
69C4×Dic101D411yescharacteristic
70C4×Dic101D411yescharacteristic
71C4×Dic101C2311yescharacteristic
72C4×Dic101C2311yescharacteristic
73C2×D4×C101—21no—
74C2×D4×C101C2311yescharacteristic
75C22×C4×C101—21no—
76C22×C4×C101C2311yescharacteristic
77C2×C8×C101C2312yes—
78C4⋊C4×C101—21no—
79C4⋊C4×C101C2312yes—
80C4⋊C4×C101C2311yescharacteristic
81C22⋊C4×C101—21no—
82C23×C101—21no—
83C23×C101C2×D411yescharacteristic
84C22×D101—24no—
85C22×D101—22no—
86C22×D101—42no—
87C22×D101—42no—
88D4×C101—22no—
89D4×C101—42no—
90C2×C4×C101—22no—
91C2×C4×C101—21no—
92C2×C4×C101—22no—
93C2×C4×C101—42no—
94C2×C4×C101C2×D411yescharacteristic
95C2×C4×C101C2411yescharacteristic
96C2×Dic101—22no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101—41no—
100C2×Dic101—42no—
101C2×Dic101C2×D411yescharacteristic
102C2×Dic101C2×D411yescharacteristic
103C2×Dic101C2×D411yescharacteristic
104C2×Dic101C2×D411yescharacteristic
105D404—24no—
106D404—22no—
107D404#4114yes—
108C4×D101—44no—
109C4×D101—44no—
110Dic202—22no—
111C8×C101—22no—
112#11717579—22no—
113C22×C101—21no—
114C22×C101—21no—
115C22×C101—41no—
116C22×C101—41no—
117C22×C101—41no—
118C22×C101AGL(1,ℤ/8)12yes—
119C22×C101C22×D411yescharacteristic
120C2×D101—24no—
121C2×D101—48no—
122C2×D101—42no—
123C2×D101—42no—
124C4×C101—21no—
125C4×C101—42no—
126C4×C101—41no—
127C4×C101C2×#4112yes—
128Dic101—21no—
129Dic101—21no—
130Dic101—21no—
131Dic101—21no—
132Dic101—41no—
133Dic101—41no—
134C2×C101—22no—
135C2×C101—81no—
136C2×C101#52511yescharacteristic
137C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C101#57511yescharacteristic
139D101—44no—
140#3114—1011nomaximal, Sylow
141C101#311411yescharacteristic, complemented, Sylow
142#529—1011no—
143C2×C4×D4—1011no—
144#483—1011no—
145#480D10111yescharacteristic, complemented
146#478—1012no—
147#463—1012no—
148#450—1011no—
149#447—1011no—
150#442—1012no—
151#440—1012no—
152#432—1011no—
153C22×D4—1011no—
154C23×C4—2021no—
155C2×SD16—1012no—
156C2×D8—1012no—
157#126—2021no—
158#125—1011no—
159#124—1011no—
160#123—2021no—
161C22⋊Q8—1011no—
162#121C2×D10111yescharacteristic
163C4×Q8—1011no—
164C4×D4—1011no—
165C4×D4—1012no—
166C4×D4—1012no—
167C4×D4—2021no—
168C4×D4—2021no—
169C4×D4—2021no—
170#117—2021no—
171C2×C4⋊C4C2×D10111yescharacteristic
172C2×C22⋊C4—2021no—
173C2×C42—1011no—
174#107C2×D10112yescomplemented
175#106—1012no—
176C4⋊C8—1012no—
177#103—1012no—
178#102—1012no—
179#102—1012no—
180#102C2×D10112yescomplemented
181C22⋊C8—1011no—
182C22⋊C8C2×D10111yescharacteristic
183#97—1012no—
184C24—2021no—
185C2×Q8—1011no—
186C2×D4—21no—
187C2×D4—1012no—
188C2×D4—2024no—
189C2×D4—2021no—
190C2×D4C22×D10111yescharacteristic
191C22×C4—21no—
192C22×C4—1011no—
193C22×C4—1011no—
194C22×C4—4044no—
195C22×C4—2022no—
196C22×C4—2022no—
197C22×C4—2021no—
198C22×C4C22×D10111yescharacteristic
199SD16—2024no—
200D8—2024no—
201C2×C8—1012no—
202C2×C8C22×D10112yes—
203C4⋊C4—21no—
204C4⋊C4—1011no—
205C4⋊C4—2021no—
206C4⋊C4—2021no—
207C4⋊C4—2021no—
208C4⋊C4—2021no—
209C4⋊C4C22×D10111yescharacteristic
210C4⋊C4C22×D10112yes—
211C22⋊C4—21no—
212C22⋊C4—2022no—
213C22⋊C4—2022no—
214C22⋊C4—2021no—
215C22⋊C4—2021no—
216C22⋊C4—2021no—
217C22⋊C4—2021no—
218C42—1011no—
219C42—1011no—
220C42—1011no—
221C42—1011no—
222C42—2021no—
223C23—21no—
224C23—4042no—
225C23—4042no—
226C23—2024no—
227C23—2022no—
228C23D4×D10111yescharacteristic
229Q8—2022no—
230D4—22no—
231D4—42no—
232D4—1014no—
233D4—2024no—
234D4—2022no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—1011no—
240C2×C4—1011no—
241C2×C4—1011no—
242C2×C4—1011no—
243C2×C4—4044no—
244C2×C4—4044no—
245C2×C4—4042no—
246C2×C4—4041no—
247C2×C4—2021no—
248C2×C4—2021no—
249C2×C4—2022no—
250C2×C4D4×D10111yescharacteristic
251C2×C4C23×D10111yescharacteristic
252C8—22no—
253C8—2022no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—4048no—
260C22—4042no—
261C22—4042no—
262C22—2024no—
263C22#1569227912yes—
264C22C2×D4×D10111yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—4041no—
269C4—4041no—
270C4—2021no—
271C4—2021no—
272C4—2021no—
273C4—2021no—
274C4#41×D10112yes—
275C2—22no—
276C2—81no—
277C2—4044no—
278C2#2026568511yescharacteristic
279C2C2×#1569227911yescharacteristic
280C2#2026597611yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#480×C101C2
socleC22×C101C22×D4
prime radical#20265310C2
hypercenter#480D101
hypocenterC101#3114
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #20265708C2semidirect
2. C2×#15692217C2semidirect
3. #20265507C2semidirect
4. #20265480C2semidirect
5. #20265391C2semidirect
6. #20265330C2semidirect
8. #20265286C2semidirect
9. #20265276C2semidirect
10. #20265231C2semidirect
11. #480×C101C2semidirect
17. #15692246C22semidirect
18. #15692242C22semidirect
21. #15692220C22semidirect
23. D4×Dic101C22semidirect
31. #15692186C22semidirect
32. #15692184C22semidirect
33. #15692178C22semidirect
37. #15692165C22semidirect
40. #107×C101C22semidirect
41. #102×C101C22semidirect
141. C101#3114semidirect
145. #480D101semidirect
174. #107C2×D101semidirect
180. #102C2×D101semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202657082
C2×#156922172
#202655072
#202654802
#202653912
#202653302
#202653102
#202652862
#202652762
#202652312
#480×C1012
#3114101