Finite Group Databasebeta

← #27181266

Subgroups — #27181266

Order 12928.

450 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×C4×D404C211yesmaximal, characteristic, complemented
2#20265505C211yesmaximal, characteristic, complemented
3#20265488C211yesmaximal, characteristic, complemented
4#20265474C212yesmaximal, complemented
5#20265389C212yesmaximal, complemented
6#20265327C211yesmaximal, characteristic, complemented
7#20265310C211yesmaximal, characteristic
8#20265287C212yesmaximal, complemented
9#20265275C212yesmaximal, complemented
10#20265234C211yesmaximal, characteristic, complemented
11#529×C101C211yesmaximal, characteristic, complemented
12C22×D404C2211yescharacteristic
13C22×C4×D101—21no—
14C2×#13108061C2212yes—
15C2×#13108059C2212yes—
16C4×D404—21no—
17C4×D404—21no—
18C4×D404C2212yes—
19C4×D404C2212yes—
20C2×C4⋊Dic101C2211yescharacteristic
21C2×#13108051—21no—
22#15692188C2212yescomplemented
23#15692186C2212yescomplemented
24C4⋊#11717579C2212yescomplemented
25#15692179C2212yescomplemented
26#15692176C2212yescomplemented
27#15692175C2212yes—
28#15692175C2212yes—
29C22⋊#11717579C2211yescharacteristic
30C22⋊#11717579C2211yescharacteristic
31#15692166C2212yescomplemented
32#126×C101—21no—
33#125×C101C2211yescharacteristic, complemented
34#124×C101C2211yescharacteristic, complemented
35#123×C101—21no—
36C22⋊Q8×C101C2211yescharacteristic
37#121×C101C2211yescharacteristic
38C4×Q8×C101C2211yescharacteristic, complemented
39C4×D4×C101—21no—
40C4×D4×C101C2211yescharacteristic, complemented
41#117×C101—21no—
42C2×C42×C101C2211yescharacteristic
43C23×D101—21no—
44C2×D404—21no—
45C2×D404—24no—
46C2×D404C2312yes—
47C22×Dic101—21no—
48C2×C4×D101—22no—
49C2×C4×D101—22no—
50C2×C4×D101—44no—
51#13108061—24no—
52#13108059—24no—
53C2×#11717579C2312yes—
54C2×#11717579C2312yes—
55C4⋊Dic101—21no—
56C4⋊Dic101C2312yes—
57#13108051—22no—
58#13108051—22no—
59C2×Q8×C101C2311yescharacteristic
60C2×D4×C101—21no—
61C2×D4×C101C2311yescharacteristic
62C22×C4×C101—21no—
63C22×C4×C101D411yescharacteristic
64C22×C4×C101D411yescharacteristic
65C22×C4×C101C2311yescharacteristic
66C4⋊C4×C101—21no—
67C4⋊C4×C101—21no—
68C4⋊C4×C101—21no—
69C4⋊C4×C101—21no—
70C4⋊C4×C101C2311yescharacteristic
71C4⋊C4×C101C2311yescharacteristic
72C22⋊C4×C101—21no—
73C22⋊C4×C101—21no—
74C22⋊C4×C101—21no—
75C22⋊C4×C101—21no—
76C22⋊C4×C101—21no—
77C42×C101—21no—
78C42×C101D411yescharacteristic
79C42×C101D411yescharacteristic
80C42×C101C2311yescharacteristic
81C42×C101C2311yescharacteristic
82C23×C101—21no—
83C23×C101C2×D411yescharacteristic
84C22×D101—24no—
85C22×D101—22no—
86C22×D101—42no—
87C22×D101—42no—
88Q8×C101—22no—
89D4×C101—22no—
90D4×C101—42no—
91C2×C4×C101—22no—
92C2×C4×C101—22no—
93C2×C4×C101—21no—
94C2×C4×C101—21no—
95C2×C4×C101—21no—
96C2×C4×C101—41no—
97C2×C4×C101—42no—
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101C2×D411yescharacteristic
102C2×C4×C101C2×D411yescharacteristic
103C2×C4×C101C2411yescharacteristic
104C2×Dic101—22no—
105C2×Dic101—42no—
106D404—24no—
107D404—22no—
108D404#4114yes—
109C4×D101—44no—
110C4×D101—44no—
111#11717579—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—21no—
126C4×C101—21no—
127C4×C101—21no—
128C4×C101—21no—
129C4×C101—41no—
130C4×C101—41no—
131C4×C101—41no—
132C4×C101C2×#4112yes—
133Dic101—42no—
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#529D10111yescharacteristic, complemented
143C2×C4×D4—1011no—
144#483—1011no—
145#480—1011no—
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—21no—
158#125C2×D10111yescharacteristic, complemented
159#124C2×D10111yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D10111yescharacteristic
162#121C2×D10111yescharacteristic
163C4×Q8C2×D10111yescharacteristic, complemented
164C4×D4—21no—
165C4×D4—1012no—
166C4×D4—1012no—
167C4×D4—2021no—
168C4×D4—2021no—
169C4×D4C2×D10111yescharacteristic, complemented
170#117—21no—
171C2×C4⋊C4—1011no—
172C2×C22⋊C4—2021no—
173C2×C42C2×D10111yescharacteristic
174#107—1012no—
175#106—1012no—
176C4⋊C8—1012no—
177#103—1012no—
178#102—1012no—
179#102—1012no—
180#102—1012no—
181C22⋊C8—1011no—
182C22⋊C8—1011no—
183#97—1012no—
184C24—2021no—
185C2×Q8C22×D10111yescharacteristic
186C2×D4—21no—
187C2×D4—1012no—
188C2×D4—2024no—
189C2×D4—2021no—
190C2×D4C22×D10111yescharacteristic
191C22×C4—21no—
192C22×C4—4044no—
193C22×C4—2021no—
194C22×C4—2022no—
195C22×C4—2022no—
196C22×C4C22⋊D10111yescharacteristic
197C22×C4C22⋊D10111yescharacteristic
198C22×C4C22×D10111yescharacteristic
199SD16—2024no—
200D8—2024no—
201C2×C8—1012no—
202C2×C8—1012no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—1012no—
208C4⋊C4—2021no—
209C4⋊C4C22×D10111yescharacteristic
210C4⋊C4C22×D10111yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—2022no—
217C22⋊C4—2022no—
218C42—21no—
219C42C22⋊D10111yescharacteristic
220C42C22⋊D10111yescharacteristic
221C42C22×D10111yescharacteristic
222C42C22×D10111yescharacteristic
223C23—21no—
224C23—4042no—
225C23—4042no—
226C23—2024no—
227C23—2022no—
228C23C2×C22⋊D10111yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—1014no—
233D4—2024no—
234D4—2022no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—4044no—
243C2×C4—4044no—
244C2×C4—4042no—
245C2×C4—2022no—
246C2×C4C2×C22⋊D10111yescharacteristic
247C2×C4C2×C22⋊D10111yescharacteristic
248C2×C4C2×C22⋊D10111yescharacteristic
249C2×C4C2×C22⋊D10111yescharacteristic
250C2×C4C2×C22⋊D10111yescharacteristic
251C2×C4C23×D10111yescharacteristic
252C8—2022no—
253C8—2022no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—4048no—
260C22—4042no—
261C22—4042no—
262C22—2024no—
263C22#1569228012yes—
264C22C22×C22⋊D10111yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—4042no—
274C4#41×D10112yes—
275C2—22no—
276C2—81no—
277C2—4044no—
278C2#2026568411yescharacteristic
279C2C2×#1569228011yescharacteristic
280C2#2026597511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#529×C101C2
socleC22×C101C22×D4
prime radical#20265310C2
hypercenter#529D101
hypocenterC101#3114
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D404C2semidirect
2. #20265505C2semidirect
3. #20265488C2semidirect
4. #20265474C2semidirect
5. #20265389C2semidirect
6. #20265327C2semidirect
8. #20265287C2semidirect
9. #20265275C2semidirect
10. #20265234C2semidirect
11. #529×C101C2semidirect
22. #15692188C22semidirect
23. #15692186C22semidirect
24. C4⋊#11717579C22semidirect
25. #15692179C22semidirect
26. #15692176C22semidirect
31. #15692166C22semidirect
33. #125×C101C22semidirect
34. #124×C101C22semidirect
38. C4×Q8×C101C22semidirect
40. C4×D4×C101C22semidirect
141. C101#3114semidirect
142. #529D101semidirect
158. #125C2×D101semidirect
159. #124C2×D101semidirect
163. C4×Q8C2×D101semidirect
169. C4×D4C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4042
#202655052
#202654882
#202654742
#202653892
#202653272
#202653102
#202652872
#202652752
#202652342
#529×C1012
#3114101