Finite Group Databasebeta

← #27151870

Subgroups — #27151870

Order 12416.

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×D388C211yesmaximal, characteristic, complemented
2#20261063C211yesmaximal, characteristic, complemented
3#20261046C211yesmaximal, characteristic, complemented
4#20261032C212yesmaximal, complemented
5#20260947C212yesmaximal, complemented
6#20260885C211yesmaximal, characteristic, complemented
7#20260868C211yesmaximal, characteristic
8#20260845C212yesmaximal, complemented
9#20260833C212yesmaximal, complemented
10#20260792C211yesmaximal, characteristic, complemented
11#529×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C4×D97—21no—
14C2×#13095498C2212yes—
15C2×#13095496C2212yes—
16C4×D388—21no—
17C4×D388—21no—
18C4×D388C2212yes—
19C4×D388C2212yes—
20C2×C4⋊Dic97C2211yescharacteristic
21C2×#13095488—21no—
22#15690743C2212yescomplemented
23#15690741C2212yescomplemented
24C4⋊#11715895C2212yescomplemented
25#15690734C2212yescomplemented
26#15690731C2212yescomplemented
27#15690730C2212yes—
28#15690730C2212yes—
29C22⋊#11715895C2211yescharacteristic
30C22⋊#11715895C2211yescharacteristic
31#15690721C2212yescomplemented
32#126×C97—21no—
33#125×C97C2211yescharacteristic, complemented
34#124×C97C2211yescharacteristic, complemented
35#123×C97—21no—
36C22⋊Q8×C97C2211yescharacteristic
37#121×C97C2211yescharacteristic
38C4×Q8×C97C2211yescharacteristic, complemented
39C4×D4×C97—21no—
40C4×D4×C97C2211yescharacteristic, complemented
41#117×C97—21no—
42C2×C42×C97C2211yescharacteristic
43C23×D97—21no—
44C2×D388—21no—
45C2×D388—24no—
46C2×D388C2312yes—
47C22×Dic97—21no—
48C2×C4×D97—22no—
49C2×C4×D97—22no—
50C2×C4×D97—44no—
51#13095498—24no—
52#13095496—24no—
53C2×#11715895C2312yes—
54C2×#11715895C2312yes—
55C4⋊Dic97—21no—
56C4⋊Dic97C2312yes—
57#13095488—22no—
58#13095488—22no—
59C2×Q8×C97C2311yescharacteristic
60C2×D4×C97—21no—
61C2×D4×C97C2311yescharacteristic
62C22×C4×C97—21no—
63C22×C4×C97D411yescharacteristic
64C22×C4×C97D411yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—21no—
67C4⋊C4×C97—21no—
68C4⋊C4×C97—21no—
69C4⋊C4×C97—21no—
70C4⋊C4×C97C2311yescharacteristic
71C4⋊C4×C97C2311yescharacteristic
72C22⋊C4×C97—21no—
73C22⋊C4×C97—21no—
74C22⋊C4×C97—21no—
75C22⋊C4×C97—21no—
76C22⋊C4×C97—21no—
77C42×C97—21no—
78C42×C97D411yescharacteristic
79C42×C97D411yescharacteristic
80C42×C97C2311yescharacteristic
81C42×C97C2311yescharacteristic
82C23×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—24no—
85C22×D97—22no—
86C22×D97—42no—
87C22×D97—42no—
88Q8×C97—22no—
89D4×C97—22no—
90D4×C97—42no—
91C2×C4×C97—22no—
92C2×C4×C97—22no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97—21no—
96C2×C4×C97—41no—
97C2×C4×C97—42no—
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2×D411yescharacteristic
101C2×C4×C97C2×D411yescharacteristic
102C2×C4×C97C2×D411yescharacteristic
103C2×C4×C97C2411yescharacteristic
104C2×Dic97—22no—
105C2×Dic97—42no—
106D388—24no—
107D388—22no—
108D388#4114yes—
109C4×D97—44no—
110C4×D97—44no—
111#11715895—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97AGL(1,ℤ/8)12yes—
119C22×C97C22×D411yescharacteristic
120C2×D97—24no—
121C2×D97—48no—
122C2×D97—42no—
123C2×D97—42no—
124C4×C97—21no—
125C4×C97—21no—
126C4×C97—21no—
127C4×C97—21no—
128C4×C97—21no—
129C4×C97—41no—
130C4×C97—41no—
131C4×C97—41no—
132C4×C97C2×#4112yes—
133Dic97—42no—
134C2×C97—22no—
135C2×C97—81no—
136C2×C97#52511yescharacteristic
137C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C97#57511yescharacteristic
139D97—44no—
140#3114—971nomaximal, Sylow
141C97#311411yescharacteristic, complemented, Sylow
142#529D9711yescharacteristic, complemented
143C2×C4×D4—971no—
144#483—971no—
145#480—971no—
146#478—972no—
147#463—972no—
148#450—971no—
149#447—971no—
150#442—972no—
151#440—972no—
152#432—971no—
153C22×D4—971no—
154C23×C4—1941no—
155C2×SD16—972no—
156C2×D8—972no—
157#126—21no—
158#125C2×D9711yescharacteristic, complemented
159#124C2×D9711yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D9711yescharacteristic
162#121C2×D9711yescharacteristic
163C4×Q8C2×D9711yescharacteristic, complemented
164C4×D4—21no—
165C4×D4—972no—
166C4×D4—972no—
167C4×D4—1941no—
168C4×D4—1941no—
169C4×D4C2×D9711yescharacteristic, complemented
170#117—21no—
171C2×C4⋊C4—971no—
172C2×C22⋊C4—1941no—
173C2×C42C2×D9711yescharacteristic
174#107—972no—
175#106—972no—
176C4⋊C8—972no—
177#103—972no—
178#102—972no—
179#102—972no—
180#102—972no—
181C22⋊C8—971no—
182C22⋊C8—971no—
183#97—972no—
184C24—1941no—
185C2×Q8C22×D9711yescharacteristic
186C2×D4—21no—
187C2×D4—972no—
188C2×D4—1944no—
189C2×D4—1941no—
190C2×D4C22×D9711yescharacteristic
191C22×C4—21no—
192C22×C4—3884no—
193C22×C4—1941no—
194C22×C4—1942no—
195C22×C4—1942no—
196C22×C4C22⋊D9711yescharacteristic
197C22×C4C22⋊D9711yescharacteristic
198C22×C4C22×D9711yescharacteristic
199SD16—1944no—
200D8—1944no—
201C2×C8—972no—
202C2×C8—972no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—972no—
208C4⋊C4—1941no—
209C4⋊C4C22×D9711yescharacteristic
210C4⋊C4C22×D9711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—1942no—
217C22⋊C4—1942no—
218C42—21no—
219C42C22⋊D9711yescharacteristic
220C42C22⋊D9711yescharacteristic
221C42C22×D9711yescharacteristic
222C42C22×D9711yescharacteristic
223C23—21no—
224C23—3882no—
225C23—3882no—
226C23—1944no—
227C23—1942no—
228C23C2×C22⋊D9711yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—974no—
233D4—1944no—
234D4—1942no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—3884no—
243C2×C4—3884no—
244C2×C4—3882no—
245C2×C4—1942no—
246C2×C4C2×C22⋊D9711yescharacteristic
247C2×C4C2×C22⋊D9711yescharacteristic
248C2×C4C2×C22⋊D9711yescharacteristic
249C2×C4C2×C22⋊D9711yescharacteristic
250C2×C4C2×C22⋊D9711yescharacteristic
251C2×C4C23×D9711yescharacteristic
252C8—1942no—
253C8—1942no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3888no—
260C22—3882no—
261C22—3882no—
262C22—1944no—
263C22#1569083512yes—
264C22C22×C22⋊D9711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—3882no—
274C4#41×D9712yes—
275C2—22no—
276C2—81no—
277C2—3884no—
278C2#2026124211yescharacteristic
279C2C2×#1569083511yescharacteristic
280C2#2026153311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#529×C97C2
socleC22×C97C22×D4
prime radical#20260868C2
hypercenter#529D97
hypocenterC97#3114
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D388C2semidirect
2. #20261063C2semidirect
3. #20261046C2semidirect
4. #20261032C2semidirect
5. #20260947C2semidirect
6. #20260885C2semidirect
8. #20260845C2semidirect
9. #20260833C2semidirect
10. #20260792C2semidirect
11. #529×C97C2semidirect
22. #15690743C22semidirect
23. #15690741C22semidirect
24. C4⋊#11715895C22semidirect
25. #15690734C22semidirect
26. #15690731C22semidirect
31. #15690721C22semidirect
33. #125×C97C22semidirect
34. #124×C97C22semidirect
38. C4×Q8×C97C22semidirect
40. C4×D4×C97C22semidirect
141. C97#3114semidirect
142. #529D97semidirect
158. #125C2×D97semidirect
159. #124C2×D97semidirect
163. C4×Q8C2×D97semidirect
169. C4×D4C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3882
#202610632
#202610462
#202610322
#202609472
#202608852
#202608682
#202608452
#202608332
#202607922
#529×C972
#311497