Finite Group Databasebeta

← #27151877

Subgroups — #27151877

Order 12416.

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#20261271C211yesmaximal, characteristic, complemented
2C2×#15690772C211yesmaximal, characteristic, complemented
3#20261048C211yesmaximal, characteristic, complemented
4#20261034C212yesmaximal, complemented
5#20260953C212yesmaximal, complemented
6#20260885C211yesmaximal, characteristic, complemented
7#20260871C211yesmaximal, characteristic
8#20260846C212yesmaximal, complemented
9#20260838C212yesmaximal, complemented
10#20260789C211yesmaximal, characteristic, complemented
11#483×C97C211yesmaximal, characteristic, complemented
12C22×D388C2211yescharacteristic
13C22×C4×D97—21no—
14C2×#13095498C2212yes—
15C2×D776C2212yes—
16#15690805—21no—
17#15690801C2211yescharacteristic, complemented
18#15690799C2211yescharacteristic, complemented
19#15690794—21no—
20#15690784C2211yescharacteristic
21Q8×Dic97C2211yescharacteristic, complemented
22#15690772—21no—
23#15690772—21no—
24#15690772C2212yes—
25#15690772C2212yes—
26#15690772C2211yescharacteristic, complemented
27#15690770—21no—
28#15690768—21no—
29C2×#13095488—21no—
30C2×C4×Dic97C2211yescharacteristic
31#15690743C2212yescomplemented
32#15690739C2212yescomplemented
33#15690732C2212yes—
34#15690730C2212yes—
35#15690730C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic
37#15690720C2212yescomplemented
38C22⋊Q8×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#106×C97C2212yescomplemented
41#103×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97—21no—
44C2×D388—21no—
45C2×D388—24no—
46C2×D388C2311yescharacteristic
47C2×D388C2312yes—
48C2×C22⋊D97—21no—
49C22×Dic97D411yescharacteristic
50C22×Dic97D411yescharacteristic
51C2×C4×D97—21no—
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97—44no—
55#13095498—24no—
56D776—24no—
57C2×#11715895C2312yes—
58C4⋊Dic97—21no—
59C4⋊Dic97—21no—
60C4⋊Dic97C2311yescharacteristic
61#13095490—21no—
62#13095488—21no—
63#13095488—22no—
64#13095488—22no—
65#13095488—21no—
66#13095488—21no—
67#13095488—21no—
68C4×Dic97—21no—
69C4×Dic97D411yescharacteristic
70C4×Dic97D411yescharacteristic
71C4×Dic97C2311yescharacteristic
72C4×Dic97C2311yescharacteristic
73C2×Q8×C97C2311yescharacteristic
74C22×C4×C97—21no—
75C22×C4×C97C2311yescharacteristic
76C2×C8×C97C2312yes—
77C4⋊C4×C97—21no—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2312yes—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C23×C97C2×D411yescharacteristic
83C22×D97—24no—
84C22×D97—22no—
85C22×D97—21no—
86C22×D97—42no—
87C22×D97—42no—
88Q8×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C22⋊D97—42no—
97C2×Dic97—22no—
98C2×Dic97—21no—
99C2×Dic97—41no—
100C2×Dic97C2×D411yescharacteristic
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C2×Dic97C2×D411yescharacteristic
104D388—24no—
105D388—22no—
106D388—22no—
107D388#4114yes—
108C4×D97—42no—
109C4×D97—44no—
110C4×D97—44no—
111C8×C97—22no—
112#11715895—22no—
113C22×C97—21no—
114C22×C97—21no—
115C22×C97AGL(1,ℤ/8)12yes—
116C22×C97C22×D411yescharacteristic
117C2×D97—24no—
118C2×D97—48no—
119C2×D97—42no—
120C2×D97—42no—
121C2×D97—41no—
122C2×D97—41no—
123C2×D97—41no—
124C4×C97—21no—
125C4×C97—41no—
126C4×C97—42no—
127C4×C97—41no—
128C4×C97C2×#4112yes—
129Dic97—21no—
130Dic97—21no—
131Dic97—21no—
132Dic97—21no—
133Dic97—41no—
134C2×C97—22no—
135C2×C97#52511yescharacteristic
136C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C97#57511yescharacteristic
138D97—44no—
139D97—81no—
140#3114—971nomaximal, Sylow
141C97#311411yescharacteristic, complemented, Sylow
142#529—971no—
143C2×C4×D4—971no—
144#483D9711yescharacteristic, complemented
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—1941no—
158#125—971no—
159#124—971no—
160#123—1941no—
161C22⋊Q8C2×D9711yescharacteristic
162#121—971no—
163C4×Q8—971no—
164C4×D4—972no—
165C4×D4—972no—
166C4×D4—971no—
167C4×D4—1941no—
168C4×D4—1941no—
169C4×D4—1941no—
170#117—1941no—
171C2×C4⋊C4C2×D9711yescharacteristic
172C2×C22⋊C4—1941no—
173C2×C42—971no—
174#107—972no—
175#106C2×D9712yescomplemented
176C4⋊C8—972no—
177#103C2×D9712yescomplemented
178#102—972no—
179#102—972no—
180#102—972no—
181C22⋊C8—971no—
182C22⋊C8C2×D9711yescharacteristic
183#97—972no—
184C24—1941no—
185C2×Q8C22×D9711yescharacteristic
186C2×D4—972no—
187C2×D4—971no—
188C2×D4—1944no—
189C2×D4—1941no—
190C2×D4—1941no—
191C22×C4—21no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—3884no—
195C22×C4—1941no—
196C22×C4—1942no—
197C22×C4—1942no—
198C22×C4C22×D9711yescharacteristic
199SD16—1944no—
200D8—1944no—
201C2×C8—972no—
202C2×C8C22×D9712yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—971no—
206C4⋊C4—1941no—
207C4⋊C4—1941no—
208C4⋊C4—1941no—
209C4⋊C4C22×D9711yescharacteristic
210C4⋊C4C22×D9712yes—
211C22⋊C4—21no—
212C22⋊C4—1942no—
213C22⋊C4—1942no—
214C22⋊C4—1941no—
215C22⋊C4—1941no—
216C22⋊C4—1941no—
217C22⋊C4—1941no—
218C42—971no—
219C42—971no—
220C42—971no—
221C42—971no—
222C42—1941no—
223C23—3882no—
224C23—3882no—
225C23—1944no—
226C23—1941no—
227C23—1942no—
228C23D4×D9711yescharacteristic
229Q8—22no—
230D4—974no—
231D4—3882no—
232D4—1944no—
233D4—1942no—
234D4—1942no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—971no—
241C2×C4—971no—
242C2×C4—971no—
243C2×C4—971no—
244C2×C4—3884no—
245C2×C4—3884no—
246C2×C4—3882no—
247C2×C4—3881no—
248C2×C4—1941no—
249C2×C4—1942no—
250C2×C4D4×D9711yescharacteristic
251C2×C4C23×D9711yescharacteristic
252C8—22no—
253C8—1942no—
254C22—21no—
255C22—21no—
256C22—3888no—
257C22—3881no—
258C22—3881no—
259C22—3881no—
260C22—3882no—
261C22—3882no—
262C22—1944no—
263C22#1569083712yes—
264C22C2×D4×D9711yescharacteristic
265C4—21no—
266C4—41no—
267C4—42no—
268C4—41no—
269C4—3881no—
270C4—1941no—
271C4—1941no—
272C4—1941no—
273C4—1941no—
274C4#41×D9712yes—
275C2—22no—
276C2—7761no—
277C2—3884no—
278C2#2026124311yescharacteristic
279C2C2×#1569083711yescharacteristic
280C2#2026153411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#483×C97C2
socleC22×C97C22×D4
prime radical#20260871C2
hypercenter#483D97
hypocenterC97#3114
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #20261271C2semidirect
2. C2×#15690772C2semidirect
3. #20261048C2semidirect
4. #20261034C2semidirect
5. #20260953C2semidirect
6. #20260885C2semidirect
8. #20260846C2semidirect
9. #20260838C2semidirect
10. #20260789C2semidirect
11. #483×C97C2semidirect
17. #15690801C22semidirect
18. #15690799C22semidirect
21. Q8×Dic97C22semidirect
26. #15690772C22semidirect
31. #15690743C22semidirect
32. #15690739C22semidirect
35. #15690730C22semidirect
37. #15690720C22semidirect
40. #106×C97C22semidirect
41. #103×C97C22semidirect
141. C97#3114semidirect
144. #483D97semidirect
175. #106C2×D97semidirect
177. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
C2×#156907722
#202610482
#202610342
#202609532
#202608852
#202608712
#202608462
#202608382
#202607892
#483×C972
#311497