Finite Group Databasebeta

← #27144874

Subgroups — #27144874

Order 12416.

688 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690778C211yesmaximal, characteristic
2#20260638C211yesmaximal, characteristic, complemented
3#20260635C212yesmaximal, complemented
4#20260629C212yesmaximal, complemented
5#20260627C212yesmaximal, complemented
6#20260624C212yesmaximal, complemented
7#20260614C212yesmaximal, complemented
8#20260590C212yesmaximal, complemented
9#386×C97C211yesmaximal, characteristic, complemented
10C24×D97C2211yescharacteristic
11C22×C22⋊D97C2212yes—
12#15690778C418yescomplemented
13C2×C4⋊Dic97C2211yescharacteristic, complemented
14C2×#13095490C2212yescomplemented
15C2×C22⋊Dic97C2212yes—
16C2×C22⋊Dic97C2211yescharacteristic
17C2×C22⋊Dic97C2212yescomplemented
18C2×#13095488C2212yes—
19C2×#13095488C2212yes—
20C2×#13095488C2212yes—
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×C4×Dic97C2211yescharacteristic, complemented
24C2×C4×Dic97C2212yescomplemented
25#15690717C2212yescomplemented
26#15690717C2212yescomplemented
27C22×D4×C97C2211yescharacteristic
28C23×C4×C97C2211yescharacteristic, complemented
29C2×C22⋊C4×C97C2212yes—
30C2×C22⋊C4×C97C2211yescharacteristic, complemented
31#95×C97C2212yescomplemented
32C23×D97—41no—
33C23×D97—41no—
34C23×D97—42no—
35C23×D97—42no—
36C23×D97C2×C414yescomplemented
37C23×D97C2312yes—
38C2×C22⋊D97—42no—
39C2×C22⋊D97—42no—
40C2×C22⋊D97C2×C418yes—
41C22×Dic97C2312yes—
42C22×Dic97C2312yes—
43C22×Dic97C2312yes—
44C4⋊Dic97—41no—
45#13095490—42no—
46C22⋊Dic97—42no—
47C22⋊Dic97—42no—
48C22⋊Dic97C2×C414yescomplemented
49#13095488—42no—
50#13095488—42no—
51#13095488—42no—
52#13095488—42no—
53#13095488C2×C418yescomplemented
54C4×Dic97—42no—
55C4×Dic97—41no—
56C24×C97C2311yescharacteristic
57C24×C97C2311yescharacteristic
58C2×D4×C97—41no—
59C2×D4×C97—41no—
60C2×D4×C97C2×C414yes—
61C22×C4×C97—42no—
62C22×C4×C97—41no—
63C22×C4×C97C2312yes—
64C22×C4×C97C2312yes—
65C22×C4×C97C2311yescharacteristic
66C22⋊C4×C97—42no—
67C22⋊C4×C97—41no—
68C23×C97—21no—
69C23×C97—22no—
70C23×C97—21no—
71C23×C97—22no—
72C23×C97—21no—
73C23×C97—21no—
74C23×C97—22no—
75C23×C97—22no—
76C23×C97C22×C412yes—
77C23×C97C22×C412yes—
78C23×C97C2411yescharacteristic
79C22×D97—22no—
80C22×D97—22no—
81C22×D97—22no—
82C22×D97—22no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—48no—
86C22×D97—44no—
87C22×D97—48no—
88C22×D97—44no—
89C22×D97—44no—
90C22×D97C22×C414yes—
91D4×C97—44no—
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97—22no—
100C2×C4×C97—44no—
101C2×C4×C97C22×C412yes—
102C22⋊D97—48no—
103C2×Dic97—22no—
104C2×Dic97—22no—
105C2×Dic97—22no—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97—22no—
111C2×Dic97C22×C414yes—
112C22×C97—24no—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97—24no—
116C22×C97—42no—
117C22×C97—41no—
118C22×C97—42no—
119C22×C97—41no—
120C22×C97—41no—
121C22×C97—41no—
122C22×C97C23×C411yescharacteristic
123C22×C97H(5,𝔽2)11yescharacteristic
124C22×C97H(5,𝔽2)12yes—
125C22×C97H(5,𝔽2)11yescharacteristic
126C22×C97H(5,𝔽2)12yes—
127C2×D97—24no—
128C2×D97—24no—
129C2×D97—24no—
130C2×D97—416no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—42no—
135C4×C97—22no—
136C4×C97—42no—
137C4×C97—42no—
138Dic97—24no—
139Dic97—42no—
140Dic97—42no—
141C2×C97—42no—
142C2×C97—42no—
143C2×C97#51812yes—
144C2×C97#51811yescharacteristic
145C2×C97#56012yes—
146C2×C97#56011yescharacteristic
147C2×C97H(3,𝔽4)11yescharacteristic
148D97—44no—
149#2375—971nomaximal, Sylow
150C97#237511yescharacteristic, complemented, Sylow
151C2×#120—971no—
152#390—971no—
153#390—972no—
154#388—972no—
155#388—972no—
156#388—972no—
157#386—972no—
158#386D9711yescharacteristic, complemented
159#379—972no—
160C25—971no—
161C22×D4—972no—
162C22×D4C2×D9711yescharacteristic
163C23×C4C2×D9711yescharacteristic, complemented
164#120—978no—
165C2×C4⋊C4—972no—
166C2×C4⋊C4—971no—
167C2×C22⋊C4—971no—
168C2×C22⋊C4—972no—
169C2×C22⋊C4—972no—
170C2×C22⋊C4—972no—
171C2×C22⋊C4—972no—
172C2×C22⋊C4—972no—
173C2×C22⋊C4—972no—
174C2×C22⋊C4—972no—
175C2×C22⋊C4C2×D9712yes—
176C2×C22⋊C4C2×D9711yescharacteristic, complemented
177C2×C42—972no—
178C2×C42—971no—
179#95—972no—
180#95—972no—
181#95C2×D9712yescomplemented
182C24—974no—
183C24—972no—
184C24—3882no—
185C24—3881no—
186C24—3881no—
187C24—3882no—
188C24C22×D9711yescharacteristic
189C24C22×D9711yescharacteristic
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—978no—
193C2×D4—3882no—
194C2×D4—3882no—
195C2×D4C4×D9714yes—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—972no—
199C22×C4—972no—
200C22×C4—972no—
201C22×C4C22×D9712yes—
202C22×C4C22×D9712yes—
203C22×C4C22×D9711yescharacteristic
204C4⋊C4—3882no—
205C4⋊C4—3881no—
206C22⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—974no—
209C22⋊C4—978no—
210C22⋊C4—3882no—
211C22⋊C4—3882no—
212C22⋊C4—3882no—
213C22⋊C4—3882no—
214C22⋊C4—3882no—
215C22⋊C4—3882no—
216C42—3881no—
217C42—3882no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—22no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—22no—
226C23—974no—
227C23—3888no—
228C23—3884no—
229C23—3888no—
230C23—3884no—
231C23—3884no—
232C23—1942no—
233C23—1942no—
234C23—1942no—
235C23—1942no—
236C23—1942no—
237C23—1942no—
238C23C2×C4×D9712yes—
239C23C2×C4×D9712yes—
240C23C23×D9711yescharacteristic
241D4—44no—
242D4—3888no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—44no—
252C2×C4—974no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4—1942no—
256C2×C4—1942no—
257C2×C4—1942no—
258C2×C4—1942no—
259C2×C4—1942no—
260C2×C4—1942no—
261C2×C4C2×C4×D9712yes—
262C22—24no—
263C22—24no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—38816no—
273C22—3882no—
274C22—3882no—
275C22—3882no—
276C22—3882no—
277C22—1944no—
278C22—1944no—
279C22—1944no—
280C22C22×C4×D9711yescharacteristic
281C22#1569085512yes—
282C22#1569085511yescharacteristic
283C22#1569085511yescharacteristic
284C22#1569085612yes—
285C4—22no—
286C4—42no—
287C4—42no—
288C4—3882no—
289C4—3882no—
290C4—1944no—
291C2—42no—
292C2—42no—
293C2—3884no—
294C2#2026120712yes—
295C2#2026120811yescharacteristic
296C2#2026147312yes—
297C2#2026147511yescharacteristic
298C2#2026147811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#386×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690778C2
hypercenter#386D97
hypocenterC97#2375
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20260638C2semidirect
3. #20260635C2semidirect
4. #20260629C2semidirect
5. #20260627C2semidirect
6. #20260624C2semidirect
7. #20260614C2semidirect
8. #20260590C2semidirect
9. #386×C97C2semidirect
12. #15690778C4semidirect
13. C2×C4⋊Dic97C22semidirect
14. C2×#13095490C22semidirect
17. C2×C22⋊Dic97C22semidirect
23. C2×C4×Dic97C22semidirect
24. C2×C4×Dic97C22semidirect
25. #15690717C22semidirect
26. #15690717C22semidirect
28. C23×C4×C97C22semidirect
30. C2×C22⋊C4×C97C22semidirect
31. #95×C97C22semidirect
36. C23×D97C2×C4semidirect
48. C22⋊Dic97C2×C4semidirect
53. #13095488C2×C4semidirect
150. C97#2375semidirect
158. #386D97semidirect
163. C23×C4C2×D97semidirect
176. C2×C22⋊C4C2×D97semidirect
181. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907782
#202606382
#202606352
#202606292
#202606272
#202606242
#202606142
#202605902
#386×C972
#237597