Finite Group Databasebeta

← #27144717

Subgroups — #27144717

Order 12416.

928 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 209 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
2C2×#15690770C212yesmaximal, complemented
3C2×C22⋊C4×D97C212yesmaximal, complemented
4#20260623C212yesmaximal, complemented
5#20260615C211yesmaximal, characteristic, complemented
6#20260611C212yesmaximal, complemented
7#20260606C212yesmaximal, complemented
8C4×#13095488C212yesmaximal, complemented
9#390×C97C211yesmaximal, characteristic, complemented
10C24×D97C2211yescharacteristic
11C22×C22⋊D97C2212yes—
12C23×Dic97C2211yescharacteristic, complemented
13C22×C4×D97C2212yes—
14C22×C4×D97C2212yes—
15#15690778C418yescomplemented
16#15690770—28no—
17C2×#13095490C2212yescomplemented
18C2×C22⋊Dic97C2212yes—
19C2×C22⋊Dic97C2211yescharacteristic
20C2×C22⋊Dic97C2211yescharacteristic, complemented
21C2×#13095488C2212yes—
22C2×#13095488C2212yes—
23C2×#13095488C2212yes—
24C22⋊C4×D97—28no—
25C2×C4×Dic97C2212yescomplemented
26#15690717C2212yescomplemented
27#15690717C2212yescomplemented
28#15690717C2212yescomplemented
29C22×D4×C97C2211yescharacteristic
30C2×C4⋊C4×C97C2211yescharacteristic, complemented
31C2×C22⋊C4×C97C2212yes—
32C2×C22⋊C4×C97C2212yescomplemented
33C2×C42×C97C2211yescharacteristic, complemented
34C23×D97—24no—
35C23×D97—24no—
36C23×D97—41no—
37C23×D97—41no—
38C23×D97C2×C414yescomplemented
39C23×D97C2312yes—
40C2×C22⋊D97—24no—
41C2×C22⋊D97—24no—
42C2×C22⋊D97C2×C418yescomplemented
43C22×Dic97—24no—
44C22×Dic97—41no—
45C22×Dic97C2312yes—
46C22×Dic97C2312yes—
47C22×Dic97C2312yes—
48C2×C4×D97—24no—
49C2×C4×D97—24no—
50C2×C4×D97—24no—
51C2×C4×D97—24no—
52C2×C4×D97—42no—
53C2×C4×D97—42no—
54#13095490—24no—
55C22⋊Dic97—24no—
56C22⋊Dic97—41no—
57C22⋊Dic97C2×C414yescomplemented
58#13095488—24no—
59#13095488C2×C418yescomplemented
60#13095488D418yescomplemented
61C4×Dic97—24no—
62C24×C97C2311yescharacteristic
63C24×C97C2311yescharacteristic
64C2×D4×C97—41no—
65C2×D4×C97—41no—
66C2×D4×C97C2×C414yescomplemented
67C22×C4×C97C2312yes—
68C22×C4×C97C2311yescharacteristic
69C22×C4×C97C2312yes—
70C4⋊C4×C97—41no—
71C22⋊C4×C97—24no—
72C22⋊C4×C97—24no—
73C42×C97—41no—
74C23×C97—22no—
75C23×C97—22no—
76C23×C97—22no—
77C23×C97—22no—
78C23×C97—21no—
79C23×C97—21no—
80C23×C97—21no—
81C23×C97—21no—
82C23×C97C22×C412yes—
83C23×C97C22×C412yes—
84C23×C97C2411yescharacteristic
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—22no—
89C22×D97—216no—
90C22×D97—44no—
91C22×D97—44no—
92C22×D97—48no—
93C22×D97—44no—
94C22×D97C22×C414yes—
95C22×D97C2×D414yes—
96C22×D97#4114yes—
97D4×C97—44no—
98C2×C4×C97—22no—
99C2×C4×C97—21no—
100C2×C4×C97—22no—
101C2×C4×C97—21no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97—22no—
105C2×C4×C97C22×C412yes—
106C2×C4×C97C2×D414yes—
107C22⋊D97—216no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97—22no—
111C2×Dic97—22no—
112C2×Dic97—22no—
113C2×Dic97—22no—
114C2×Dic97—22no—
115C2×Dic97—44no—
116C2×Dic97C22×C414yes—
117C2×Dic97C2×D414yes—
118C4×D97—48no—
119C4×D97—48no—
120C22×C97—24no—
121C22×C97—24no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—41no—
125C22×C97—41no—
126C22×C97—42no—
127C22×C97—42no—
128C22×C97—41no—
129C22×C97—41no—
130C22×C97C23×C411yescharacteristic
131C22×C97C22×D412yes—
132C22×C97C2×#4112yes—
133C22×C97H(5,𝔽2)11yescharacteristic
134C22×C97H(5,𝔽2)11yescharacteristic
135C2×D97—24no—
136C2×D97—24no—
137C2×D97—24no—
138C2×D97—24no—
139C2×D97—24no—
140C2×D97—24no—
141C2×D97—416no—
142C2×D97C4×D418yescomplemented
143C4×C97—22no—
144C4×C97—24no—
145C4×C97—42no—
146Dic97—24no—
147Dic97—24no—
148Dic97—42no—
149C2×C97—42no—
150C2×C97—42no—
151C2×C97C2×C4×D412yes—
152C2×C97#51811yescharacteristic
153C2×C97D4211yescharacteristic
154C2×C97#54612yes—
155C2×C97#55111yescharacteristic
156D97—28no—
157#2358—971nomaximal, Sylow
158C97#235811yescharacteristic, complemented, Sylow
159C2×#120—971no—
160C2×C4×D4—972no—
161C22×C22⋊C4—972no—
162#390D9711yescharacteristic, complemented
163#388—972no—
164#386—971no—
165#386—972no—
166#385—972no—
167C4×C22⋊C4—972no—
168C25—971no—
169C22×D4—972no—
170C22×D4C2×D9711yescharacteristic
171C23×C4—971no—
172C23×C4—972no—
173C23×C4—972no—
174#120—978no—
175C4×D4—1948no—
176C2×C4⋊C4—972no—
177C2×C4⋊C4C2×D9711yescharacteristic, complemented
178C2×C22⋊C4—971no—
179C2×C22⋊C4—972no—
180C2×C22⋊C4—972no—
181C2×C22⋊C4—971no—
182C2×C22⋊C4—972no—
183C2×C22⋊C4—972no—
184C2×C22⋊C4—1948no—
185C2×C22⋊C4C2×D9712yes—
186C2×C22⋊C4C2×D9712yescomplemented
187C2×C42—972no—
188C2×C42C2×D9711yescharacteristic, complemented
189#95—972no—
190#95—972no—
191#95—972no—
192C24—972no—
193C24—974no—
194C24—3881no—
195C24—3881no—
196C24—1944no—
197C24—1944no—
198C24C22×D9711yescharacteristic
199C24C22×D9711yescharacteristic
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—978no—
203C2×D4—1944no—
204C2×D4—1944no—
205C2×D4C4×D9714yescomplemented
206C22×C4—972no—
207C22×C4—972no—
208C22×C4—972no—
209C22×C4—3882no—
210C22×C4—3882no—
211C22×C4—3881no—
212C22×C4—1944no—
213C22×C4—1944no—
214C22×C4—1944no—
215C22×C4—1944no—
216C22×C4—1944no—
217C22×C4C22×D9712yes—
218C22×C4C22×D9711yescharacteristic
219C22×C4C22×D9712yes—
220C4⋊C4—41no—
221C4⋊C4—1944no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—974no—
225C22⋊C4—978no—
226C22⋊C4—978no—
227C22⋊C4—3881no—
228C22⋊C4—1944no—
229C22⋊C4—1944no—
230C42—41no—
231C42—1944no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—974no—
241C23—974no—
242C23—974no—
243C23—3884no—
244C23—3884no—
245C23—3884no—
246C23—3888no—
247C23—19416no—
248C23—1942no—
249C23—1942no—
250C23—1942no—
251C23—1942no—
252C23C2×C4×D9712yes—
253C23C2×C4×D9712yes—
254C23C23×D9711yescharacteristic
255D4—44no—
256D4—19416no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—974no—
265C2×C4—974no—
266C2×C4—3888no—
267C2×C4—3888no—
268C2×C4—3884no—
269C2×C4—1942no—
270C2×C4—1942no—
271C2×C4—1942no—
272C2×C4—1942no—
273C2×C4—1942no—
274C2×C4—1942no—
275C2×C4—1942no—
276C2×C4C2×C4×D9712yes—
277C2×C4D4×D9714yes—
278C22—24no—
279C22—24no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—42no—
287C22—42no—
288C22—978no—
289C22—38816no—
290C22—1944no—
291C22—1944no—
292C22—1944no—
293C22—1944no—
294C22—1944no—
295C22—1944no—
296C22C22×C4×D9711yescharacteristic
297C22C2×D4×D9712yes—
298C22#41×D9712yes—
299C22#1569085511yescharacteristic
300C22#1569085511yescharacteristic
301C4—22no—
302C4—24no—
303C4—42no—
304C4—3882no—
305C4—1944no—
306C4—1944no—
307C2—42no—
308C2—42no—
309C2—1948no—
310C2C4×D4×D9712yes—
311C2#2026120811yescharacteristic
312C2#2026137011yescharacteristic
313C2#2026137412yes—
314C2#2026141511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#390×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690778C2
hypercenter#390D97
hypocenterC97#2358
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#15690770C2semidirect
3. C2×C22⋊C4×D97C2semidirect
4. #20260623C2semidirect
5. #20260615C2semidirect
6. #20260611C2semidirect
7. #20260606C2semidirect
8. C4×#13095488C2semidirect
9. #390×C97C2semidirect
12. C23×Dic97C22semidirect
15. #15690778C4semidirect
17. C2×#13095490C22semidirect
20. C2×C22⋊Dic97C22semidirect
25. C2×C4×Dic97C22semidirect
26. #15690717C22semidirect
27. #15690717C22semidirect
28. #15690717C22semidirect
30. C2×C4⋊C4×C97C22semidirect
32. C2×C22⋊C4×C97C22semidirect
33. C2×C42×C97C22semidirect
38. C23×D97C2×C4semidirect
42. C2×C22⋊D97C2×C4semidirect
57. C22⋊Dic97C2×C4semidirect
59. #13095488C2×C4semidirect
60. #13095488D4semidirect
66. C2×D4×C97C2×C4semidirect
142. C2×D97C4×D4semidirect
158. C97#2358semidirect
162. #390D97semidirect
177. C2×C4⋊C4C2×D97semidirect
186. C2×C22⋊C4C2×D97semidirect
188. C2×C42C2×D97semidirect
205. C2×D4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907782
C2×#156907702
C2×C22⋊C4×D972
#202606232
#202606152
#202606112
#202606062
C4×#130954882
#390×C972
#235897