Finite Group Databasebeta

← #397×D97 #27144959

Subgroups — #397×D97 #27144959

Order 12416.

820 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D97C212yesmaximal
3C2×C22⋊C4×D97C211yesmaximal, characteristic
4#20260664C212yesmaximal, complemented
5#20260663C212yesmaximal, complemented
6#20260662C211yesmaximal, characteristic, complemented
7#20260661C211yesmaximal, characteristic, complemented
8#20260660C211yesmaximal, characteristic, complemented
9#95×D97C212yesmaximal, complemented
10#95×D97C211yesmaximal, characteristic, complemented
11#397×C97C211yesmaximal, characteristic, complemented
12C24×D97C2211yescharacteristic
13C22×C4×D97C2212yes—
14C22×C4×D97C2212yes—
15C22×C4×D97C2211yescharacteristic
16C22×C4×D97C2211yescharacteristic
17C2×C4⋊Dic97C2211yescharacteristic, complemented
18C2×#13095490C2212yescomplemented
19C4⋊C4×D97—24no—
20C2×C22⋊Dic97C2212yes—
21C2×C22⋊Dic97C2211yescharacteristic
22C2×#13095488C2211yescharacteristic
23C2×#13095488C2212yes—
24C2×#13095488C2212yes—
25C2×#13095488C2211yescharacteristic
26C22⋊C4×D97—28no—
27C22⋊C4×D97—24no—
28#15690717C2211yescharacteristic, complemented
29#15690717C2212yescomplemented
30#15690717C2212yescomplemented
31#15690717C2212yescomplemented
32#15690717C2212yescomplemented
33C2×C4⋊C4×C97C2211yescharacteristic, complemented
34C2×C22⋊C4×C97C2211yescharacteristic
35C2×C22⋊C4×C97C2212yes—
36#95×C97C2211yescharacteristic, complemented
37#95×C97C2212yescomplemented
38C23×D97—22no—
39C23×D97—22no—
40C23×D97—24no—
41C23×D97—24no—
42C23×D97D414yes—
43C23×D97C2311yescharacteristic
44C23×D97C2311yescharacteristic
45C22×Dic97C2312yes—
46C22×Dic97C2311yescharacteristic
47C22×Dic97C2312yes—
48C22×Dic97C2311yescharacteristic
49C2×C4×D97—24no—
50C2×C4×D97—24no—
51C2×C4×D97—24no—
52C2×C4×D97—22no—
53C2×C4×D97—24no—
54C2×C4×D97—24no—
55C2×C4×D97—24no—
56C2×C4×D97—24no—
57C2×C4×D97—24no—
58C2×C4×D97D414yes—
59C4⋊Dic97—22no—
60#13095490—24no—
61C22⋊Dic97—24no—
62C22⋊Dic97—22no—
63#13095488—24no—
64#13095488—24no—
65#13095488—22no—
66#13095488—22no—
67C24×C97C2311yescharacteristic
68C22×C4×C97C2312yes—
69C22×C4×C97C2312yes—
70C22×C4×C97C2311yescharacteristic
71C22×C4×C97C2311yescharacteristic
72C4⋊C4×C97—22no—
73C22⋊C4×C97—24no—
74C22⋊C4×C97—22no—
75C23×C97—21no—
76C23×C97—21no—
77C23×C97—22no—
78C23×C97—22no—
79C23×C97C2×D412yes—
80C23×C97C2411yescharacteristic
81C22×D97—21no—
82C22×D97—21no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—28no—
86C22×D97—216no—
87C22×D97—44no—
88C22×D97—44no—
89C22×D97—48no—
90C22×D97C2×D412yes—
91C22×D97C2×D412yes—
92C22×D97C2×D412yes—
93C22×D97#4112yes—
94C22×D97#4114yes—
95C22×D97#4114yes—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97—21no—
100C2×C4×C97—22no—
101C2×C4×C97—22no—
102C2×C4×C97—22no—
103C2×C4×C97—22no—
104C2×C4×C97—22no—
105C2×C4×C97C2×D412yes—
106C2×Dic97—22no—
107C2×Dic97—22no—
108C2×Dic97—22no—
109C2×Dic97—21no—
110C2×Dic97—22no—
111C2×Dic97—22no—
112C2×Dic97—22no—
113C2×Dic97—22no—
114C2×Dic97—22no—
115C2×Dic97C2×D412yes—
116C4×D97—28no—
117C4×D97—48no—
118C4×D97—48no—
119C4×D97—44no—
120C22×C97—22no—
121C22×C97—24no—
122C22×C97—41no—
123C22×C97—41no—
124C22×C97—42no—
125C22×C97C22×D411yescharacteristic
126C22×C97C22×D411yescharacteristic
127C22×C97C2×#4111yescharacteristic
128C22×C97C2×#4112yes—
129C22×C97C2×#4112yes—
130C2×D97—22no—
131C2×D97—24no—
132C2×D97—41no—
133C2×D97—41no—
134C2×D97—42no—
135C2×D97—416no—
136C2×D97#12114yes—
137C2×D97#12314yes—
138C2×D97#12318yes—
139C2×D97#12414yes—
140C2×D97#12618yes—
141C4×C97—22no—
142C4×C97—42no—
143C4×C97—42no—
144C4×C97—41no—
145Dic97—22no—
146Dic97—42no—
147Dic97—42no—
148Dic97—41no—
149C2×C97—42no—
150C2×C97C2×#12111yescharacteristic
151C2×C97C2×#12311yescharacteristic
152C2×C97C2×#12312yes—
153C2×C97C2×#12411yescharacteristic
154C2×C97C2×#12612yes—
155D97—42no—
156D97#39718yesdirect product factor
157C2×#397—971nomaximal, Sylow
158C97C2×#39711yescharacteristic, complemented, Sylow
159C22×C4⋊C4—971no—
160C22×C22⋊C4—971no—
161C22×C22⋊C4—972no—
162#397—971no—
163#397—971no—
164#397—972no—
165#397—971no—
166#397—972no—
167#397D9711yescharacteristic, direct product factor
168C2×#95—971no—
169C2×#95—972no—
170C25—971no—
171C23×C4—971no—
172C23×C4—972no—
173C23×C4—971no—
174C23×C4—972no—
175C2×C4⋊C4—972no—
176C2×C4⋊C4—971no—
177C2×C4⋊C4—1944no—
178C2×C4⋊C4C2×D9711yescharacteristic, complemented
179C2×C22⋊C4—971no—
180C2×C22⋊C4—972no—
181C2×C22⋊C4—971no—
182C2×C22⋊C4—972no—
183C2×C22⋊C4—972no—
184C2×C22⋊C4—971no—
185C2×C22⋊C4—1944no—
186C2×C22⋊C4—1948no—
187C2×C22⋊C4C2×D9712yes—
188C2×C22⋊C4C2×D9711yescharacteristic
189#95—972no—
190#95—972no—
191#95—971no—
192#95—972no—
193#95—972no—
194#95C2×D9712yescomplemented
195#95C2×D9711yescharacteristic, complemented
196C24—971no—
197C24—974no—
198C24—971no—
199C24—1942no—
200C24—1942no—
201C24—1944no—
202C24—1944no—
203C24C22×D9711yescharacteristic
204C22×C4—971no—
205C22×C4—974no—
206C22×C4—972no—
207C22×C4—971no—
208C22×C4—972no—
209C22×C4—1942no—
210C22×C4—1944no—
211C22×C4—1944no—
212C22×C4—1944no—
213C22×C4—1944no—
214C22×C4—1944no—
215C22×C4—1944no—
216C22×C4—1944no—
217C22×C4—1944no—
218C22×C4C22×D9712yes—
219C22×C4C22×D9711yescharacteristic
220C22×C4C22×D9712yes—
221C22×C4C22×D9711yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—1944no—
224C4⋊C4—1942no—
225C22⋊C4—24no—
226C22⋊C4—22no—
227C22⋊C4—1942no—
228C22⋊C4—1944no—
229C22⋊C4—1944no—
230C22⋊C4—1942no—
231C22⋊C4—1944no—
232C22⋊C4—1942no—
233C23—21no—
234C23—21no—
235C23—22no—
236C23—22no—
237C23—972no—
238C23—972no—
239C23—972no—
240C23—974no—
241C23—974no—
242C23—972no—
243C23—3884no—
244C23—3884no—
245C23—3888no—
246C23—1948no—
247C23—19416no—
248C23—1941no—
249C23—1942no—
250C23—1942no—
251C23—1941no—
252C23D4×D9712yes—
253C23C23×D9711yescharacteristic
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—972no—
264C2×C4—3888no—
265C2×C4—3888no—
266C2×C4—3884no—
267C2×C4—1948no—
268C2×C4—1942no—
269C2×C4—1942no—
270C2×C4—1941no—
271C2×C4—1942no—
272C2×C4—1942no—
273C2×C4—1942no—
274C2×C4—1942no—
275C2×C4—1942no—
276C2×C4—1942no—
277C2×C4D4×D9712yes—
278C22—22no—
279C22—24no—
280C22—41no—
281C22—41no—
282C22—42no—
283C22—974no—
284C22—978no—
285C22—974no—
286C22—974no—
287C22—978no—
288C22—38816no—
289C22—3881no—
290C22—3881no—
291C22—3882no—
292C22—1942no—
293C22—1944no—
294C22C2×D4×D9711yescharacteristic
295C22C2×D4×D9711yescharacteristic
296C22#41×D9711yescharacteristic
297C22#41×D9712yes—
298C22#41×D9712yes—
299C4—22no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—3882no—
304C4—3882no—
305C4—3881no—
306C4—1942no—
307C2—42no—
308C2—978no—
309C2—3882no—
310C2#121×D9711yescharacteristic
311C2#123×D9711yescharacteristic
312C2#123×D9712yes—
313C2#124×D9711yescharacteristic
314C2#126×D9712yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#397×C97C2
prime radicalC24×D97C22
hypercenter#397D97
hypocenterC97C2×#397
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×D97C2semidirect
4. #20260664C2semidirect
5. #20260663C2semidirect
6. #20260662C2semidirect
7. #20260661C2semidirect
8. #20260660C2semidirect
9. #95×D97C2semidirect
10. #95×D97C2semidirect
11. #397×C97C2semidirect
17. C2×C4⋊Dic97C22semidirect
18. C2×#13095490C22semidirect
28. #15690717C22semidirect
29. #15690717C22semidirect
30. #15690717C22semidirect
31. #15690717C22semidirect
32. #15690717C22semidirect
33. C2×C4⋊C4×C97C22semidirect
36. #95×C97C22semidirect
37. #95×C97C22semidirect
156. D97#397direct (N × H)
158. C97C2×#397semidirect
167. #397D97direct (N × H)
178. C2×C4⋊C4C2×D97semidirect
194. #95C2×D97semidirect
195. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D972
C2×C22⋊C4×D972
#202606642
#202606632
#202606622
#202606612
#202606602
#95×D972
#397×C972
C2×#39797