Finite Group Databasebeta

← #27150015

Subgroups — #27150015

Order 12416.

1224 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D97C211yesmaximal, characteristic, complemented
2C2×#15690781C212yesmaximal, complemented
3C2×#15690778C212yesmaximal, complemented
4#20260615C212yesmaximal, complemented
5#20260611C214yesmaximal, complemented
6#20260595C211yesmaximal, characteristic, complemented
7#20260592C212yesmaximal, complemented
8C2×#121×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×C22⋊D97C2214yes—
11C23×Dic97C2211yescharacteristic
12C22×C4×D97—28no—
13C22×C4×D97—24no—
14C22×C4×D97C2212yes—
15C22×C4×D97C2212yes—
16#15690781—28no—
17#15690778—28no—
18C2×C4⋊Dic97C2212yescomplemented
19C2×C22⋊Dic97C2212yes—
20C2×C22⋊Dic97C2214yescomplemented
21C2×#13095488C2214yes—
22#15690717C2212yescomplemented
23#15690717C2214yescomplemented
24C22×D4×C97C2212yes—
25C22×D4×C97C2211yescharacteristic, complemented
26C23×C4×C97C2211yescharacteristic
27#121×C97—24no—
28C2×C4⋊C4×C97C2211yescharacteristic, complemented
29C2×C22⋊C4×C97C2212yescomplemented
30C23×D97—22no—
31C23×D97—22no—
32C23×D97—24no—
33C23×D97—24no—
34C23×D97D414yescomplemented
35C23×D97C2312yes—
36C2×C22⋊D97—28no—
37C2×C22⋊D97—28no—
38C2×C22⋊D97—28no—
39C22×Dic97—22no—
40C22×Dic97—24no—
41C22×Dic97C2314yes—
42C22×Dic97C2312yes—
43C2×C4×D97—232no—
44C2×C4×D97—24no—
45C2×C4×D97—24no—
46C2×C4×D97—24no—
47C2×C4×D97—24no—
48C2×C4×D97—24no—
49C2×C4×D97D418yescomplemented
50C4⋊Dic97—24no—
51C22⋊Dic97—24no—
52C22⋊Dic97—28no—
53#13095488—28no—
54C24×C97C2312yes—
55C24×C97C2311yescharacteristic
56C2×D4×C97—24no—
57C2×D4×C97—24no—
58C2×D4×C97—24no—
59C2×D4×C97—24no—
60C2×D4×C97—22no—
61C22×C4×C97—24no—
62C22×C4×C97D414yescomplemented
63C22×C4×C97C2312yes—
64C22×C4×C97C2311yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C4⋊C4×C97—22no—
67C22⋊C4×C97—24no—
68C23×C97—22no—
69C23×C97—22no—
70C23×C97—22no—
71C23×C97—24no—
72C23×C97—22no—
73C23×C97—22no—
74C23×C97—24no—
75C23×C97C2×D412yes—
76C23×C97C2×D412yes—
77C23×C97#4112yes—
78C23×C97C2411yescharacteristic
79C22×D97—28no—
80C22×D97—216no—
81C22×D97—28no—
82C22×D97—28no—
83C22×D97—22no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97—216no—
88C22×D97C2×D414yes—
89C22×D97C2×D414yes—
90C22×D97#4114yes—
91D4×C97—48no—
92D4×C97—44no—
93C2×C4×C97—22no—
94C2×C4×C97—28no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97C2×D414yes—
99C2×C4×C97C2×D412yes—
100C2×C4×C97C2×D412yes—
101C22⋊D97—416no—
102C2×Dic97—28no—
103C2×Dic97—22no—
104C2×Dic97—22no—
105C2×Dic97—24no—
106C2×Dic97—24no—
107C2×Dic97—24no—
108C2×Dic97C2×D414yes—
109C4×D97—216no—
110C4×D97—216no—
111C22×C97—22no—
112C22×C97—24no—
113C22×C97—22no—
114C22×C97—24no—
115C22×C97—42no—
116C22×C97—44no—
117C22×C97—42no—
118C22×C97—42no—
119C22×C97—44no—
120C22×C97#12114yescomplemented
121C22×C97C22×D412yes—
122C22×C97C22×D411yescharacteristic
123C22×C97C22×D411yescharacteristic
124C22×C97C2×#4112yes—
125C22×C97C2×#4111yescharacteristic
126C2×D97—232no—
127C2×D97—24no—
128C2×D97—24no—
129C2×D97—24no—
130C2×D97—24no—
131C2×D97—24no—
132C2×D97—24no—
133C2×D97#12118yescomplemented
134C4×C97—22no—
135C4×C97—42no—
136C4×C97#12014yescomplemented
137Dic97—24no—
138Dic97—44no—
139C2×C97—24no—
140C2×C97—82no—
141C2×C97C2×#12011yescharacteristic
142C2×C97C2×#12112yes—
143C2×C97C2×#12111yescharacteristic
144C2×C97#52511yescharacteristic
145C2×C97#52512yes—
146D97—28no—
147#2847—971nomaximal, Sylow
148C97#284711yescharacteristic, complemented, Sylow
149C24×C4—971no—
150C2×#121—972no—
151C2×#121D9711yescharacteristic, complemented
152C2×#120—972no—
153#386—972no—
154#386—974no—
155#380—971no—
156#380—972no—
157C25—971no—
158C22×D4—974no—
159C22×D4C2×D9712yes—
160C22×D4C2×D9711yescharacteristic, complemented
161C23×C4—972no—
162C23×C4—971no—
163C23×C4—972no—
164C23×C4—1948no—
165C23×C4—1944no—
166C23×C4C2×D9711yescharacteristic
167#121—24no—
168#121—1948no—
169#120—1948no—
170C2×C4⋊C4—972no—
171C2×C4⋊C4C2×D9711yescharacteristic, complemented
172C2×C22⋊C4—972no—
173C2×C22⋊C4—974no—
174C2×C22⋊C4—974no—
175C2×C22⋊C4C2×D9712yescomplemented
176#95—974no—
177#95—972no—
178C24—972no—
179C24—974no—
180C24—1944no—
181C24—1944no—
182C24—1942no—
183C24—1942no—
184C24C22×D9712yes—
185C24C22×D9711yescharacteristic
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—1948no—
192C2×D4—1948no—
193C2×D4—1948no—
194C22×C4—24no—
195C22×C4—972no—
196C22×C4—978no—
197C22×C4—974no—
198C22×C4—1944no—
199C22×C4—1944no—
200C22×C4—19432no—
201C22×C4—1944no—
202C22×C4—1942no—
203C22×C4—1944no—
204C22×C4—1944no—
205C22×C4—1944no—
206C22×C4C22⋊D9714yescomplemented
207C22×C4C22×D9712yes—
208C22×C4C22×D9711yescharacteristic
209C22×C4C22×D9711yescharacteristic
210C4⋊C4—22no—
211C4⋊C4—1944no—
212C22⋊C4—24no—
213C22⋊C4—1948no—
214C22⋊C4—1948no—
215C22⋊C4—1944no—
216C23—22no—
217C23—22no—
218C23—24no—
219C23—22no—
220C23—22no—
221C23—24no—
222C23—22no—
223C23—974no—
224C23—974no—
225C23—974no—
226C23—19416no—
227C23—1948no—
228C23—1948no—
229C23—19416no—
230C23—1948no—
231C23—1942no—
232C23—1942no—
233C23—1942no—
234C23—1942no—
235C23D4×D9712yes—
236C23C2×C22⋊D9712yes—
237C23#1309551012yes—
238C23C23×D9711yescharacteristic
239D4—48no—
240D4—44no—
241D4—38816no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—974no—
248C2×C4—19416no—
249C2×C4—1944no—
250C2×C4—1944no—
251C2×C4—1944no—
252C2×C4—1948no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4—19416no—
256C2×C4D4×D9714yes—
257C2×C4C2×C22⋊D9712yes—
258C2×C4C2×C22⋊D9712yes—
259C22—22no—
260C22—24no—
261C22—22no—
262C22—24no—
263C22—42no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—44no—
268C22—978no—
269C22—19432no—
270C22—1944no—
271C22—1944no—
272C22—1944no—
273C22—1944no—
274C22—1944no—
275C22—1944no—
276C22#1569078114yescomplemented
277C22C2×D4×D9712yes—
278C22C2×D4×D9711yescharacteristic
279C22C22×C22⋊D9711yescharacteristic
280C22#41×D9712yes—
281C22C2×#1309551011yescharacteristic
282C4—22no—
283C4—42no—
284C4—3884no—
285C4—1944no—
286C4#1569077814yescomplemented
287C2—24no—
288C2—82no—
289C2—1948no—
290C2C2×#1569077811yescharacteristic
291C2#121×D9712yes—
292C2C2×#1569078111yescharacteristic
293C2#2026124211yescharacteristic
294C2#2026124512yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC2×#121×C97C2
hypercenterC2×#121D97
hypocenterC97#2847
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×C4×D97C2semidirect
2. C2×#15690781C2semidirect
3. C2×#15690778C2semidirect
4. #20260615C2semidirect
5. #20260611C2semidirect
6. #20260595C2semidirect
7. #20260592C2semidirect
8. C2×#121×C97C2semidirect
18. C2×C4⋊Dic97C22semidirect
20. C2×C22⋊Dic97C22semidirect
22. #15690717C22semidirect
23. #15690717C22semidirect
25. C22×D4×C97C22semidirect
28. C2×C4⋊C4×C97C22semidirect
29. C2×C22⋊C4×C97C22semidirect
34. C23×D97D4semidirect
49. C2×C4×D97D4semidirect
62. C22×C4×C97D4semidirect
120. C22×C97#121semidirect
133. C2×D97#121semidirect
136. C4×C97#120semidirect
148. C97#2847semidirect
151. C2×#121D97semidirect
160. C22×D4C2×D97semidirect
171. C2×C4⋊C4C2×D97semidirect
175. C2×C22⋊C4C2×D97semidirect
206. C22×C4C22⋊D97semidirect
276. C22#15690781semidirect
286. C4#15690778semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D972
C2×#156907812
C2×#156907782
#202606152
#202606112
#202605952
#202605922
C2×#121×C972
#284797