Finite Group Databasebeta

← #27145086

Subgroups — #27145086

Order 12416.

1232 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 205 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×#15690781C212yesmaximal, complemented
2C2×#15690779C212yesmaximal, complemented
3C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
4#20260649C212yesmaximal, complemented
5#20260643C212yesmaximal, complemented
6#20260638C212yesmaximal, complemented
7#20260595C211yesmaximal, characteristic, complemented
8#20260591C212yesmaximal, complemented
9C23×D4×C97C211yesmaximal, characteristic, complemented
10C22×C22⋊D97C2211yescharacteristic
11C22×C22⋊D97C2214yescomplemented
12C22×C4×D97C2211yescharacteristic, complemented
13#15690781—28no—
14#15690779—28no—
15C4×C22⋊D97—24no—
16C2×C4⋊Dic97C2212yescomplemented
17C2×#13095490C2211yescharacteristic, complemented
18C2×C22⋊Dic97C2211yescharacteristic
19C2×C22⋊Dic97C2212yes—
20C2×C22⋊Dic97C2214yescomplemented
21C2×C22⋊Dic97C2214yescomplemented
22C2×C22⋊Dic97C2214yescomplemented
23C2×#13095488C2211yescharacteristic, complemented
24C2×C4×Dic97C2211yescharacteristic, complemented
25#15690717C2212yescomplemented
26C25×C97C2212yes—
27C22×D4×C97—28no—
28C22×D4×C97—24no—
29C22×D4×C97C2212yes—
30C22×D4×C97C2212yes—
31C23×C4×C97C2211yescharacteristic
32C23×D97C2311yescharacteristic
33C2×C22⋊D97—22no—
34C2×C22⋊D97—22no—
35C2×C22⋊D97—28no—
36C2×C22⋊D97—28no—
37C2×C22⋊D97—44no—
38C2×C22⋊D97D414yescomplemented
39C22×Dic97C2311yescharacteristic
40C22×Dic97C2311yescharacteristic
41C22×Dic97C2314yes—
42C22×Dic97C2311yescharacteristic
43C2×C4×D97—22no—
44C2×C4×D97—24no—
45C4⋊Dic97—24no—
46#13095490—22no—
47C22⋊Dic97—22no—
48C22⋊Dic97—24no—
49C22⋊Dic97—28no—
50C22⋊Dic97—28no—
51C22⋊Dic97—44no—
52#13095488—22no—
53C4×Dic97—22no—
54C24×C97—28no—
55C24×C97—24no—
56C24×C97—24no—
57C24×C97—24no—
58C24×C97—44no—
59C24×C97C2314yes—
60C24×C97C2311yescharacteristic
61C2×D4×C97—232no—
62C2×D4×C97—28no—
63C2×D4×C97—24no—
64C2×D4×C97—24no—
65C2×D4×C97—24no—
66C2×D4×C97D418yescomplemented
67C22×C4×C97—24no—
68C22×C4×C97D414yescomplemented
69C22×C4×C97C2311yescharacteristic
70C22×C4×C97C2311yescharacteristic
71C23×C97—22no—
72C23×C97—22no—
73C23×C97—216no—
74C23×C97—24no—
75C23×C97—24no—
76C23×C97—24no—
77C23×C97—24no—
78C23×C97—24no—
79C23×C97—24no—
80C23×C97—416no—
81C23×C97—416no—
82C23×C97—48no—
83C23×C97—48no—
84C23×C97C2×D412yes—
85C23×C97C2×D418yes—
86C23×C97C2×D412yes—
87C23×C97#4112yes—
88C23×C97C2411yescharacteristic
89C22×D97—21no—
90C22×D97—22no—
91C22×D97—21no—
92C22×D97—22no—
93C22×D97C2×D412yescomplemented
94D4×C97—216no—
95D4×C97—216no—
96C2×C4×C97—22no—
97C2×C4×C97—28no—
98C2×C4×C97C2×D412yes—
99C2×C4×C97C2×D414yes—
100C2×C4×C97C2×D412yes—
101C22⋊D97—28no—
102C22⋊D97—416no—
103C2×Dic97—21no—
104C2×Dic97—21no—
105C2×Dic97—21no—
106C2×Dic97—22no—
107C2×Dic97—24no—
108C2×Dic97—21no—
109C2×Dic97—24no—
110C2×Dic97—22no—
111C2×Dic97—24no—
112C2×Dic97C2×D412yescomplemented
113C4×D97—44no—
114C22×C97—22no—
115C22×C97—22no—
116C22×C97—24no—
117C22×C97—24no—
118C22×C97—28no—
119C22×C97—28no—
120C22×C97—28no—
121C22×C97—432no—
122C22×C97—44no—
123C22×C97—44no—
124C22×C97—44no—
125C22×C97—44no—
126C22×C97#12114yescomplemented
127C22×C97C22×D411yescharacteristic
128C22×C97C22×D411yescharacteristic
129C22×C97C22×D412yes—
130C22×C97C2×#4111yescharacteristic
131C22×C97H(5,𝔽2)12yes—
132C2×D97—22no—
133C2×D97—24no—
134C2×D97—41no—
135C2×D97—41no—
136C2×D97—42no—
137C4×C97—22no—
138C4×C97#12014yescomplemented
139Dic97—22no—
140Dic97—41no—
141Dic97—41no—
142Dic97—44no—
143C2×C97—24no—
144C2×C97—48no—
145C2×C97C2×#12011yescharacteristic
146C2×C97C2×#12111yescharacteristic
147C2×C97#53511yescharacteristic
148C2×C97D4212yes—
149C2×C97#54612yes—
150D97—42no—
151#2408—971nomaximal, Sylow
152C97#240811yescharacteristic, complemented, Sylow
153C23×D4D9711yescharacteristic, complemented
154C2×#121—972no—
155C2×#120—972no—
156C2×C4×D4—971no—
157#394—972no—
158#392—972no—
159#390—972no—
160#380—971no—
161#379—972no—
162C25C2×D9712yes—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—971no—
166C22×D4—974no—
167C22×D4C2×D9712yes—
168C22×D4C2×D9712yes—
169C23×C4—971no—
170C23×C4C2×D9711yescharacteristic
171#121—1948no—
172#120—1948no—
173C4×D4—1944no—
174C2×C4⋊C4—971no—
175C2×C4⋊C4—972no—
176C2×C22⋊C4—974no—
177C2×C22⋊C4—971no—
178C2×C22⋊C4—972no—
179C2×C22⋊C4—974no—
180C2×C22⋊C4—974no—
181C2×C22⋊C4—971no—
182C2×C42—971no—
183#95—972no—
184C24—28no—
185C24—24no—
186C24—24no—
187C24—24no—
188C24—44no—
189C24—971no—
190C24C22×D9714yes—
191C24C22×D9711yescharacteristic
192C2×D4—232no—
193C2×D4—28no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—974no—
198C2×D4—3884no—
199C2×D4—1948no—
200C2×D4—1948no—
201C2×D4—1942no—
202C2×D4—1942no—
203C2×D4C22⋊D9718yescomplemented
204C22×C4—24no—
205C22×C4—971no—
206C22×C4—974no—
207C22×C4—971no—
208C22×C4—971no—
209C22×C4—1942no—
210C22×C4—1944no—
211C22×C4C22⋊D9714yescomplemented
212C22×C4C22×D9711yescharacteristic
213C22×C4C22×D9711yescharacteristic
214C4⋊C4—1942no—
215C4⋊C4—1944no—
216C22⋊C4—3884no—
217C22⋊C4—1948no—
218C22⋊C4—1944no—
219C22⋊C4—1942no—
220C22⋊C4—1948no—
221C22⋊C4—1942no—
222C42—1942no—
223C23—22no—
224C23—22no—
225C23—216no—
226C23—24no—
227C23—24no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—24no—
232C23—416no—
233C23—416no—
234C23—48no—
235C23—48no—
236C23—972no—
237C23—1941no—
238C23—1942no—
239C23—1941no—
240C23—1942no—
241C23D4×D9712yes—
242C23C2×C22⋊D9718yes—
243C23C2×C22⋊D9712yes—
244C23#1309551012yes—
245C23C23×D9711yescharacteristic
246D4—216no—
247D4—216no—
248D4—38816no—
249D4—1948no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—972no—
253C2×C4—3884no—
254C2×C4—1944no—
255C2×C4—1941no—
256C2×C4—1942no—
257C2×C4—1941no—
258C2×C4—1944no—
259C2×C4—1942no—
260C2×C4—1944no—
261C2×C4—1941no—
262C2×C4—1941no—
263C2×C4C2×C22⋊D9712yes—
264C2×C4C2×C22⋊D9712yes—
265C2×C4C2×C22⋊D9714yes—
266C22—24no—
267C22—22no—
268C22—22no—
269C22—24no—
270C22—28no—
271C22—28no—
272C22—28no—
273C22—432no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—44no—
278C22—3882no—
279C22—3881no—
280C22—3881no—
281C22—1942no—
282C22—1944no—
283C22#1569078114yescomplemented
284C22C2×D4×D9711yescharacteristic
285C22C22×C22⋊D9712yes—
286C22C22×C22⋊D9711yescharacteristic
287C22C2×#1309551011yescharacteristic
288C22#1569085512yes—
289C4—22no—
290C4—3881no—
291C4—3881no—
292C4—3884no—
293C4—1942no—
294C4#1569077914yescomplemented
295C2—24no—
296C2—48no—
297C2—3882no—
298C2C2×#1569077911yescharacteristic
299C2C2×#1569078111yescharacteristic
300C2#2026130211yescharacteristic
301C2D4×C22⋊D9712yes—
302C2#2026137912yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC23×D4×C97C2
hypercenterC23×D4D97
hypocenterC97#2408
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690781C2semidirect
2. C2×#15690779C2semidirect
3. C2×C4×C22⋊D97C2semidirect
4. #20260649C2semidirect
5. #20260643C2semidirect
6. #20260638C2semidirect
7. #20260595C2semidirect
8. #20260591C2semidirect
9. C23×D4×C97C2semidirect
11. C22×C22⋊D97C22semidirect
12. C22×C4×D97C22semidirect
16. C2×C4⋊Dic97C22semidirect
17. C2×#13095490C22semidirect
20. C2×C22⋊Dic97C22semidirect
21. C2×C22⋊Dic97C22semidirect
22. C2×C22⋊Dic97C22semidirect
23. C2×#13095488C22semidirect
24. C2×C4×Dic97C22semidirect
25. #15690717C22semidirect
38. C2×C22⋊D97D4semidirect
66. C2×D4×C97D4semidirect
68. C22×C4×C97D4semidirect
93. C22×D97C2×D4semidirect
112. C2×Dic97C2×D4semidirect
126. C22×C97#121semidirect
138. C4×C97#120semidirect
152. C97#2408semidirect
153. C23×D4D97semidirect
203. C2×D4C22⋊D97semidirect
211. C22×C4C22⋊D97semidirect
283. C22#15690781semidirect
294. C4#15690779semidirect

Maximal subgroups

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

subgroupindex
C2×#156907812
C2×#156907792
C2×C4×C22⋊D972
#202606492
#202606432
#202606382
#202605952
#202605912
C23×D4×C972
#240897