Finite Group Databasebeta

← #27145063

Subgroups — #27145063

Order 12416.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 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×C22⋊D97C211yesmaximal, characteristic
2C2×#15690793C212yesmaximal, complemented
3#20260650C214yesmaximal, complemented
4#20260614C212yesmaximal, complemented
5#20260607C212yesmaximal, complemented
6#20260591C211yesmaximal, characteristic, complemented
7#20260590C212yesmaximal, complemented
8C22×C22⋊C4×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×C22⋊D97—28no—
11C22×C22⋊D97—24no—
12C22×C22⋊D97C2212yes—
13C22×C22⋊D97C2212yes—
14C23×Dic97C2211yescharacteristic
15#15690793—28no—
16C2×#13095490C2214yescomplemented
17C2×C22⋊Dic97C2212yes—
18C2×C22⋊Dic97C2212yescomplemented
19C2×C22⋊Dic97C2212yescomplemented
20C2×#13095488C2214yescomplemented
21C2×#13095488C2214yescomplemented
22#15690717C2214yescomplemented
23C25×C97C2211yescharacteristic
24C23×C4×C97C2211yescharacteristic, complemented
25C23×C4×C97C2211yescharacteristic, complemented
26C2×C22⋊C4×C97—24no—
27C2×C22⋊C4×C97C2212yes—
28C2×C22⋊C4×C97C2212yescomplemented
29C23×D97—22no—
30C23×D97—24no—
31C23×D97—24no—
32C23×D97—42no—
33C23×D97C2312yes—
34C2×C22⋊D97—232no—
35C2×C22⋊D97—24no—
36C2×C22⋊D97—24no—
37C2×C22⋊D97—24no—
38C2×C22⋊D97—24no—
39C2×C22⋊D97—24no—
40C2×C22⋊D97D418yes—
41C22×Dic97—22no—
42C22×Dic97—24no—
43C22×Dic97C2314yes—
44C22×Dic97C2312yes—
45#13095490—28no—
46C22⋊Dic97—24no—
47C22⋊Dic97—24no—
48C22⋊Dic97—42no—
49#13095488—28no—
50#13095488—44no—
51C24×C97—24no—
52C24×C97—24no—
53C24×C97—42no—
54C24×C97D414yes—
55C24×C97C2311yescharacteristic
56C24×C97C2311yescharacteristic
57C24×C97C2311yescharacteristic
58C22×C4×C97—24no—
59C22×C4×C97—22no—
60C22×C4×C97—22no—
61C22×C4×C97—42no—
62C22×C4×C97C2312yes—
63C22×C4×C97C2312yes—
64C22⋊C4×C97—24no—
65C22⋊C4×C97—24no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—22no—
69C23×C97—22no—
70C23×C97—28no—
71C23×C97—28no—
72C23×C97—22no—
73C23×C97—22no—
74C23×C97—22no—
75C23×C97—44no—
76C23×C97—416no—
77C23×C97C2×D414yes—
78C23×C97C2×D412yes—
79C23×C97C2×D412yes—
80C23×C97C2×D412yes—
81C23×C97#4114yes—
82C23×C97C2411yescharacteristic
83C22×D97—28no—
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—24no—
87C22×D97—24no—
88C22×D97—416no—
89C22×D97—44no—
90C22×D97—44no—
91C22×D97C2×D414yes—
92C2×C4×C97—22no—
93C2×C4×C97—22no—
94C2×C4×C97—24no—
95C2×C4×C97—24no—
96C2×C4×C97—44no—
97C2×C4×C97—44no—
98C22⋊D97—216no—
99C22⋊D97—216no—
100C2×Dic97—28no—
101C2×Dic97—22no—
102C2×Dic97—22no—
103C2×Dic97—24no—
104C2×Dic97—24no—
105C2×Dic97—24no—
106C2×Dic97C2×D414yes—
107C22×C97—22no—
108C22×C97—22no—
109C22×C97—22no—
110C22×C97—22no—
111C22×C97—28no—
112C22×C97—28no—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97—22no—
116C22×C97—416no—
117C22×C97—44no—
118C22×C97#12014yes—
119C22×C97#12314yescomplemented
120C22×C97C22×D412yes—
121C22×C97C22×D411yescharacteristic
122C22×C97C2×#4112yes—
123C22×C97H(5,𝔽2)12yes—
124C2×D97—24no—
125C2×D97—28no—
126C2×D97—416no—
127C2×D97—42no—
128C2×D97—42no—
129C2×D97—44no—
130C4×C97—42no—
131C4×C97—42no—
132Dic97—24no—
133Dic97—44no—
134C2×C97—24no—
135C2×C97—24no—
136C2×C97—42no—
137C2×C97C2×#12011yescharacteristic
138C2×C97C2×#12311yescharacteristic
139C2×C97#53411yescharacteristic
140C2×C97#54614yes—
141D97—44no—
142#2405—971nomaximal, Sylow
143C97#240511yescharacteristic, complemented, Sylow
144C23×D4—971no—
145C2×#123—972no—
146C22×C22⋊C4D9711yescharacteristic, complemented
147#394—974no—
148#386—972no—
149#385—972no—
150#379—971no—
151#379—972no—
152C25—971no—
153C25C2×D9711yescharacteristic
154C22×D4—972no—
155C22×D4—972no—
156C22×D4—1948no—
157C22×D4—1944no—
158C23×C4—971no—
159C23×C4C2×D9711yescharacteristic, complemented
160C23×C4C2×D9711yescharacteristic, complemented
161#123—1948no—
162C2×C4⋊C4—974no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—972no—
165C2×C22⋊C4—974no—
166C2×C22⋊C4—972no—
167C2×C22⋊C4—972no—
168C2×C22⋊C4—974no—
169C2×C22⋊C4C2×D9712yes—
170C2×C22⋊C4C2×D9712yescomplemented
171#95—974no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—972no—
176C24—3882no—
177C24—1944no—
178C24—1944no—
179C24—1942no—
180C24C22⋊D9714yes—
181C24C22×D9711yescharacteristic
182C24C22×D9711yescharacteristic
183C24C22×D9711yescharacteristic
184C2×D4—978no—
185C2×D4—1944no—
186C2×D4—1944no—
187C2×D4—19432no—
188C2×D4—1944no—
189C2×D4—1944no—
190C2×D4—1944no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—972no—
196C22×C4—974no—
197C22×C4—1944no—
198C22×C4—1942no—
199C22×C4C22×D9712yes—
200C22×C4C22×D9712yes—
201C4⋊C4—1948no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—3884no—
205C22⋊C4—3882no—
206C22⋊C4—1948no—
207C22⋊C4—1944no—
208C22⋊C4—1944no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—974no—
221C23—3884no—
222C23—38816no—
223C23—3884no—
224C23—1948no—
225C23—1942no—
226C23—1942no—
227C23—1944no—
228C23—1944no—
229C23D4×D9714yes—
230C23C2×C22⋊D9712yes—
231C23C2×C22⋊D9712yes—
232C23C2×C22⋊D9712yes—
233C23#1309551114yes—
234C23C23×D9711yescharacteristic
235D4—19416no—
236D4—19416no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—974no—
244C2×C4—1944no—
245C2×C4—1944no—
246C2×C4—1944no—
247C2×C4—1948no—
248C2×C4—1942no—
249C2×C4—1942no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—22no—
254C22—28no—
255C22—28no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—38816no—
262C22—3882no—
263C22—3882no—
264C22—3884no—
265C22—1944no—
266C22—1948no—
267C22#1569077814yes—
268C22#1569079314yescomplemented
269C22C2×D4×D9712yes—
270C22C22×C22⋊D9711yescharacteristic
271C22C2×#1309551112yes—
272C22#1569085512yes—
273C4—42no—
274C4—42no—
275C4—3884no—
276C4—1944no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—3884no—
281C2C2×#1569077811yescharacteristic
282C2C2×#1569079311yescharacteristic
283C2#2026129411yescharacteristic
284C2#2026137514yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC22×C22⋊C4×C97C2
prime radicalC23×C22⋊D97C2
hypercenterC22×C22⋊C4D97
hypocenterC97#2405
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15690793C2semidirect
3. #20260650C2semidirect
4. #20260614C2semidirect
5. #20260607C2semidirect
6. #20260591C2semidirect
7. #20260590C2semidirect
8. C22×C22⋊C4×C97C2semidirect
16. C2×#13095490C22semidirect
18. C2×C22⋊Dic97C22semidirect
19. C2×C22⋊Dic97C22semidirect
20. C2×#13095488C22semidirect
21. C2×#13095488C22semidirect
22. #15690717C22semidirect
24. C23×C4×C97C22semidirect
25. C23×C4×C97C22semidirect
28. C2×C22⋊C4×C97C22semidirect
119. C22×C97#123semidirect
143. C97#2405semidirect
146. C22×C22⋊C4D97semidirect
159. C23×C4C2×D97semidirect
160. C23×C4C2×D97semidirect
170. C2×C22⋊C4C2×D97semidirect
268. C22#15690793semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D972
C2×#156907932
#202606502
#202606142
#202606072
#202605912
#202605902
C22×C22⋊C4×C972
#240597