Finite Group Databasebeta

← #27145051

Subgroups — #27145051

Order 12416.

1056 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 185 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×#15690778C212yesmaximal, complemented
2C22×#13095488C211yesmaximal, characteristic, complemented
3#20260646C212yesmaximal, complemented
4#20260643C212yesmaximal, complemented
5#20260615C212yesmaximal, complemented
6#20260614C214yesmaximal, complemented
7#20260598C211yesmaximal, characteristic, complemented
8C22×#41×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C22×C22⋊D97C2214yes—
11C23×Dic97C2211yescharacteristic
12#15690778—28no—
13C2×C22⋊Dic97C2212yes—
14C2×C22⋊Dic97C2214yescomplemented
15C2×#13095488—24no—
16C2×#13095488C2212yes—
17C2×#13095488C2214yes—
18C2×#13095488C2212yes—
19#15690717C2212yescomplemented
20#15690717C2214yescomplemented
21#15690717C2212yescomplemented
22C2×#41×C97—28no—
23C2×#41×C97—24no—
24C22×Q8×C97C2211yescharacteristic, complemented
25C22×D4×C97C2212yes—
26C22×D4×C97C2211yescharacteristic, complemented
27C23×C4×C97C2211yescharacteristic
28C23×C4×C97C2212yescomplemented
29C23×D97—22no—
30C23×D97—24no—
31C23×D97—24no—
32C23×D97—42no—
33C23×D97C2312yes—
34C2×C22⋊D97—28no—
35C2×C22⋊D97—28no—
36C2×C22⋊D97—44no—
37C22×Dic97—22no—
38C22×Dic97—42no—
39C22×Dic97C2312yes—
40C22×Dic97C2314yes—
41C22⋊Dic97—24no—
42C22⋊Dic97—44no—
43#13095488—24no—
44#13095488—24no—
45#13095488—28no—
46C24×C97C2312yes—
47C24×C97C2311yescharacteristic
48#41×C97—232no—
49C2×Q8×C97—24no—
50C2×Q8×C97—22no—
51C2×D4×C97—24no—
52C2×D4×C97—24no—
53C2×D4×C97—24no—
54C2×D4×C97—22no—
55C2×D4×C97D418yescomplemented
56C22×C4×C97—24no—
57C22×C4×C97—24no—
58C22×C4×C97—24no—
59C22×C4×C97—24no—
60C22×C4×C97D414yescomplemented
61C22×C4×C97C2311yescharacteristic
62C22×C4×C97C2311yescharacteristic
63C22×C4×C97C2312yes—
64C23×C97—22no—
65C23×C97—22no—
66C23×C97—22no—
67C23×C97—22no—
68C23×C97—24no—
69C23×C97—24no—
70C23×C97C2×D414yes—
71C23×C97C2×D414yes—
72C23×C97C2×D412yes—
73C23×C97C2411yescharacteristic
74C22×D97—28no—
75C22×D97—22no—
76C22×D97—24no—
77C22×D97—22no—
78C22×D97—24no—
79C22×D97—416no—
80C22×D97—44no—
81C22×D97—44no—
82C22×D97#4114yescomplemented
83Q8×C97—28no—
84D4×C97—28no—
85D4×C97—216no—
86C2×C4×C97—216no—
87C2×C4×C97—28no—
88C2×C4×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97—22no—
92C2×C4×C97C2×D412yes—
93C2×C4×C97C2×D414yes—
94C2×C4×C97C2×D412yes—
95C22⋊D97—416no—
96C2×Dic97—22no—
97C2×Dic97—24no—
98C2×Dic97—24no—
99C2×Dic97—24no—
100C2×Dic97—24no—
101C2×Dic97—44no—
102C22×C97—22no—
103C22×C97—22no—
104C22×C97—28no—
105C22×C97—24no—
106C22×C97—28no—
107C22×C97—42no—
108C22×C97—42no—
109C22×C97—44no—
110C22×C97#12014yes—
111C22×C97C22×D411yescharacteristic
112C22×C97C22×D412yes—
113C22×C97C2×#4112yes—
114C22×C97H(5,𝔽2)12yes—
115C2×D97—24no—
116C2×D97—28no—
117C2×D97—416no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—44no—
121C4×C97—22no—
122C4×C97—24no—
123C4×C97—22no—
124Dic97—42no—
125Dic97—44no—
126C2×C97—24no—
127C2×C97—44no—
128C2×C97C2×#12011yescharacteristic
129C2×C97#52511yescharacteristic
130C2×C97#53511yescharacteristic
131C2×C97#54614yes—
132D97—44no—
133#2404—971nomaximal, Sylow
134C97#240411yescharacteristic, complemented, Sylow
135C22×#41D9711yescharacteristic, complemented
136C2×#120—972no—
137C22×C22⋊C4—971no—
138#393—972no—
139#392—972no—
140#386—972no—
141#386—974no—
142#381—971no—
143C25—971no—
144C2×#41—28no—
145C2×#41—24no—
146C22×Q8C2×D9711yescharacteristic, complemented
147C22×D4—974no—
148C22×D4C2×D9712yes—
149C22×D4C2×D9711yescharacteristic, complemented
150C23×C4—971no—
151C23×C4C2×D9711yescharacteristic
152C23×C4C2×D9712yescomplemented
153#120—1948no—
154C2×C22⋊C4—974no—
155C2×C22⋊C4—972no—
156C2×C22⋊C4—972no—
157C2×C22⋊C4—972no—
158C2×C22⋊C4—974no—
159C2×C22⋊C4—1944no—
160#95—972no—
161#95—972no—
162#95—974no—
163C24—972no—
164C24—3882no—
165C24—1944no—
166C24—1944no—
167C24—1942no—
168C24C22×D9712yes—
169C24C22×D9711yescharacteristic
170#41—232no—
171C2×Q8—24no—
172C2×Q8—22no—
173C2×D4—24no—
174C2×D4—24no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—3884no—
178C2×D4—1948no—
179C2×D4—1948no—
180C2×D4C22⋊D9718yescomplemented
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—974no—
186C22×C4—972no—
187C22×C4—3882no—
188C22×C4—1942no—
189C22×C4C22⋊D9714yescomplemented
190C22×C4C22×D9711yescharacteristic
191C22×C4C22×D9711yescharacteristic
192C22×C4C22×D9712yes—
193C22⋊C4—3884no—
194C22⋊C4—1944no—
195C22⋊C4—1944no—
196C22⋊C4—1944no—
197C22⋊C4—1948no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—974no—
205C23—3884no—
206C23—38816no—
207C23—3884no—
208C23—1948no—
209C23—1942no—
210C23—1944no—
211C23—1942no—
212C23—1944no—
213C23C2×C22⋊D9714yes—
214C23C2×C22⋊D9714yes—
215C23C2×C22⋊D9712yes—
216C23C23×D9711yescharacteristic
217Q8—28no—
218D4—28no—
219D4—216no—
220D4—38816no—
221C2×C4—216no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3884no—
228C2×C4—1942no—
229C2×C4—1944no—
230C2×C4—1944no—
231C2×C4—1944no—
232C2×C4—1944no—
233C2×C4C2×C22⋊D9712yes—
234C2×C4C2×C22⋊D9714yes—
235C2×C4C2×C22⋊D9712yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—24no—
240C22—28no—
241C22—42no—
242C22—42no—
243C22—44no—
244C22—38816no—
245C22—3882no—
246C22—3882no—
247C22—3884no—
248C22—1944no—
249C22—1948no—
250C22#1569077914yes—
251C22C22×C22⋊D9711yescharacteristic
252C22C22×C22⋊D9712yes—
253C22#41×D9712yes—
254C22#1569085612yes—
255C4—22no—
256C4—24no—
257C4—22no—
258C4—3882no—
259C4—3884no—
260C2—24no—
261C2—44no—
262C2—3884no—
263C2C2×#1569077911yescharacteristic
264C2#2026124211yescharacteristic
265C2#2026129911yescharacteristic
266C2#2026137314yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC22×#41×C97C2
hypercenterC22×#41D97
hypocenterC97#2404
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#15690778C2semidirect
2. C22×#13095488C2semidirect
3. #20260646C2semidirect
4. #20260643C2semidirect
5. #20260615C2semidirect
6. #20260614C2semidirect
7. #20260598C2semidirect
8. C22×#41×C97C2semidirect
14. C2×C22⋊Dic97C22semidirect
19. #15690717C22semidirect
20. #15690717C22semidirect
21. #15690717C22semidirect
24. C22×Q8×C97C22semidirect
26. C22×D4×C97C22semidirect
28. C23×C4×C97C22semidirect
55. C2×D4×C97D4semidirect
60. C22×C4×C97D4semidirect
82. C22×D97#41semidirect
134. C97#2404semidirect
135. C22×#41D97semidirect
146. C22×Q8C2×D97semidirect
149. C22×D4C2×D97semidirect
152. C23×C4C2×D97semidirect
180. C2×D4C22⋊D97semidirect
189. C22×C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907782
C22×#130954882
#202606462
#202606432
#202606152
#202606142
#202605982
C22×#41×C972
#240497