Finite Group Databasebeta

← #27145057

Subgroups — #27145057

Order 12416.

1056 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#13095511C211yesmaximal, characteristic, complemented
2C2×#15690780C212yesmaximal, complemented
3C22×#13095488C211yesmaximal, characteristic, complemented
4#20260648C212yesmaximal, complemented
5#20260645C212yesmaximal, complemented
6#20260613C214yesmaximal, complemented
7#20260612C212yesmaximal, complemented
8#381×C97C211yesmaximal, characteristic, complemented
9C24×D97C2211yescharacteristic
10C2×#13095511—28no—
11C2×#13095511—24no—
12C22×Dic194C2211yescharacteristic, complemented
13C22×D388C2214yes—
14C22×D388C2211yescharacteristic, complemented
15C22×C22⋊D97C2212yes—
16C23×Dic97C2211yescharacteristic
17C22×C4×D97C2212yescomplemented
18#15690780—28no—
19C2×#13095488—24no—
20C2×#13095488C2212yes—
21C2×#13095488C2214yes—
22C2×#13095488C2212yes—
23C2×#13095488C2214yescomplemented
24#15690717C2214yescomplemented
25C23×C4×C97C2211yescharacteristic
26C2×C22⋊C4×C97C2212yes—
27#95×C97C2212yescomplemented
28#95×C97C2212yescomplemented
29C23×D97—22no—
30C23×D97—24no—
31C23×D97—24no—
32C23×D97—42no—
33C23×D97C2312yes—
34C23×D97C2312yes—
35#13095511—232no—
36C2×Dic194—22no—
37C2×Dic194—24no—
38C2×D388—24no—
39C2×D388—22no—
40C2×D388—28no—
41C2×D388—28no—
42C2×D388—44no—
43C2×C22⋊D97—24no—
44C2×C22⋊D97—24no—
45C2×C22⋊D97D418yescomplemented
46C22×Dic97—22no—
47C22×Dic97—42no—
48C22×Dic97C2312yes—
49C22×Dic97C2312yes—
50C2×C4×D97—24no—
51C2×C4×D97—24no—
52C2×C4×D97—24no—
53#13095488—24no—
54#13095488—24no—
55#13095488—28no—
56#13095488—44no—
57C24×C97C2311yescharacteristic
58C22×C4×C97—24no—
59C22×C4×C97D414yescomplemented
60C22×C4×C97C2314yes—
61C22×C4×C97C2311yescharacteristic
62C22×C4×C97C2311yescharacteristic
63C22⋊C4×C97—24no—
64C23×C97—22no—
65C23×C97—22no—
66C23×C97C2×D414yes—
67C23×C97C2×D412yes—
68C23×C97C2411yescharacteristic
69C22×D97—28no—
70C22×D97—22no—
71C22×D97—24no—
72C22×D97—22no—
73C22×D97—22no—
74C22×D97—24no—
75C22×D97—22no—
76C22×D97—24no—
77C22×D97—24no—
78C22×D97—416no—
79C22×D97—44no—
80C22×D97—44no—
81C22×D97C2×D414yes—
82C22×D97#4114yescomplemented
83C2×C4×C97—28no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—24no—
87C2×C4×C97—24no—
88C2×C4×C97—24no—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D412yes—
91C22⋊D97—216no—
92C2×Dic97—22no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—24no—
96C2×Dic97—44no—
97C2×Dic97C2×D414yes—
98D388—28no—
99D388—416no—
100C4×D97—216no—
101Dic194—28no—
102C22×C97—22no—
103C22×C97—22no—
104C22×C97—28no—
105C22×C97#12014yes—
106C22×C97C22×D412yes—
107C22×C97C22×D411yescharacteristic
108C22×C97C2×#4112yes—
109C22×C97H(5,𝔽2)12yes—
110C2×D97—24no—
111C2×D97—24no—
112C2×D97—28no—
113C2×D97—28no—
114C2×D97—416no—
115C2×D97—42no—
116C2×D97—42no—
117C2×D97—42no—
118C2×D97—42no—
119C2×D97—44no—
120C2×D97—44no—
121C4×C97—22no—
122C4×C97—22no—
123C4×C97—44no—
124Dic97—24no—
125Dic97—42no—
126C2×C97—24no—
127C2×C97C2×#12011yescharacteristic
128C2×C97#52511yescharacteristic
129C2×C97#53511yescharacteristic
130C2×C97#54614yes—
131D97—44no—
132D97—44no—
133#2404—971nomaximal, Sylow
134C97#240411yescharacteristic, complemented, Sylow
135C22×#41—971no—
136C2×#120—972no—
137C22×C22⋊C4—971no—
138#393—972no—
139#392—972no—
140#386—972no—
141#386—974no—
142#381D9711yescharacteristic, complemented
143C25—971no—
144C2×#41—1948no—
145C2×#41—1944no—
146C22×Q8—971no—
147C22×D4—972no—
148C22×D4—974no—
149C22×D4—971no—
150C23×C4—971no—
151C23×C4—972no—
152C23×C4C2×D9711yescharacteristic
153#120—1948no—
154C2×C22⋊C4—974no—
155C2×C22⋊C4—972no—
156C2×C22⋊C4—972no—
157C2×C22⋊C4—974no—
158C2×C22⋊C4—1944no—
159C2×C22⋊C4C2×D9712yes—
160#95—974no—
161#95C2×D9712yescomplemented
162#95C2×D9712yescomplemented
163C24—972no—
164C24—972no—
165C24—3882no—
166C24—1944no—
167C24—1944no—
168C24—1942no—
169C24C22×D9711yescharacteristic
170#41—19432no—
171C2×Q8—1944no—
172C2×Q8—1942no—
173C2×D4—978no—
174C2×D4—3884no—
175C2×D4—1948no—
176C2×D4—1948no—
177C2×D4—1944no—
178C2×D4—1944no—
179C2×D4—1944no—
180C2×D4—1942no—
181C22×C4—24no—
182C22×C4—972no—
183C22×C4—972no—
184C22×C4—3882no—
185C22×C4—1944no—
186C22×C4—1944no—
187C22×C4—1942no—
188C22×C4—1944no—
189C22×C4D38814yescomplemented
190C22×C4C22×D9714yes—
191C22×C4C22×D9711yescharacteristic
192C22×C4C22×D9711yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—3884no—
195C22⋊C4—1948no—
196C22⋊C4—1944no—
197C22⋊C4—1944no—
198C23—22no—
199C23—22no—
200C23—974no—
201C23—974no—
202C23—3884no—
203C23—38816no—
204C23—3884no—
205C23—1948no—
206C23—1944no—
207C23—1942no—
208C23—1942no—
209C23—1942no—
210C23—1944no—
211C23—1942no—
212C23—1944no—
213C23—1944no—
214C23D4×D9714yes—
215C23C2×D38812yes—
216C23C23×D9711yescharacteristic
217Q8—1948no—
218D4—38816no—
219D4—1948no—
220D4—19416no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—974no—
228C2×C4—3884no—
229C2×C4—19416no—
230C2×C4—1942no—
231C2×C4—1944no—
232C2×C4—1942no—
233C2×C4—1942no—
234C2×C4C2×D38812yes—
235C2×C4C2×D38812yes—
236C22—28no—
237C22—22no—
238C22—22no—
239C22—38816no—
240C22—3884no—
241C22—3884no—
242C22—3882no—
243C22—3882no—
244C22—3882no—
245C22—3882no—
246C22—1948no—
247C22—1948no—
248C22—1944no—
249C22—1944no—
250C22#1569078014yes—
251C22C2×D4×D9712yes—
252C22C22×D38811yescharacteristic
253C22#41×D9712yes—
254C22#1569085612yes—
255C4—22no—
256C4—22no—
257C4—44no—
258C4—3882no—
259C4—1944no—
260C2—24no—
261C2—3884no—
262C2—3884no—
263C2C2×#1569078011yescharacteristic
264C2#2026124711yescharacteristic
265C2#2026130411yescharacteristic
266C2#2026137714yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
Fitting#381×C97C2
hypercenter#381D97
hypocenterC97#2404
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C2×#15690780C2semidirect
3. C22×#13095488C2semidirect
4. #20260648C2semidirect
5. #20260645C2semidirect
6. #20260613C2semidirect
7. #20260612C2semidirect
8. #381×C97C2semidirect
12. C22×Dic194C22semidirect
14. C22×D388C22semidirect
17. C22×C4×D97C22semidirect
23. C2×#13095488C22semidirect
24. #15690717C22semidirect
27. #95×C97C22semidirect
28. #95×C97C22semidirect
45. C2×C22⋊D97D4semidirect
59. C22×C4×C97D4semidirect
82. C22×D97#41semidirect
134. C97#2404semidirect
142. #381D97semidirect
161. #95C2×D97semidirect
162. #95C2×D97semidirect
189. C22×C4D388semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156907802
C22×#130954882
#202606482
#202606452
#202606132
#202606122
#381×C972
#240497