Finite Group Databasebeta

← #27145055

Subgroups — #27145055

Order 12416.

1056 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 173 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×#15690779C212yesmaximal, complemented
3#20260647C212yesmaximal, complemented
4#20260644C212yesmaximal, complemented
5#20260614C212yesmaximal, complemented
6#20260611C214yesmaximal, complemented
7#20260598C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C97C211yesmaximal, characteristic, complemented
9C2×#13095511—28no—
10C2×#13095511—24no—
11C22×Dic194C2211yescharacteristic, complemented
12C22×D388C2211yescharacteristic, complemented
13C22×C22⋊D97C2214yes—
14C22×C22⋊D97C2212yes—
15C22×C4×D97C2212yescomplemented
16#15690779—28no—
17C2×C22⋊Dic97C2214yes—
18C2×C22⋊Dic97C2212yes—
19C2×#13095488C2214yescomplemented
20#15690717C2212yescomplemented
21#15690717C2214yescomplemented
22#15690717C2212yescomplemented
23C25×C97C2211yescharacteristic
24C23×C4×C97C2211yescharacteristic
25C23×C4×C97C2211yescharacteristic
26C2×C22⋊C4×C97—24no—
27C2×C22⋊C4×C97C2212yes—
28C2×C22⋊C4×C97C2212yes—
29C23×D97C2312yes—
30#13095511—232no—
31C2×Dic194—22no—
32C2×Dic194—24no—
33C2×D388—24no—
34C2×D388—22no—
35C2×C22⋊D97—24no—
36C2×C22⋊D97—24no—
37C2×C22⋊D97—28no—
38C2×C22⋊D97—28no—
39C2×C22⋊D97—44no—
40C2×C22⋊D97D418yescomplemented
41C22×Dic97C2314yes—
42C22×Dic97C2312yes—
43C2×C4×D97—24no—
44C2×C4×D97—24no—
45C2×C4×D97—24no—
46C22⋊Dic97—24no—
47C22⋊Dic97—28no—
48#13095488—44no—
49C24×C97—22no—
50C24×C97—24no—
51C24×C97—24no—
52C24×C97—42no—
53C24×C97C2312yes—
54C24×C97C2311yescharacteristic
55C22×C4×C97—24no—
56C22×C4×C97—22no—
57C22×C4×C97—42no—
58C22×C4×C97D414yescomplemented
59C22×C4×C97C2312yes—
60C22×C4×C97C2311yescharacteristic
61C22×C4×C97C2311yescharacteristic
62C22⋊C4×C97—24no—
63C22⋊C4×C97—24no—
64C23×C97—22no—
65C23×C97—22no—
66C23×C97—28no—
67C23×C97—22no—
68C23×C97—24no—
69C23×C97—22no—
70C23×C97—24no—
71C23×C97—416no—
72C23×C97—44no—
73C23×C97—44no—
74C23×C97C2×D414yes—
75C23×C97C2×D412yes—
76C23×C97#4114yescomplemented
77C23×C97C2411yescharacteristic
78C22×D97—22no—
79C22×D97—22no—
80C22×D97—24no—
81C22×D97—24no—
82C22×D97C2×D414yes—
83C2×C4×C97—28no—
84C2×C4×C97—22no—
85C2×C4×C97—22no—
86C2×C4×C97—22no—
87C2×C4×C97—24no—
88C2×C4×C97—44no—
89C2×C4×C97C2×D412yes—
90C2×C4×C97C2×D412yes—
91C22⋊D97—216no—
92C22⋊D97—416no—
93C2×Dic97—22no—
94C2×Dic97—22no—
95C2×Dic97—24no—
96C2×Dic97—24no—
97C2×Dic97—24no—
98C2×Dic97C2×D414yes—
99D388—28no—
100C4×D97—216no—
101Dic194—28no—
102C22×C97—22no—
103C22×C97—22no—
104C22×C97—28no—
105C22×C97—24no—
106C22×C97—28no—
107C22×C97—416no—
108C22×C97—42no—
109C22×C97—42no—
110C22×C97—44no—
111C22×C97#12014yes—
112C22×C97C22×D412yes—
113C22×C97C22×D411yescharacteristic
114C22×C97C2×#4112yes—
115C22×C97H(5,𝔽2)12yes—
116C2×D97—24no—
117C2×D97—28no—
118C2×D97—42no—
119C2×D97—42no—
120C2×D97—44no—
121C4×C97—22no—
122C4×C97—22no—
123C4×C97—42no—
124Dic97—24no—
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×#41—971no—
136C2×#120—972no—
137C22×C22⋊C4D9711yescharacteristic, complemented
138#393—972no—
139#392—972no—
140#386—972no—
141#386—974no—
142#381—971no—
143C25C2×D9711yescharacteristic
144C2×#41—1948no—
145C2×#41—1944no—
146C22×Q8—971no—
147C22×D4—972no—
148C22×D4—974no—
149C22×D4—971no—
150C23×C4—972no—
151C23×C4C2×D9711yescharacteristic
152C23×C4C2×D9711yescharacteristic
153#120—1948no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—974no—
156C2×C22⋊C4—972no—
157C2×C22⋊C4—974no—
158C2×C22⋊C4C2×D9712yes—
159C2×C22⋊C4C2×D9712yes—
160#95—972no—
161#95—972no—
162#95—974no—
163C24—22no—
164C24—24no—
165C24—24no—
166C24—42no—
167C24—972no—
168C24C22×D9712yes—
169C24C22×D9711yescharacteristic
170#41—19432no—
171C2×Q8—1944no—
172C2×Q8—1942no—
173C2×D4—978no—
174C2×D4—3884no—
175C2×D4—1944no—
176C2×D4—1948no—
177C2×D4—1948no—
178C2×D4—1944no—
179C2×D4—1944no—
180C2×D4—1942no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—42no—
184C22×C4—972no—
185C22×C4—974no—
186C22×C4—1944no—
187C22×C4—1944no—
188C22×C4—1944no—
189C22×C4C22⋊D9714yescomplemented
190C22×C4C22×D9712yes—
191C22×C4C22×D9711yescharacteristic
192C22×C4C22×D9711yescharacteristic
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—3884no—
196C22⋊C4—1948no—
197C22⋊C4—1944no—
198C23—22no—
199C23—22no—
200C23—28no—
201C23—24no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—416no—
206C23—44no—
207C23—44no—
208C23—974no—
209C23—1942no—
210C23—1942no—
211C23—1944no—
212C23—1944no—
213C23D4×D9714yes—
214C23C2×C22⋊D9712yes—
215C23#1309551114yescomplemented
216C23C23×D9711yescharacteristic
217Q8—1948no—
218D4—38816no—
219D4—1948no—
220D4—19416no—
221C2×C4—28no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—974no—
228C2×C4—19416no—
229C2×C4—1944no—
230C2×C4—1944no—
231C2×C4—1944no—
232C2×C4—1942no—
233C2×C4—1942no—
234C2×C4C2×C22⋊D9712yes—
235C2×C4C2×C22⋊D9712yes—
236C22—22no—
237C22—22no—
238C22—28no—
239C22—24no—
240C22—28no—
241C22—416no—
242C22—42no—
243C22—42no—
244C22—44no—
245C22—3882no—
246C22—3882no—
247C22—3884no—
248C22—1944no—
249C22—1948no—
250C22#1569077814yes—
251C22C2×D4×D9712yes—
252C22C22×C22⋊D9711yescharacteristic
253C22C2×#1309551112yes—
254C22#1569085512yes—
255C4—22no—
256C4—22no—
257C4—42no—
258C4—3884no—
259C4—1944no—
260C2—24no—
261C2—44no—
262C2—3884no—
263C2C2×#1569077811yescharacteristic
264C2#2026124611yescharacteristic
265C2#2026130211yescharacteristic
266C2#2026137514yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC22×C22⋊C4×C97C2
hypercenterC22×C22⋊C4D97
hypocenterC97#2404
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C2×#15690779C2semidirect
3. #20260647C2semidirect
4. #20260644C2semidirect
5. #20260614C2semidirect
6. #20260611C2semidirect
7. #20260598C2semidirect
8. C22×C22⋊C4×C97C2semidirect
11. C22×Dic194C22semidirect
12. C22×D388C22semidirect
15. C22×C4×D97C22semidirect
19. C2×#13095488C22semidirect
20. #15690717C22semidirect
21. #15690717C22semidirect
22. #15690717C22semidirect
40. C2×C22⋊D97D4semidirect
58. C22×C4×C97D4semidirect
76. C23×C97#41semidirect
134. C97#2404semidirect
137. C22×C22⋊C4D97semidirect
189. C22×C4C22⋊D97semidirect
215. C23#13095511semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156907792
#202606472
#202606442
#202606142
#202606112
#202605982
C22×C22⋊C4×C972
#240497