Finite Group Databasebeta

← C2×#20260869 #27150912

Subgroups — C2×#20260869 #27150912

Order 12416.

1048 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 197 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×#13095496C212yesmaximal, complemented
2C2×#15690784C211yesmaximal, characteristic, complemented
3#20260869C218yesmaximal, direct product factor
4C2×#15690731C212yesmaximal, complemented
5C2×C22⋊#11715895C211yesmaximal, characteristic, complemented
6C23×D4×C97C211yesmaximal, characteristic, complemented
7C22×D388C2211yescharacteristic, complemented
8C22×C22⋊D97—21no—
9C2×#13095496—28no—
10C2×#13095496C2218yescomplemented
11C22×#11715895C2212yescomplemented
12#15690784—22no—
13#15690784C2214yescomplemented
14C2×C4⋊Dic97C2211yescharacteristic, complemented
15C2×#13095488—21no—
16#15690731C2218yescomplemented
17C22⋊#11715895C2214yescomplemented
18C25×C97—21no—
19C22×D4×C97—28no—
20C22×D4×C97—21no—
21C22×D4×C97—22no—
22C22×D4×C97C2212yes—
23C22×D4×C97C2214yescomplemented
24C23×C4×C97C2211yescharacteristic
25C23×D97—21no—
26C2×D388—24no—
27C2×D388—21no—
28C2×D388C2312yescomplemented
29C2×C22⋊D97—24no—
30C2×C22⋊D97—44no—
31C22×Dic97—21no—
32#13095496—216no—
33C2×#11715895—24no—
34C2×#11715895C2314yescomplemented
35C4⋊Dic97—21no—
36C4⋊Dic97C2312yescomplemented
37#13095488—24no—
38C24×C97—28no—
39C24×C97—22no—
40C24×C97—24no—
41C24×C97—44no—
42C24×C97—44no—
43C24×C97D411yescharacteristic
44C2×D4×C97—28no—
45C2×D4×C97—216no—
46C2×D4×C97—216no—
47C2×D4×C97—22no—
48C2×D4×C97—24no—
49C2×D4×C97—24no—
50C2×D4×C97D418yescomplemented
51C2×D4×C97C2314yes—
52C22×C4×C97—24no—
53C22×C4×C97D411yescharacteristic
54C22×C4×C97D412yes—
55C22×C4×C97C2312yes—
56C22×C4×C97C2311yescharacteristic
57C23×C97—22no—
58C23×C97—22no—
59C23×C97—216no—
60C23×C97—28no—
61C23×C97—24no—
62C23×C97—48no—
63C23×C97—432no—
64C23×C97—44no—
65C23×C97—48no—
66C23×C97—44no—
67C23×C97D814yescomplemented
68C23×C97C2×D412yes—
69C23×C97C2×D411yescharacteristic
70C22×D97—22no—
71C22×D97—42no—
72C22×D97—42no—
73C22×D97—42no—
74D4×C97—216no—
75D4×C97—28no—
76D4×C97—24no—
77D4×C97C2×D418yescomplemented
78C2×C4×C97—22no—
79C2×C4×C97—24no—
80C2×C4×C97—24no—
81C2×C4×C97C2×D412yes—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D414yes—
84C2×C4×C97C2411yescharacteristic
85C22⋊D97—48no—
86C2×Dic97—22no—
87C2×Dic97—42no—
88D388—24no—
89D388—42no—
90#11715895—24no—
91C22×C97—22no—
92C22×C97—22no—
93C22×C97—28no—
94C22×C97—28no—
95C22×C97—432no—
96C22×C97—44no—
97C22×C97—44no—
98C22×C97—416no—
99C22×C97#12012yes—
100C22×C97C2×D812yes—
101C22×C97C2×D814yescomplemented
102C22×C97AGL(1,ℤ/8)12yes—
103C22×C97C22×D411yescharacteristic
104C2×D97—42no—
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—84no—
108C4×C97—22no—
109C4×C97#12012yes—
110C4×C97C22×D412yes—
111Dic97—42no—
112C2×C97—24no—
113C2×C97—48no—
114C2×C97#44714yescomplemented
115C2×C97C2×#12011yescharacteristic
116C2×C97C22×D811yescharacteristic
117C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
118D97—82no—
119C2×#447—971nomaximal, Sylow
120C97C2×#44711yescharacteristic, complemented, Sylow
121C23×D4D9711yescharacteristic, complemented
122C22×D8—972no—
123C2×#121—971no—
124#447—978no—
125C2×#102—972no—
126C2×C22⋊C8—971no—
127C25—21no—
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—971no—
132C22×D4—1941no—
133C22×D4C2×D9712yes—
134C22×D4C2×D9714yescomplemented
135C23×C4C2×D9711yescharacteristic
136C2×D8—978no—
137C2×D8—1948no—
138C22×C8—972no—
139#121—974no—
140#121—1942no—
141C2×C4⋊C4—971no—
142C2×C22⋊C4—1941no—
143#102—978no—
144C22⋊C8—974no—
145C24—28no—
146C24—24no—
147C24—22no—
148C24—44no—
149C24—44no—
150C24—1941no—
151C24C22⋊D9711yescharacteristic
152C2×D4—28no—
153C2×D4—216no—
154C2×D4—216no—
155C2×D4—22no—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—972no—
159C2×D4—3884no—
160C2×D4—1944no—
161C2×D4—1941no—
162C2×D4—1944no—
163C2×D4C22⋊D9718yescomplemented
164C2×D4C22×D9714yes—
165C22×C4—24no—
166C22×C4—1941no—
167C22×C4C22⋊D9712yes—
168C22×C4C22⋊D9711yescharacteristic
169C22×C4C22×D9712yes—
170C22×C4C22×D9711yescharacteristic
171D8—19416no—
172C2×C8—974no—
173C2×C8—1944no—
174C4⋊C4—972no—
175C4⋊C4—1941no—
176C22⋊C4—1944no—
177C23—22no—
178C23—22no—
179C23—216no—
180C23—28no—
181C23—24no—
182C23—432no—
183C23—48no—
184C23—44no—
185C23—48no—
186C23—44no—
187C23—3882no—
188C23—3882no—
189C23—3882no—
190C23—1942no—
191C23#1309549614yescomplemented
192C23C2×C22⋊D9712yes—
193C23C2×C22⋊D9711yescharacteristic
194D4—216no—
195D4—28no—
196D4—24no—
197D4—3888no—
198D4—3882no—
199D4—1944no—
200D4C2×C22⋊D9718yescomplemented
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—3882no—
205C2×C4—1942no—
206C2×C4C2×C22⋊D9712yes—
207C2×C4C2×C22⋊D9711yescharacteristic
208C2×C4C2×C22⋊D9714yes—
209C2×C4C23×D9711yescharacteristic
210C8—1944no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—28no—
215C22—432no—
216C22—416no—
217C22—44no—
218C22—44no—
219C22—7764no—
220C22—3882no—
221C22—3882no—
222C22—3882no—
223C22#1569077912yes—
224C22C2×#1309549612yes—
225C22C2×#1309549614yescomplemented
226C22#1569083612yes—
227C22C22×C22⋊D9711yescharacteristic
228C4—22no—
229C4—3882no—
230C4#1569077912yes—
231C4C22×C22⋊D9712yes—
232C2—24no—
233C2—48no—
234C2—7762no—
235C2#2026086914yesdirect product factor
236C2C2×#1569077911yescharacteristic
237C2C22×#1309549611yescharacteristic
238C2C2×#1569083611yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC23×D4×C97C2
socleC23×C97C2×D4
hypercenterC23×D4D97
hypocenterC97C2×#447
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#13095496C2semidirect
2. C2×#15690784C2semidirect
3. #20260869C2direct (N × H)
4. C2×#15690731C2semidirect
5. C2×C22⋊#11715895C2semidirect
6. C23×D4×C97C2semidirect
7. C22×D388C22semidirect
10. C2×#13095496C22semidirect
11. C22×#11715895C22semidirect
13. #15690784C22semidirect
14. C2×C4⋊Dic97C22semidirect
16. #15690731C22semidirect
17. C22⋊#11715895C22semidirect
23. C22×D4×C97C22semidirect
28. C2×D388C23semidirect
34. C2×#11715895C23semidirect
36. C4⋊Dic97C23semidirect
50. C2×D4×C97D4semidirect
67. C23×C97D8semidirect
77. D4×C97C2×D4semidirect
101. C22×C97C2×D8semidirect
114. C2×C97#447semidirect
120. C97C2×#447semidirect
121. C23×D4D97semidirect
134. C22×D4C2×D97semidirect
163. C2×D4C22⋊D97semidirect
191. C23#13095496semidirect
200. D4C2×C22⋊D97semidirect
225. C22C2×#13095496semidirect
235. C2#20260869direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954962
C2×#156907842
#202608692
C2×#156907312
C2×C22⋊#117158952
C23×D4×C972
C2×#44797