Finite Group Databasebeta

← C2×#20261121 #27152167

Subgroups — C2×#20261121 #27152167

Order 12416.

668 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13095498C211yesmaximal, characteristic, complemented
2C22×D776C211yesmaximal, characteristic, complemented
3C22×#13095496C211yesmaximal, characteristic, complemented
4C2×#15690806C211yesmaximal, characteristic, complemented
5C2×#15690799C211yesmaximal, characteristic, complemented
6#20261121C218yesmaximal, direct product factor
7C2×#15690720C211yesmaximal, characteristic, complemented
8C22×SD16×C97C211yesmaximal, characteristic, complemented
9C22×D388C2211yescharacteristic
10C22×D388C2211yescharacteristic
11C22×C22⋊D97—22no—
12C2×#13095498—24no—
13C2×#13095498C2214yescomplemented
14C2×D776—24no—
15C2×D776C2214yescomplemented
16C2×#13095496—24no—
17C2×#13095496C2214yescomplemented
18C22×#11715895C2211yescharacteristic
19#15690806—22no—
20#15690806C2214yescomplemented
21#15690799—22no—
22#15690799C2214yescomplemented
23C2×#13095488—22no—
24C2×C4×Dic97C2211yescharacteristic
25#15690720C2214yescomplemented
26C22×Q8×C97C2211yescharacteristic
27C22×D4×C97C2211yescharacteristic
28C2×SD16×C97—24no—
29C2×SD16×C97C2214yescomplemented
30C22×C8×C97C2211yescharacteristic
31C23×D97—21no—
32C23×D97—21no—
33C2×D388—22no—
34C2×D388—22no—
35C2×D388—22no—
36C2×D388—22no—
37C2×D388—21no—
38C2×D388—21no—
39C2×D388C2312yes—
40C2×D388C2312yes—
41C2×C22⋊D97—28no—
42C2×C22⋊D97—48no—
43C22×Dic97D412yes—
44#13095498—28no—
45D776—28no—
46#13095496—28no—
47C2×#11715895D414yescomplemented
48C2×#11715895C2312yes—
49#13095488—28no—
50C4×Dic97D412yes—
51C4×Dic97C2312yes—
52C24×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—22no—
55C2×Q8×C97—22no—
56C2×Q8×C97C2312yes—
57C2×D4×C97—22no—
58C2×D4×C97—22no—
59C2×D4×C97—21no—
60C2×D4×C97C2312yes—
61C22×C4×C97—21no—
62C22×C4×C97C2311yescharacteristic
63SD16×C97—28no—
64C2×C8×C97D414yescomplemented
65C2×C8×C97C2312yes—
66C23×C97—22no—
67C23×C97—42no—
68C23×C97—44no—
69C23×C97C2×D411yescharacteristic
70C22×D97—22no—
71C22×D97—22no—
72C22×D97—42no—
73C22×D97—42no—
74C22×D97—44no—
75C22×D97—44no—
76Q8×C97—22no—
77Q8×C97—22no—
78Q8×C97—42no—
79D4×C97—22no—
80D4×C97—22no—
81D4×C97—42no—
82C2×C4×C97—22no—
83C2×C4×C97—42no—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D412yes—
87C2×C4×C97C2411yescharacteristic
88C22⋊D97—416no—
89C2×Dic97—24no—
90C2×Dic97C2×D414yes—
91D388—22no—
92D388—22no—
93D388—22no—
94D388—22no—
95D388—42no—
96D388—42no—
97C8×C97C2×D414yescomplemented
98#11715895C2×D414yescomplemented
99C22×C97—42no—
100C22×C97—44no—
101C22×C97—84no—
102C22×C97#12712yes—
103C22×C97AGL(1,ℤ/8)14yes—
104C22×C97C22×D411yescharacteristic
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—44no—
108C2×D97—44no—
109C2×D97—84no—
110C2×D97—84no—
111C4×C97—42no—
112C4×C97#12712yes—
113C4×C97C22×D411yescharacteristic
114C4×C97C22×D411yescharacteristic
115Dic97—24no—
116C2×C97—82no—
117C2×C97#49614yescomplemented
118C2×C97C2×#12711yescharacteristic
119C2×C97C2×AGL(1,ℤ/8)12yes—
120D97—82no—
121D97—82no—
122C2×#496—971nomaximal, Sylow
123C97C2×#49611yescharacteristic, complemented, Sylow
124C22×SD16—971no—
125C22×SD16D9711yescharacteristic, complemented
126C22×D8—971no—
127C22×D8—971no—
128C2×#127—971no—
129C2×#124—971no—
130#496—978no—
131C2×#97—971no—
132C22×Q8C2×D9711yescharacteristic
133C22×D4—971no—
134C22×D4—971no—
135C22×D4—1942no—
136C22×D4C2×D9711yescharacteristic
137C2×SD16—24no—
138C2×SD16—974no—
139C2×SD16—1944no—
140C2×SD16C2×D9714yescomplemented
141C2×D8—974no—
142C2×D8—974no—
143C2×D8—1944no—
144C2×D8—1944no—
145C22×C8—971no—
146C22×C8C2×D9711yescharacteristic
147#127—974no—
148#127—1942no—
149#124—974no—
150#124—1942no—
151C2×C22⋊C4—1942no—
152C2×C42—971no—
153#97—974no—
154C24—21no—
155C24—1941no—
156C24—1941no—
157C2×Q8—21no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8C22×D9712yes—
161C2×D4—22no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—972no—
165C2×D4—972no—
166C2×D4—3888no—
167C2×D4—1948no—
168C2×D4—1941no—
169C2×D4—1941no—
170C2×D4—1942no—
171C2×D4—1942no—
172C2×D4—1942no—
173C2×D4—1942no—
174C2×D4C22×D9712yes—
175C22×C4—21no—
176C22×C4—972no—
177C22×C4C22×D9711yescharacteristic
178SD16—28no—
179SD16—1948no—
180D8—1948no—
181D8—1948no—
182C2×C8—974no—
183C2×C8—972no—
184C2×C8C22⋊D9714yescomplemented
185C2×C8C22×D9712yes—
186C22⋊C4—1948no—
187C42—972no—
188C42—972no—
189C23—22no—
190C23—42no—
191C23—44no—
192C23—3882no—
193C23—3882no—
194C23—3884no—
195C23—3884no—
196C23—1942no—
197C23—1942no—
198C23D4×D9711yescharacteristic
199Q8—22no—
200Q8—22no—
201Q8—42no—
202D4—22no—
203D4—22no—
204D4—42no—
205D4—38816no—
206D4—3882no—
207D4—3882no—
208D4—1942no—
209D4—1942no—
210D4—1942no—
211D4—1942no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—974no—
215C2×C4—1944no—
216C2×C4D4×D9711yescharacteristic
217C2×C4D4×D9712yes—
218C2×C4C2×C22⋊D9712yes—
219C2×C4C23×D9711yescharacteristic
220C8—974no—
221C8C2×C22⋊D9714yescomplemented
222C22—44no—
223C22—42no—
224C22—84no—
225C22—7764no—
226C22—7764no—
227C22—3884no—
228C22—3884no—
229C22—3882no—
230C22—3882no—
231C22#1569080612yes—
232C22#1569083714yes—
233C22C2×D4×D9711yescharacteristic
234C4—42no—
235C4—1944no—
236C4#1569080612yes—
237C4C2×D4×D9711yescharacteristic
238C4C22×C22⋊D9711yescharacteristic
239C2—82no—
240C2—7762no—
241C2—7762no—
242C2#2026112114yesdirect product factor
243C2C2×#1569080611yescharacteristic
244C2C2×#1569083712yes—
subgroupG/H
centerC23D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC22×SD16×C97C2
socleC23×C97C2×D4
hypercenterC22×SD16D97
hypocenterC97C2×#496
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#13095498C2semidirect
2. C22×D776C2semidirect
3. C22×#13095496C2semidirect
4. C2×#15690806C2semidirect
5. C2×#15690799C2semidirect
6. #20261121C2direct (N × H)
7. C2×#15690720C2semidirect
8. C22×SD16×C97C2semidirect
13. C2×#13095498C22semidirect
15. C2×D776C22semidirect
17. C2×#13095496C22semidirect
20. #15690806C22semidirect
22. #15690799C22semidirect
25. #15690720C22semidirect
29. C2×SD16×C97C22semidirect
47. C2×#11715895D4semidirect
64. C2×C8×C97D4semidirect
97. C8×C97C2×D4semidirect
98. #11715895C2×D4semidirect
117. C2×C97#496semidirect
123. C97C2×#496semidirect
125. C22×SD16D97semidirect
140. C2×SD16C2×D97semidirect
184. C2×C8C22⋊D97semidirect
221. C8C2×C22⋊D97semidirect
242. C2#20261121direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130954982
C22×D7762
C22×#130954962
C2×#156908062
C2×#156907992
#202611212
C2×#156907202
C22×SD16×C972
C2×#49697