Finite Group Databasebeta

← C22×#908 #87155

Subgroups — C22×#908 #87155

Order 384.

1720 conjugacy classes of subgroups; only the 565 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 565 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×#235C211yesmaximal, characteristic, complemented
2C23×D12C211yesmaximal, characteristic, complemented
3C22×C3×M16C211yesmaximal, characteristic, complemented
4C2×#908C2124yesmaximal, direct product factor
5C22×D24C212yesmaximal, complemented
6C22×#204C212yesmaximal, complemented
7C22×AGL(1,ℤ/8)—31nomaximal, Sylow
8C24×D3—21no—
9C23×C3×C4C2211yescharacteristic
10C22×#67—21no—
11C2×#235—26no—
12C2×#235C22112yescomplemented
13C22×D12—26no—
14C22×D12—21no—
15C22×D12C2212yes—
16C22×D12C22112yescomplemented
17C22×D12C2211yescharacteristic, complemented
18C22×C4×D3—21no—
19C22×Dic6C2211yescharacteristic, complemented
20C2×C3×M16C22112yescomplemented
21C22×C3×C8C2212yescomplemented
22#908C22164yesdirect product factor
23C2×D24C22124yescomplemented
24C2×#204C22124yescomplemented
25C22×#41—31no—
26C23×D4—31no—
27C2×AGL(1,ℤ/8)—324no—
28C22×SD16—32no—
29C22×D8—32no—
30C22×M16D311yescharacteristic, complemented
31C24×C3D411yescharacteristic
32C23×D3—21no—
33C23×D3—22no—
34C23×D3—212no—
35C23×D3—48no—
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D416yes—
38C22×C3×C4C2311yescharacteristic
39C22×C3×C4C2316yes—
40C2×#67—212no—
41C22×Dic3—21no—
42#235—224no—
43#235C23116yescomplemented
44C2×D12—26no—
45C2×D12—224no—
46C2×D12—212no—
47C2×D12—23no—
48C2×D12C23112yes—
49C2×D12C23116yescomplemented
50C2×D12C2316yescomplemented
51C2×C4×D3—212no—
52C2×Dic6—23no—
53C2×Dic6C2316yescomplemented
54C3×M16C23116yescomplemented
55C2×C3×C8C23112yescomplemented
56D24C23132yescomplemented
57#204C23132yescomplemented
58C25—61no—
59C2×#41—312no—
60C2×#41—66no—
61C22×Q8—31no—
62C22×D4—31no—
63C22×D4—32no—
64C22×D4—312no—
65C22×D4—61no—
66C22×D4—66no—
67C22×D4—61no—
68C23×C4—61no—
69C23×C4C2×D311yescharacteristic
70AGL(1,ℤ/8)—364no—
71C2×SD16—324no—
72C2×D8—324no—
73C2×M16C2×D3112yescomplemented
74C22×C8C2×D312yescomplemented
75C23×C3—24no—
76C23×C3C2×D416yes—
77C23×C3C2×D411yescharacteristic
78C22×D3—26no—
79C22×D3—212no—
80C22×D3—216no—
81C22×D3—448no—
82C22×D3—44no—
83C22×D3—48no—
84C2×C3×C4C2×D413yes—
85C2×C3×C4C2×D416yes—
86C2×C3×C4C2×D4112yes—
87C2×C3×C4C2413yes—
88C2×C3×C4C2414yes—
89#67—216no—
90C2×Dic3—26no—
91D12—212no—
92D12—216no—
93D12—26no—
94D12C2418yes—
95D12C2414yescomplemented
96C4×D3—216no—
97Dic6—26no—
98Dic6C2414yescomplemented
99C3×C8C2418yescomplemented
100C24—612no—
101C24—61no—
102C24—62no—
103C24—128no—
104C24D1211yescharacteristic
105#41—316no—
106#41—624no—
107C2×Q8—36no—
108C2×Q8—63no—
109C2×D4—36no—
110C2×D4—312no—
111C2×D4—316no—
112C2×D4—612no—
113C2×D4—612no—
114C2×D4—66no—
115C2×D4—624no—
116C2×D4—63no—
117C22×C4—61no—
118C22×C4—612no—
119C22×C4D1216yes—
120C22×C4D1211yescharacteristic
121C22×C4C22×D316yes—
122C22×C4C22×D311yescharacteristic
123SD16—332no—
124D8—332no—
125M16C22×D3116yescomplemented
126C2×C8C22×D3112yescomplemented
127C22×C3—212no—
128C22×C3AGL(1,ℤ/8)14yescomplemented
129C22×C3C22×D414yes—
130C22×C3C22×D413yes—
131C2×D3—24no—
132C2×D3—28no—
133C2×D3—432no—
134C2×D3—412no—
135C2×D3—424no—
136C3×C4C22×D413yes—
137C3×C4C22×D414yes—
138C3×C4C2511yescharacteristic
139Dic3—24no—
140C23—24no—
141C23—66no—
142C23—616no—
143C23—612no—
144C23—1248no—
145C23—128no—
146C23—124no—
147C23C2×D1216yes—
148C23C2×D1211yescharacteristic
149Q8—34no—
150Q8—66no—
151D4—34no—
152D4—38no—
153D4—616no—
154D4—616no—
155D4—612no—
156D4—66no—
157C2×C4—66no—
158C2×C4—616no—
159C2×C4C2×D1216yes—
160C2×C4C2×D1213yes—
161C2×C4C2×D12112yes—
162C2×C4C23×D313yes—
163C2×C4C23×D314yes—
164C8C23×D318yescomplemented
165C2×C3—24no—
166C2×C3C2×AGL(1,ℤ/8)16yescomplemented
167C2×C3C23×D411yescharacteristic
168D3—44no—
169D3—48no—
170C22—212no—
171C22—64no—
172C22—68no—
173C22—1232no—
174C22—1224no—
175C22—1212no—
176C22#90814yesdirect product factor
177C22C22×D1214yes—
178C22C22×D1213yes—
179C4—64no—
180C4C22×D1214yes—
181C4C22×D1213yes—
182C4C24×D311yescharacteristic
183C3C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
184C2—24no—
185C2—128no—
186C2—124no—
187C2C2×#90816yesdirect product factor
188C2C23×D1211yescharacteristic
subgroupG/H
centerC23C2×D12
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC22×C3×M16C2
socleC23×C3C2×D4
hypercenterC22×M16D3
hypocenterC3C22×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. C22×#235C2semidirect
2. C23×D12C2semidirect
3. C22×C3×M16C2semidirect
4. C2×#908C2direct (N × H)
5. C22×D24C2semidirect
6. C22×#204C2semidirect
12. C2×#235C22semidirect
16. C22×D12C22semidirect
17. C22×D12C22semidirect
19. C22×Dic6C22semidirect
20. C2×C3×M16C22semidirect
21. C22×C3×C8C22semidirect
22. #908C22direct (N × H)
23. C2×D24C22semidirect
24. C2×#204C22semidirect
30. C22×M16D3semidirect
43. #235C23semidirect
49. C2×D12C23semidirect
50. C2×D12C23semidirect
53. C2×Dic6C23semidirect
54. C3×M16C23semidirect
55. C2×C3×C8C23semidirect
56. D24C23semidirect
57. #204C23semidirect
73. C2×M16C2×D3semidirect
74. C22×C8C2×D3semidirect
95. D12C24semidirect
98. Dic6C24semidirect
99. C3×C8C24semidirect
125. M16C22×D3semidirect
126. C2×C8C22×D3semidirect
128. C22×C3AGL(1,ℤ/8)semidirect
164. C8C23×D3semidirect
166. C2×C3C2×AGL(1,ℤ/8)semidirect
176. C22#908direct (N × H)
183. C3C22×AGL(1,ℤ/8)semidirect
187. C2C2×#908direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2352
C23×D122
C22×C3×M162
C2×#9082
C22×D242
C22×#2042
C22×AGL(1,ℤ/8)3