Finite Group Databasebeta

← C2×C3×D3×D4 #64344

Subgroups — C2×C3×D3×D4 #64344

Order 288.

499 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 194 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
1C23×C3×D3C212yesmaximal, complemented
2C2×C32×D4C211yesmaximal, characteristic, complemented
3C2×C3×#67C212yesmaximal, complemented
4C3×D3×D4C218yesmaximal, direct product factor
5C2×C3×D12C211yesmaximal, characteristic, complemented
6C2×C3×C4×D3C211yesmaximal, characteristic, complemented
7C22×C3×D4—31nomaximal
8C2×D3×D4C311yesmaximal, characteristic, direct product factor
9C23×C32C2212yescomplemented
10C22×C3×D3—28no—
11C22×C3×D3C2211yescharacteristic
12C22×C3×D3C2218yescomplemented
13C22×C3×D3C2212yescomplemented
14C32×D4C2214yescomplemented
15C2×C32×C4C2211yescharacteristic, complemented
16C3×#67C2218yescomplemented
17C2×C3×Dic3C2211yescharacteristic, complemented
18C3×D12C2214yescomplemented
19C3×C4×D3C2214yescomplemented
20C24×C3—32no—
21C23×D3C2×C312yescomplemented
22C2×C3×D4—21no—
23C2×C3×D4—32no—
24C2×C3×D4—38no—
25C2×C3×D4—31no—
26C2×C3×D4D311yescharacteristic, direct product factor
27C2×C3×D4C2×C311yescharacteristic, complemented
28C22×C3×C4—31no—
29C2×#67C2×C312yescomplemented
30D3×D4C2×C318yesdirect product factor
31C2×D12C2×C311yescharacteristic, complemented
32C2×C4×D3C2×C311yescharacteristic, complemented
33C22×C32—24no—
34C22×C32C2311yescharacteristic
35C22×C32C2314yescomplemented
36C2×C3×D3—24no—
37C2×C3×D3—216no—
38C2×C3×D3D414yesdirect product factor
39C2×C3×D3C2312yes—
40C2×C3×D3C2314yescomplemented
41C32×C4C2312yescomplemented
42C3×Dic3C2312yescomplemented
43C22×D4—31noSylow
44C23×C3—22no—
45C23×C3—31no—
46C23×C3—32no—
47C23×C3—38no—
48C23×C3—68no—
49C23×C3C2×D312yescomplemented
50C23×C3C22×C312yescomplemented
51C22×D3—28no—
52C22×D3C22×C311yescharacteristic
53C22×D3C22×C318yescomplemented
54C22×D3C22×C312yescomplemented
55C3×D4—24no—
56C3×D4—38no—
57C3×D4—34no—
58C3×D4C2×D314yesdirect product factor
59C3×D4C22×C314yescomplemented
60C2×C3×C4—21no—
61C2×C3×C4—31no—
62C2×C3×C4—34no—
63C2×C3×C4C2×D311yescharacteristic, complemented
64C2×C3×C4C22×C311yescharacteristic, complemented
65#67C22×C318yescomplemented
66C2×Dic3C22×C311yescharacteristic, complemented
67D12C22×C314yescomplemented
68C4×D3C22×C314yescomplemented
69C2×C32—24no—
70C2×C32C2×D412yescomplemented
71C2×C32C2411yescharacteristic
72C3×D3—24no—
73C3×D3C2×D414yesdirect product factor
74C24—32no—
75C2×D4—32no—
76C2×D4—38no—
77C2×D4—31no—
78C2×D4C3×D311yescharacteristic, direct product factor
79C22×C4—31no—
80C22×C3—24no—
81C22×C3—24no—
82C22×C3—21no—
83C22×C3—24no—
84C22×C3—34no—
85C22×C3—32no—
86C22×C3—34no—
87C22×C3—44no—
88C22×C3—616no—
89C22×C3—64no—
90C22×C3C22×D311yescharacteristic
91C22×C3C22×D314yescomplemented
92C22×C3C23×C311yescharacteristic
93C22×C3C23×C314yescomplemented
94C2×D3—24no—
95C2×D3—216no—
96C2×D3C3×D414yesdirect product factor
97C2×D3C23×C312yes—
98C2×D3C23×C314yescomplemented
99C3×C4—22no—
100C3×C4—32no—
101C3×C4C22×D312yescomplemented
102C3×C4C23×C312yescomplemented
103Dic3C23×C312yescomplemented
104C32C22×D411yescharacteristic, complemented, Sylow
105C23—31no—
106C23—32no—
107C23—38no—
108C23—68no—
109C23C2×C3×D312yescomplemented
110D4—38no—
111D4—34no—
112D4C2×C3×D314yesdirect product factor
113C2×C4—31no—
114C2×C4—34no—
115C2×C4C2×C3×D311yescharacteristic, complemented
116C2×C3—24no—
117C2×C3—24no—
118C2×C3—21no—
119C2×C3—22no—
120C2×C3—34no—
121C2×C3—44no—
122C2×C3—64no—
123C2×C3D3×D412yesdirect product factor
124C2×C3C2×C3×D412yescomplemented
125C2×C3C23×D311yescharacteristic
126C2×C3C24×C311yescharacteristic
127D3—24no—
128D3C2×C3×D414yesdirect product factor
129C22—24no—
130C22—32no—
131C22—34no—
132C22—34no—
133C22—616no—
134C22—64no—
135C22C22×C3×D311yescharacteristic
136C22C22×C3×D314yescomplemented
137C4—32no—
138C4C22×C3×D312yescomplemented
139C3—21no—
140C3C2×D3×D411yescharacteristic, direct product factor
141C3C22×C3×D411yescharacteristic, complemented
142C2—24no—
143C2—34no—
144C2—64no—
145C2C3×D3×D412yesdirect product factor
146C2C23×C3×D311yescharacteristic
subgroupG/H
centerC22×C3C22×D3
commutator, cosocleC2×C3C24×C3
FrattiniC2C23×C3×D3
FittingC2×C32×D4C2
socleC22×C32C23
hypercenterC2×C3×D4D3
hypocenterC3C22×C3×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C23×C3×D3C2semidirect
2. C2×C32×D4C2semidirect
3. C2×C3×#67C2semidirect
4. C3×D3×D4C2direct (N × H)
5. C2×C3×D12C2semidirect
6. C2×C3×C4×D3C2semidirect
8. C2×D3×D4C3direct (N × H)
9. C23×C32C22semidirect
12. C22×C3×D3C22semidirect
13. C22×C3×D3C22semidirect
14. C32×D4C22semidirect
15. C2×C32×C4C22semidirect
16. C3×#67C22semidirect
17. C2×C3×Dic3C22semidirect
18. C3×D12C22semidirect
19. C3×C4×D3C22semidirect
21. C23×D3C2×C3semidirect
26. C2×C3×D4D3direct (N × H)
27. C2×C3×D4C2×C3semidirect
29. C2×#67C2×C3semidirect
30. D3×D4C2×C3direct (N × H)
31. C2×D12C2×C3semidirect
32. C2×C4×D3C2×C3semidirect
35. C22×C32C23semidirect
38. C2×C3×D3D4direct (N × H)
40. C2×C3×D3C23semidirect
41. C32×C4C23semidirect
42. C3×Dic3C23semidirect
49. C23×C3C2×D3semidirect
50. C23×C3C22×C3semidirect
53. C22×D3C22×C3semidirect
54. C22×D3C22×C3semidirect
58. C3×D4C2×D3direct (N × H)
59. C3×D4C22×C3semidirect
63. C2×C3×C4C2×D3semidirect
64. C2×C3×C4C22×C3semidirect
65. #67C22×C3semidirect
66. C2×Dic3C22×C3semidirect
67. D12C22×C3semidirect
68. C4×D3C22×C3semidirect
70. C2×C32C2×D4semidirect
73. C3×D3C2×D4direct (N × H)
78. C2×D4C3×D3direct (N × H)
91. C22×C3C22×D3semidirect
93. C22×C3C23×C3semidirect
96. C2×D3C3×D4direct (N × H)
98. C2×D3C23×C3semidirect
101. C3×C4C22×D3semidirect
102. C3×C4C23×C3semidirect
103. Dic3C23×C3semidirect
104. C32C22×D4semidirect
109. C23C2×C3×D3semidirect
112. D4C2×C3×D3direct (N × H)
115. C2×C4C2×C3×D3semidirect
123. C2×C3D3×D4direct (N × H)
124. C2×C3C2×C3×D4semidirect
128. D3C2×C3×D4direct (N × H)
136. C22C22×C3×D3semidirect
138. C4C22×C3×D3semidirect
140. C3C2×D3×D4direct (N × H)
141. C3C22×C3×D4semidirect
145. C2C3×D3×D4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×D32
C2×C32×D42
C2×C3×#672
C3×D3×D42
C2×C3×D122
C2×C3×C4×D32
C22×C3×D43
C2×D3×D43