Finite Group Databasebeta

← C2×C3×#237 #64345

Subgroups — C2×C3×#237 #64345

Order 288.

355 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 178 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
1C2×C32×D4C211yesmaximal, characteristic, complemented
2C2×C3×#67C212yesmaximal, complemented
3C22×C3×Dic3C212yesmaximal, complemented
4C3×#237C218yesmaximal, direct product factor
5C2×C3×C4×D3C211yesmaximal, characteristic, complemented
6C2×C3×Dic6C211yesmaximal, characteristic, complemented
7C2×C3×#41—31nomaximal
8C2×#237C311yesmaximal, characteristic, direct product factor
9C23×C32C2212yes—
10C22×C3×D3C2211yescharacteristic
11C32×D4C2214yescomplemented
12C2×C32×C4C2211yescharacteristic
13C3×#67C2218yescomplemented
14C2×C3×Dic3C2212yes—
15C2×C3×Dic3C2211yescharacteristic
16C2×C3×Dic3C2218yescomplemented
17C3×C4×D3C2214yescomplemented
18C3×Dic6C2214yescomplemented
19C3×#41—38no—
20C2×C3×Q8—31no—
21C2×C3×D4—21no—
22C2×C3×D4—32no—
23C2×C3×D4D311yescharacteristic, complemented
24C2×C3×D4C2×C311yescharacteristic, complemented
25C22×C3×C4—32no—
26C22×C3×C4—31no—
27C2×#67C2×C312yescomplemented
28C22×Dic3C2×C312yescomplemented
29#237C2×C318yesdirect product factor
30C2×C4×D3C2×C311yescharacteristic, complemented
31C2×Dic6C2×C311yescharacteristic, complemented
32C22×C32—24no—
33C22×C32C2314yes—
34C22×C32C2311yescharacteristic
35C2×C3×D3—22no—
36C2×C3×D3C2312yes—
37C32×C4C2312yes—
38C3×Dic3C2314yes—
39C3×Dic3C2312yes—
40C2×#41—31noSylow
41C23×C3—22no—
42C23×C3—31no—
43C23×C3C2×D312yes—
44C23×C3C22×C312yes—
45C22×D3C22×C311yescharacteristic
46C3×Q8—34no—
47C3×D4—24no—
48C3×D4—38no—
49C3×D4C2×D314yescomplemented
50C3×D4C22×C314yescomplemented
51C2×C3×C4—21no—
52C2×C3×C4—38no—
53C2×C3×C4—31no—
54C2×C3×C4—32no—
55C2×C3×C4—34no—
56C2×C3×C4C2×D311yescharacteristic
57C2×C3×C4C22×C311yescharacteristic
58#67C22×C318yescomplemented
59C2×Dic3C22×C312yes—
60C2×Dic3C22×C311yescharacteristic
61C2×Dic3C22×C318yescomplemented
62C4×D3C22×C314yescomplemented
63Dic6C22×C314yescomplemented
64C2×C32—24no—
65C2×C32#4112yescomplemented
66C2×C32C2411yescharacteristic
67C3×D3—22no—
68#41—38no—
69C2×Q8—31no—
70C2×D4—32no—
71C2×D4C3×D311yescharacteristic, complemented
72C22×C4—32no—
73C22×C4—31no—
74C22×C3—24no—
75C22×C3—24no—
76C22×C3—21no—
77C22×C3—24no—
78C22×C3—32no—
79C22×C3—44no—
80C22×C3—62no—
81C22×C3C22×D314yes—
82C22×C3C22×D311yescharacteristic
83C22×C3C23×C314yes—
84C22×C3C23×C311yescharacteristic
85C2×D3—22no—
86C2×D3C23×C312yes—
87C3×C4—22no—
88C3×C4—32no—
89C3×C4—34no—
90C3×C4C22×D312yes—
91C3×C4C23×C312yes—
92Dic3C23×C314yes—
93Dic3C23×C312yes—
94C32C2×#4111yescharacteristic, complemented, Sylow
95C23—31no—
96C23C2×C3×D312yes—
97Q8—34no—
98D4—38no—
99D4C2×C3×D314yescomplemented
100C2×C4—31no—
101C2×C4—32no—
102C2×C4—38no—
103C2×C4—34no—
104C2×C4C2×C3×D311yescharacteristic
105C2×C3—24no—
106C2×C3—24no—
107C2×C3—21no—
108C2×C3—22no—
109C2×C3—44no—
110C2×C3—62no—
111C2×C3#23712yesdirect product factor
112C2×C3C3×#4112yescomplemented
113C2×C3C23×D311yescharacteristic
114C2×C3C24×C311yescharacteristic
115D3—22no—
116C22—24no—
117C22—32no—
118C22—62no—
119C22C22×C3×D314yes—
120C22C22×C3×D311yescharacteristic
121C4—32no—
122C4—34no—
123C4C22×C3×D312yes—
124C3—21no—
125C3C2×#23711yescharacteristic, direct product factor
126C3C2×C3×#4111yescharacteristic, complemented
127C2—24no—
128C2—62no—
129C2C3×#23712yesdirect product factor
130C2C23×C3×D311yescharacteristic
subgroupG/H
centerC22×C3C22×D3
commutator, cosocleC2×C3C24×C3
FrattiniC2C23×C3×D3
FittingC2×C32×D4C2
socleC22×C32C23
hypercenterC2×C3×D4D3
hypocenterC3C2×C3×#41
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. C2×C32×D4C2semidirect
2. C2×C3×#67C2semidirect
3. C22×C3×Dic3C2semidirect
4. C3×#237C2direct (N × H)
5. C2×C3×C4×D3C2semidirect
6. C2×C3×Dic6C2semidirect
8. C2×#237C3direct (N × H)
11. C32×D4C22semidirect
13. C3×#67C22semidirect
16. C2×C3×Dic3C22semidirect
17. C3×C4×D3C22semidirect
18. C3×Dic6C22semidirect
23. C2×C3×D4D3semidirect
24. C2×C3×D4C2×C3semidirect
27. C2×#67C2×C3semidirect
28. C22×Dic3C2×C3semidirect
29. #237C2×C3direct (N × H)
30. C2×C4×D3C2×C3semidirect
31. C2×Dic6C2×C3semidirect
49. C3×D4C2×D3semidirect
50. C3×D4C22×C3semidirect
58. #67C22×C3semidirect
61. C2×Dic3C22×C3semidirect
62. C4×D3C22×C3semidirect
63. Dic6C22×C3semidirect
65. C2×C32#41semidirect
71. C2×D4C3×D3semidirect
94. C32C2×#41semidirect
99. D4C2×C3×D3semidirect
111. C2×C3#237direct (N × H)
112. C2×C3C3×#41semidirect
125. C3C2×#237direct (N × H)
126. C3C2×C3×#41semidirect
129. C2C3×#237direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×D42
C2×C3×#672
C22×C3×Dic32
C3×#2372
C2×C3×C4×D32
C2×C3×Dic62
C2×C3×#413
C2×#2373