Finite Group Databasebeta

← C23×#4207 #13049355

Subgroups — C23×#4207 #13049355

Order 1296.

1149 conjugacy classes of subgroups; only the 310 normal subgroups are shown.

normal subgroups only

This group has 1149 conjugacy classes of subgroups — too many to draw. Only its 310 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×#737C211yesmaximal, characteristic, complemented
2C22×#4207C2114yesmaximal, direct product factor
3C23×C3⋊D9C311yesmaximal, characteristic, complemented
4C23×AGL(1,ℤ/9)—32nomaximal
5C23×(C3⋊D3)⋊C3—31nomaximal
6C23×C3×D9—31nomaximal
7C22×#737C2217yescomplemented
8C2×#4207C22128yesdirect product factor
9C23×3−1+2—21no—
10C23×3−1+2—22no—
11C23×3−1+2D312yescomplemented
12C23×H(3,𝔽3)D311yescharacteristic
13C23×C3×C9—21no—
14C23×C3×C9D311yescharacteristic, complemented
15C23×C3×C9C2×C311yescharacteristic, complemented
16C22×C3⋊D9C2×C3114yescomplemented
17C22×AGL(1,ℤ/9)—328no—
18C22×(C3⋊D3)⋊C3—314no—
19C22×C3×D9—314no—
20C2×#737C2317yescomplemented
21#4207C2318yesdirect product factor
22C23×C3⋊D3—31no—
23C23×C3×D3—91no—
24C23×D9—31no—
25C23×D9—32no—
26C22×3−1+2—27no—
27C22×3−1+2—214no—
28C22×3−1+2C2×D3114yescomplemented
29C22×H(3,𝔽3)C2×D317yes—
30C22×C3×C9—27no—
31C22×C3×C9C2×D317yescomplemented
32C22×C3×C9C22×C317yescomplemented
33C2×C3⋊D9C22×C3128yescomplemented
34C2×AGL(1,ℤ/9)—356no—
35C2×(C3⋊D3)⋊C3—328no—
36C2×C3×D9—328no—
37#737C2411yescharacteristic, complemented, Sylow
38C23×C32—21no—
39C23×C32C3×D311yescharacteristic
40C23×C32C3⋊D311yescharacteristic
41C22×C3⋊D3—314no—
42C22×C3×D3—914no—
43C23×C9—22no—
44C23×C9—21no—
45C23×C9C3×D311yescharacteristic
46C23×C9C3×D312yes—
47C22×D9—314no—
48C22×D9—328no—
49C2×3−1+2—27no—
50C2×3−1+2—214no—
51C2×3−1+2C22×D3114yescomplemented
52C2×H(3,𝔽3)C22×D317yes—
53C2×C3×C9—27no—
54C2×C3×C9C22×D317yescomplemented
55C2×C3×C9C23×C317yescomplemented
56C3⋊D9C23×C318yescomplemented
57AGL(1,ℤ/9)—316no—
58(C3⋊D3)⋊C3—38no—
59C3×D9—38no—
60C24×C3—271no—
61C23×D3—91no—
62C23×D3—271no—
63C22×C32—27no—
64C22×C32C2×C3×D317yes—
65C22×C32C2×C3⋊D317yes—
66C2×C3⋊D3—328no—
67C2×C3×D3—928no—
68C22×C9—214no—
69C22×C9—27no—
70C22×C9C2×C3×D317yes—
71C22×C9C2×C3×D3114yes—
72C2×D9—328no—
73C2×D9—356no—
743−1+2—21no—
753−1+2—22no—
763−1+2C23×D312yescomplemented
77H(3,𝔽3)C23×D311yescharacteristic
78C3×C9—21no—
79C3×C9C23×D311yescharacteristic, complemented
80C3×C9C24×C311yescharacteristic, complemented
81C23×C3—31no—
82C23×C3—31no—
83C23×C3—61no—
84C23×C3—2714no—
85C23×C3C3×C3⋊D311yescharacteristic
86C22×D3—914no—
87C22×D3—2714no—
88C2×C32—27no—
89C2×C32C22×C3×D317yes—
90C2×C32C22×C3⋊D317yes—
91C3⋊D3—38no—
92C3×D3—98no—
93C2×C9—214no—
94C2×C9—27no—
95C2×C9C22×C3×D317yes—
96C2×C9C22×C3×D3114yes—
97D9—38no—
98D9—316no—
99C24—271noSylow
100C22×C3—37no—
101C22×C3—37no—
102C22×C3—67no—
103C22×C3—2728no—
104C22×C3C2×C3×C3⋊D317yes—
105C2×D3—928no—
106C2×D3—2728no—
107C32—21no—
108C32C23×C3×D311yescharacteristic
109C32C23×C3⋊D311yescharacteristic
110C9—22no—
111C9—21no—
112C9C23×C3×D311yescharacteristic
113C9C23×C3×D312yes—
114C23—2714no—
115C23#420711yescharacteristic, direct product factor
116C2×C3—37no—
117C2×C3—37no—
118C2×C3—67no—
119C2×C3—278no—
120C2×C3C22×C3×C3⋊D317yes—
121D3—98no—
122D3—278no—
123C22—2728no—
124C22C2×#420717yesdirect product factor
125C3—31no—
126C3—31no—
127C3—61no—
128C3C23×C3×C3⋊D311yescharacteristic
129C2—278no—
130C2C22×#420717yesdirect product factor
subgroupG/H
center, hypercenterC23#4207
commutator, cosocle, hypocenterC3×C9C24×C3
FrattiniC3C23×C3×C3⋊D3
FittingC23×#737C2
socleC23×C3C3×C3⋊D3
LCSG ▷ C3×C9
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×#737C2semidirect
2. C22×#4207C2direct (N × H)
3. C23×C3⋊D9C3semidirect
7. C22×#737C22semidirect
8. C2×#4207C22direct (N × H)
11. C23×3−1+2D3semidirect
14. C23×C3×C9D3semidirect
15. C23×C3×C9C2×C3semidirect
16. C22×C3⋊D9C2×C3semidirect
20. C2×#737C23semidirect
21. #4207C23direct (N × H)
28. C22×3−1+2C2×D3semidirect
31. C22×C3×C9C2×D3semidirect
32. C22×C3×C9C22×C3semidirect
33. C2×C3⋊D9C22×C3semidirect
37. #737C24semidirect
51. C2×3−1+2C22×D3semidirect
54. C2×C3×C9C22×D3semidirect
55. C2×C3×C9C23×C3semidirect
56. C3⋊D9C23×C3semidirect
76. 3−1+2C23×D3semidirect
79. C3×C9C23×D3semidirect
80. C3×C9C24×C3semidirect
115. C23#4207direct (N × H)
124. C22C2×#4207direct (N × H)
130. C2C22×#4207direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#7372
C22×#42072
C23×C3⋊D93
C23×AGL(1,ℤ/9)3
C23×(C3⋊D3)⋊C33
C23×C3×D93