Finite Group Databasebeta

← #13047984

Subgroups — #13047984

Order 1296.

234 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C3×Q8×3−1+2C211yesmaximal, characteristic
2#10619962C211yesmaximal, characteristic
3C3⋊(3−1+2⋊C8)C211yesmaximal, characteristic
4C3×#3756—31nomaximal
5#88547C311yesmaximal, characteristic, complemented
6#3674⋊C3—33nomaximal
7C3×C4×3−1+2—21no—
8C3×C4×3−1+2C2211yescharacteristic
9C3⋊(3−1+2⋊C4)—21no—
10C33×Q8D311yescharacteristic
11C3×#634—31no—
12C3×C3⋊(C3⋊C8)—31no—
13Q8×3−1+2—23no—
14Q8×3−1+2D313yes—
15C3×Q8×C9—21no—
16C3×Q8×C9C2×C311yescharacteristic
17#6226C2×C311yescharacteristic
183−1+2⋊Q8—33no—
19C3⋊(C9⋊C8)C2×C311yescharacteristic
203−1+2⋊C8—33no—
21C2×C3×3−1+2D411yescharacteristic
22#3756—31no—
23C3×#216—93no—
24C3×#216—91no—
25#3674—33no—
26C33×C4—21no—
27C33×C4C2×D311yescharacteristic
28C3×C3⋊Dic3—61no—
29C4×3−1+2—26no—
30C4×3−1+2—23no—
31C4×3−1+2—23no—
32C4×3−1+2C2×D313yes—
33C3×C4×C9—22no—
34C3×C4×C9—21no—
35C3×C4×C9—21no—
36C3×C4×C9C22×C311yescharacteristic
37C3⋊Dic9—21no—
383−1+2⋊C4—63no—
39C3×3−1+2Q1611yescharacteristic, complemented, Sylow
40C32×Q8—21no—
41C32×Q8—33no—
42C32×Q8C3×D311yescharacteristic
43C32×Q8C3⋊D311yescharacteristic
44#634—31no—
45C3×Dic6—93no—
46C3×Dic6—91no—
47C3⋊(C3⋊C8)—31no—
48C3×C3⋊C8—93no—
49C3×C3⋊C8—91no—
50Q8×C9—23no—
51Q8×C9C3×D313yes—
52Dic18—33no—
53C9⋊C8—33no—
54C2×C33#6711yescharacteristic
55C2×3−1+2—23no—
56C2×3−1+2#6713yes—
57C2×C3×C9—21no—
58C2×C3×C9C3×D411yescharacteristic
59C3×Q16—271no—
60#216—93no—
61#216—91no—
62C32×C4—21no—
63C32×C4—22no—
64C32×C4—21no—
65C32×C4—21no—
66C32×C4—33no—
67C32×C4—63no—
68C32×C4C2×C3×D311yescharacteristic
69C32×C4C2×C3⋊D311yescharacteristic
70C3⋊Dic3—61no—
71C3×Dic3—183no—
72C3×Dic3—181no—
73C4×C9—26no—
74C4×C9—23no—
75C4×C9—23no—
76C4×C9C2×C3×D313yes—
77Dic9—63no—
78C33#21611yescharacteristic
793−1+2—23no—
803−1+2#21613yescomplemented
81C3×C9—21no—
82C3×C9C3×Q1611yescharacteristic, complemented
83C3×Q8—31no—
84C3×Q8—61no—
85C3×Q8—271no—
86C3×Q8AGL(1,ℤ/9)13yes—
87C3×Q8C3×C3⋊D311yescharacteristic
88Dic6—93no—
89Dic6—91no—
90C3×C8—271no—
91C3⋊C8—93no—
92C3⋊C8—91no—
93C2×C32—21no—
94C2×C32—33no—
95C2×C32C3×#6711yescharacteristic
96C2×C32#63811yescharacteristic
97C2×C9—23no—
98C2×C9C3×#6713yes—
99Q16—271noSylow
100C3×C4—23no—
101C3×C4—21no—
102C3×C4—31no—
103C3×C4—61no—
104C3×C4—62no—
105C3×C4—61no—
106C3×C4—541no—
107C3×C4C2×AGL(1,ℤ/9)13yes—
108C3×C4C2×C3×C3⋊D311yescharacteristic
109Dic3—183no—
110Dic3—181no—
111C32—21no—
112C32—33no—
113C32C3×#21611yescharacteristic
114C32#375611yescharacteristic
115C9—23no—
116C9C3×#21613yescomplemented
117Q8—271no—
118Q8C3⋊AGL(1,ℤ/9)11yescharacteristic
119C8—271no—
120C2×C3—31no—
121C2×C3—61no—
122C2×C3#611⋊C313yes—
123C2×C3C3×#63811yescharacteristic
124C4—21no—
125C4—541no—
126C4C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
127C3—31no—
128C3—61no—
129C3#3674⋊C313yescomplemented
130C3C3×#375611yescharacteristic
131C2#1061998011yescharacteristic
subgroupG/H
centerC2#10619980
commutator, cosocleC3×C4×C9C22×C3
FrattiniC3×C4C2×C3×C3⋊D3
FittingC3×Q8×3−1+2C2
socleC2×C32C3×#67
prime radicalC2×C33#67
hypercenterQ8C3⋊AGL(1,ℤ/9)
hypocenterC3×C9C3×Q16
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
5. #88547C3semidirect
39. C3×3−1+2Q16semidirect
80. 3−1+2#216semidirect
82. C3×C9C3×Q16semidirect
116. C9C3×#216semidirect
129. C3#3674⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×3−1+22
#106199622
C3⋊(3−1+2⋊C8)2
C3×#37563
#885473
#3674⋊C33