Finite Group Databasebeta

← #13046825

Subgroups — #13046825

Order 1296.

180 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 35 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
1Q8×AΣL(1,𝔽27)C211yesmaximal, characteristic
2#10619660C211yesmaximal, characteristic
3#10619509C211yesmaximal, characteristic
4C3×#3756—31nomaximal
5#3674⋊C3—32nomaximal
6#3756⋊C3—31nomaximal
7C4×AΣL(1,𝔽27)—21no—
8C4×AΣL(1,𝔽27)C2211yescharacteristic
9#66330—21no—
10C33×Q8D311yescharacteristic
11C3×#634—31no—
12C3×C3⋊(C3⋊C8)—31no—
13Q8×3−1+2D312yes—
14Q8×H(3,𝔽3)D311yescharacteristic
153−1+2⋊Q8—32no—
16#634⋊C3—31no—
173−1+2⋊C8—32no—
18(C3⋊(C3⋊C8))⋊C3—31no—
19C2×AΣL(1,𝔽27)D411yescharacteristic
20#3756—31no—
21#3756—31no—
22C3×#216—93no—
23C3×#216—91no—
24#3674—32no—
25C33×C4—21no—
26C33×C4C2×D311yescharacteristic
27C3×C3⋊Dic3—61no—
28C4×3−1+2—22no—
29C4×3−1+2C2×D312yes—
30C4×H(3,𝔽3)—21no—
31C4×H(3,𝔽3)C2×D311yescharacteristic
323−1+2⋊C4—62no—
33(C3⋊Dic3)⋊C3—61no—
34AΣL(1,𝔽27)Q1611yescharacteristic, complemented, Sylow
35C32×Q8—31no—
36C32×Q8—33no—
37C32×Q8—31no—
38C32×Q8C3⋊D311yescharacteristic
39#634—31no—
40#634—31no—
41C3×Dic6—93no—
42C3×Dic6—91no—
43C3⋊(C3⋊C8)—31no—
44C3⋊(C3⋊C8)—31no—
45C3×C3⋊C8—93no—
46C3×C3⋊C8—91no—
47Q8×C9—32no—
48Dic18—32no—
49C9⋊C8—32no—
50C2×C33#6711yescharacteristic
51C2×3−1+2#6712yes—
52C2×H(3,𝔽3)#6711yescharacteristic
53C3×Q16—271no—
54#216—93no—
55#216—91no—
56#216—271no—
57C32×C4—21no—
58C32×C4—31no—
59C32×C4—33no—
60C32×C4—31no—
61C32×C4—61no—
62C32×C4—61no—
63C32×C4—63no—
64C32×C4C2×C3⋊D311yescharacteristic
65C3⋊Dic3—61no—
66C3⋊Dic3—61no—
67C3×Dic3—183no—
68C3×Dic3—181no—
69C4×C9—32no—
70C4×C9—62no—
71Dic9—62no—
72C33#21611yescharacteristic, complemented
733−1+2#21612yescomplemented
74H(3,𝔽3)#21611yescharacteristic, complemented
75C3×Q8—31no—
76C3×Q8—33no—
77C3×Q8—91no—
78C3×Q8—271no—
79C3×Q8#27211yescharacteristic
80Dic6—93no—
81Dic6—91no—
82Dic6—271no—
83C3×C8—271no—
84C3⋊C8—93no—
85C3⋊C8—91no—
86C3⋊C8—271no—
87C2×C32—31no—
88C2×C32—33no—
89C2×C32—31no—
90C2×C32#63811yescharacteristic
91C2×C9—32no—
92Q16—271noSylow
93C3×C4—21no—
94C3×C4—33no—
95C3×C4—31no—
96C3×C4—61no—
97C3×C4—63no—
98C3×C4—91no—
99C3×C4—181no—
100C3×C4—541no—
101C3×C4C2×#27211yescharacteristic
102Dic3—183no—
103Dic3—181no—
104Dic3—541no—
105C32—33no—
106C32—31no—
107C32—31no—
108C32#375611yescharacteristic
109C9—32no—
110Q8—271no—
111Q8#418311yescharacteristic
112C8—271no—
113C2×C3—33no—
114C2×C3—31no—
115C2×C3—91no—
116C2×C3#623511yescharacteristic
117C4—21no—
118C4—541no—
119C4C2×#418311yescharacteristic
120C3—33no—
121C3—31no—
122C3—91no—
123C3#8855011yescharacteristic
124C2#1061967911yescharacteristic
subgroupG/H
centerC2#10619679
commutator, cosocleC4×AΣL(1,𝔽27)C22
FrattiniC32×C4C2×C3⋊D3
FittingQ8×AΣL(1,𝔽27)C2
socleC2×C3#6235
prime radicalC2×AΣL(1,𝔽27)D4
hypercenterQ8#4183
hypocenterAΣL(1,𝔽27)Q16
LCSG ▷ C4×AΣL(1,𝔽27) ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
34. AΣL(1,𝔽27)Q16semidirect
72. C33#216semidirect
73. 3−1+2#216semidirect
74. H(3,𝔽3)#216semidirect

Maximal subgroups

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

subgroupindex
Q8×AΣL(1,𝔽27)2
#106196602
#106195092
C3×#37563
#3674⋊C33
#3756⋊C33