Finite Group Databasebeta

← #13046413

Subgroups — #13046413

Order 1296.

192 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 24 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
1C4⋊#4183C211yesmaximal, characteristic, complemented
2C4⋊#4174C211yesmaximal, characteristic, complemented
3#10619501C211yesmaximal, characteristic, complemented
4#88789—31nomaximal
5#88436—31nomaximal
6C2×#4183—21no—
7C2×#4174—21no—
8C4×AΣL(1,𝔽27)C2211yescharacteristic
9C3×C4⋊(C3⋊D3)—31no—
10C32×D12—31no—
11C3⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
12C4⋊#272D311yescharacteristic, complemented
13C4⋊AGL(1,ℤ/9)—61no—
14C4⋊((C3⋊D3)⋊C3)—31no—
15(C3⋊(C3⋊C8))⋊C3—31no—
16C2×AΣL(1,𝔽27)D411yescharacteristic
17#4183—41no—
18#4174—41no—
19#3753—91no—
20#3714—91no—
21#3713—91no—
22#3713—93no—
23C2×C3×C3⋊D3—61no—
24C2×C32×D3—61no—
25C33×C4C2×D311yescharacteristic
26C2×#272—21no—
27C2×AGL(1,ℤ/9)—62no—
28C2×(C3⋊D3)⋊C3—61no—
29C4×3−1+2—21no—
30C4×H(3,𝔽3)C2×D311yescharacteristic
31AΣL(1,𝔽27)D811yescharacteristic, complemented, Sylow
32C32×D4—91no—
33C4⋊(C3⋊D3)—31no—
34C4⋊(C3⋊D3)—31no—
35C3×D12—33no—
36C3×D12—31no—
37C3×D12—91no—
38C3×D12—93no—
39C3×D12—91no—
40C3⋊(C3⋊C8)—31no—
41C3⋊(C3⋊C8)—91no—
42C3⋊(C3⋊C8)—93no—
43C3×C3⋊C8—91no—
44D36—61no—
45C2×C33D1211yescharacteristic
46C3×C3⋊D3—121no—
47C32×D3—121no—
48C2×3−1+2—21no—
49C2×H(3,𝔽3)#6711yescharacteristic
50#272—41no—
51AGL(1,ℤ/9)—122no—
52(C3⋊D3)⋊C3—121no—
53#213—271no—
54#213—271no—
55#213—273no—
56D24—271no—
57C22×C32—181no—
58C2×C3⋊D3—61no—
59C2×C3⋊D3—61no—
60C2×C3×D3—63no—
61C2×C3×D3—61no—
62C2×C3×D3—181no—
63C2×C3×D3—183no—
64C2×C3×D3—181no—
65C32×C4—31no—
66C32×C4—33no—
67C32×C4—31no—
68C32×C4D3211yescharacteristic
69C2×D9—62no—
70C4×C9—61no—
71C33D2411yescharacteristic, complemented
723−1+2—21no—
73H(3,𝔽3)#21311yescharacteristic, complemented
74C3×D4—93no—
75C3×D4—91no—
76C3×D4—271no—
77D12—31no—
78D12—91no—
79D12—93no—
80D12—91no—
81D12—271no—
82C3×C8—271no—
83C3⋊C8—91no—
84C3⋊C8—271no—
85C3⋊C8—273no—
86C2×C32—31no—
87C2×C32—33no—
88C2×C32—31no—
89C2×C32—361no—
90C2×C32#62611yescharacteristic
91C3⋊D3—121no—
92C3⋊D3—121no—
93C3×D3—123no—
94C3×D3—121no—
95C3×D3—361no—
96C3×D3—363no—
97C3×D3—361no—
98C2×C9—61no—
99D9—122no—
100D8—811noSylow
101C22×C3—183no—
102C22×C3—181no—
103C22×C3—541no—
104C2×D3—61no—
105C2×D3—181no—
106C2×D3—183no—
107C2×D3—181no—
108C2×D3—541no—
109C3×C4—31no—
110C3×C4—33no—
111C3×C4—91no—
112C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
113C32—33no—
114C32—31no—
115C32—31no—
116C32#371411yescharacteristic
117C9—61no—
118D4—91no—
119D4—271no—
120C8—271no—
121C2×C3—31no—
122C2×C3—33no—
123C2×C3—91no—
124C2×C3—363no—
125C2×C3—361no—
126C2×C3—1081no—
127C2×C3#620011yescharacteristic
128D3—121no—
129D3—361no—
130D3—363no—
131D3—361no—
132D3—1081no—
133C22—181no—
134C22—541no—
135C4#4174⋊C211yescharacteristic
136C3—31no—
137C3—33no—
138C3—91no—
139C3#8843611yescharacteristic
140C2—361no—
141C2—1081no—
142C2#1061954911yescharacteristic
subgroupG/H
centerC2#10619549
commutator, Fitting, cosocleC4×AΣL(1,𝔽27)C22
FrattiniC32×C4D32
socleC2×C3#6200
hypercenterC4#4174⋊C2
hypocenterAΣL(1,𝔽27)D8
LCSG ▷ C4×AΣL(1,𝔽27) ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27)
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4⋊#4183C2semidirect
2. C4⋊#4174C2semidirect
3. #10619501C2semidirect
11. C3⋊(C3⋊(C3⋊C8))D3semidirect
12. C4⋊#272D3semidirect
31. AΣL(1,𝔽27)D8semidirect
71. C33D24semidirect
73. H(3,𝔽3)#213semidirect

Maximal subgroups

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

subgroupindex
C4⋊#41832
C4⋊#41742
#106195012
#887893
#884363