Finite Group Databasebeta

← #13047462

Subgroups — #13047462

Order 1296.

400 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 59 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, complemented
2C4⋊#4183C213yesmaximal, complemented
3C4×#4183C213yesmaximal, complemented
4C3×#3832—31nomaximal
5#88724—32nomaximal
6#88722—31nomaximal
7C2×#4183C2213yes—
8C4×AΣL(1,𝔽27)C2213yes—
9#66330C2211yescharacteristic
10C33×Q8D311yescharacteristic, complemented
11C3×C4⋊(C3⋊D3)—33no—
12C3×C4×C3⋊D3—33no—
13Q8×3−1+2D312yescomplemented
14Q8×H(3,𝔽3)D311yescharacteristic, complemented
15C4⋊AGL(1,ℤ/9)—36no—
16C4×AGL(1,ℤ/9)—36no—
17C4⋊((C3⋊D3)⋊C3)—33no—
18C4×(C3⋊D3)⋊C3—33no—
19C2×AΣL(1,𝔽27)C2311yescharacteristic
20#4183—23no—
21#3832—31no—
22#3832—31no—
23C3×#239—93no—
24C3×#239—91no—
25#3701—32no—
26C2×C3×C3⋊D3—33no—
27C33×C4C2×D313yes—
28C3×C3⋊Dic3—31no—
29C2×AGL(1,ℤ/9)—36no—
30C2×(C3⋊D3)⋊C3—33no—
31C4×3−1+2C2×D316yes—
32C4×H(3,𝔽3)C2×D313yes—
333−1+2⋊C4—32no—
34(C3⋊Dic3)⋊C3—31no—
35AΣL(1,𝔽27)#4111yescharacteristic, complemented, Sylow
36C32×Q8—31no—
37C32×Q8—33no—
38C32×Q8—31no—
39C32×Q8C3⋊D311yescharacteristic
40C4⋊(C3⋊D3)—33no—
41C4⋊(C3⋊D3)—33no—
42C4×C3⋊D3—33no—
43C4×C3⋊D3—33no—
44C3×D12—99no—
45C3×D12—93no—
46C3×C4×D3—99no—
47C3×C4×D3—93no—
48Q8×C9—32no—
49D36—36no—
50C4×D9—36no—
51C2×C33C22×D311yescharacteristic
52C3×C3⋊D3—63no—
53C2×3−1+2C22×D312yes—
54C2×H(3,𝔽3)C22×D311yescharacteristic
55AGL(1,ℤ/9)—66no—
56(C3⋊D3)⋊C3—63no—
57C3×#41—271no—
58#239—93no—
59#239—91no—
60#239—271no—
61C2×C3⋊D3—33no—
62C2×C3⋊D3—33no—
63C2×C3×D3—99no—
64C2×C3×D3—93no—
65C32×C4—33no—
66C32×C4—39no—
67C32×C4—33no—
68C32×C4C2×C3⋊D313yes—
69C3⋊Dic3—31no—
70C3⋊Dic3—31no—
71C3×Dic3—93no—
72C3×Dic3—91no—
73C2×D9—36no—
74C4×C9—36no—
75Dic9—32no—
76C33#23911yescharacteristic, complemented
773−1+2#23912yescomplemented
78H(3,𝔽3)#23911yescharacteristic, complemented
79C3×Q8—31no—
80C3×Q8—33no—
81C3×Q8—91no—
82C3×Q8#27211yescharacteristic
83C3×D4—273no—
84C2×C3×C4—273no—
85D12—99no—
86D12—93no—
87D12—273no—
88C4×D3—99no—
89C4×D3—93no—
90C4×D3—273no—
91C2×C32—31no—
92C2×C32—33no—
93C2×C32—31no—
94C2×C32C22×C3⋊D311yescharacteristic
95C3⋊D3—63no—
96C3⋊D3—63no—
97C3×D3—189no—
98C3×D3—183no—
99C2×C9—32no—
100D9—66no—
101#41—271noSylow
102C22×C3—273no—
103C2×D3—99no—
104C2×D3—93no—
105C2×D3—273no—
106C3×C4—39no—
107C3×C4—33no—
108C3×C4—93no—
109C3×C4—271no—
110C3×C4C2×#27213yes—
111Dic3—93no—
112Dic3—91no—
113Dic3—271no—
114C32—33no—
115C32—31no—
116C32—31no—
117C32#383211yescharacteristic
118C9—32no—
119Q8#418311yescharacteristic, complemented
120D4—273no—
121C2×C4—273no—
122C2×C3—33no—
123C2×C3—31no—
124C2×C3—91no—
125C2×C3—543no—
126C2×C3C22×#27211yescharacteristic
127D3—189no—
128D3—183no—
129D3—543no—
130C22—273no—
131C4—271no—
132C4C2×#418313yes—
133C3—33no—
134C3—31no—
135C3—91no—
136C3#8874811yescharacteristic
137C2—543no—
138C2C22×#418311yescharacteristic
subgroupG/H
centerC2C22×#4183
commutator, cosocleC2×AΣL(1,𝔽27)C23
FrattiniC2×C32C22×C3⋊D3
FittingQ8×AΣL(1,𝔽27)C2
socleC2×C3C22×#272
hypercenterQ8#4183
hypocenterAΣL(1,𝔽27)#41
LCSG ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27)
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. Q8×AΣL(1,𝔽27)C2semidirect
2. C4⋊#4183C2semidirect
3. C4×#4183C2semidirect
10. C33×Q8D3semidirect
13. Q8×3−1+2D3semidirect
14. Q8×H(3,𝔽3)D3semidirect
35. AΣL(1,𝔽27)#41semidirect
76. C33#239semidirect
77. 3−1+2#239semidirect
78. H(3,𝔽3)#239semidirect
119. Q8#4183semidirect

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
C4⋊#41832
C4×#41832
C3×#38323
#887243
#887223