Finite Group Databasebeta

← #13047049

Subgroups — #13047049

Order 1296.

141 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 141 conjugacy classes of subgroups — too many to draw. Only its 17 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
1#10619580C211yesmaximal, characteristic, complemented
2C3×C3⋊(#41⋊C3)—31nomaximal
3#88748C311yesmaximal, characteristic, complemented
4#66321—41nomaximal
5C32×SL(2,𝔽3)D311yescharacteristic, complemented
6Q8×H(3,𝔽3)C2×C311yescharacteristic
7C4⋊#272—31no—
8C4×#272—31no—
9#6204—21no—
10C2×AΣL(1,𝔽27)—41no—
11C3×#239—31no—
12C3×#239—91no—
13C3×#41⋊C3—91no—
14C3⋊(#41⋊C3)—33no—
15C32×Dic3—121no—
16C2×#272—31no—
17C4×H(3,𝔽3)—31no—
18#1084A411yescharacteristic
19AΣL(1,𝔽27)—41noSylow
20C32×Q8—31no—
21C32×Q8C3×D311yescharacteristic
22C3×D12—91no—
23C3×D12—93no—
24C3×C4×D3—91no—
25C3×C4×D3—93no—
26C3×SL(2,𝔽3)—31no—
27C3×SL(2,𝔽3)—33no—
28Q8⋊C9—61no—
29C2×C33—41no—
30C2×3−1+2—81no—
31C2×H(3,𝔽3)C2×A411yescharacteristic
32#272—61no—
33C3×#41—91no—
34#239—31no—
35#239—91no—
36#41⋊C3—93no—
37C2×C3×D3—93no—
38C2×C3×D3—91no—
39C32×C4—33no—
40C32×C4—31no—
41C32×C4—361no—
42C3×Dic3—31no—
43C3×Dic3—91no—
44C3×Dic3—123no—
45C33—41no—
463−1+2—81no—
47H(3,𝔽3)#41⋊C311yescharacteristic, complemented
48C3×Q8—31no—
49C3×Q8—91no—
50C3×Q8(C3⋊D3)⋊C311yescharacteristic
51C3×D4—271no—
52C2×C3×C4—271no—
53D12—91no—
54D12—93no—
55C4×D3—91no—
56C4×D3—93no—
57SL(2,𝔽3)—33no—
58C2×C32—31no—
59C2×C32—123no—
60C2×C32—121no—
61C2×C32D3×A411yescharacteristic
62C3×D3—183no—
63C3×D3—181no—
64C2×C9—241no—
65#41—91noSylow
66C22×C3—271no—
67C2×D3—91no—
68C2×D3—93no—
69C3×C4—31no—
70C3×C4—91no—
71C3×C4—93no—
72C3×C4—91no—
73C3×C4—363no—
74Dic3—31no—
75Dic3—91no—
76C32—31no—
77C32—123no—
78C32—121no—
79C32C3⋊(#41⋊C3)11yescharacteristic
80C9—241no—
81Q8#417411yescharacteristic
82D4—271no—
83C2×C4—271no—
84C2×C3—31no—
85C2×C3—91no—
86C2×C3—123no—
87C2×C3—541no—
88C2×C3(C3⋊D3)⋊A411yescharacteristic
89D3—181no—
90D3—183no—
91C22—271no—
92C4—31no—
93C4—91no—
94C3—31no—
95C3—91no—
96C3—123no—
97C3#8862211yescharacteristic
98C2—541no—
99C2#1061977311yescharacteristic
subgroupG/H
center, socle, hypercenterC2×C3(C3⋊D3)⋊A4
commutator, Fitting, cosocle, hypocenterQ8×H(3,𝔽3)C2×C3
FrattiniC2×C32D3×A4
LCSG ▷ Q8×H(3,𝔽3)
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. #10619580C2semidirect
3. #88748C3semidirect
5. C32×SL(2,𝔽3)D3semidirect
47. H(3,𝔽3)#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
#106195802
C3×C3⋊(#41⋊C3)3
#887483
#663214