Finite Group Databasebeta

← #13048390

Subgroups — #13048390

Order 1296.

146 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 28 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
1SL(2,𝔽3)×3−1+2C211yesmaximal, characteristic
2C3×C3⋊#226—31nomaximal
3C9⋊#226C311yesmaximal, characteristic, complemented
4#3688⋊C3—32nomaximal
5#3674⋊C3—31nomaximal
6C3⋊(3−1+2⋊C4)—41nomaximal
7C32×SL(2,𝔽3)D311yescharacteristic
8Q8×3−1+2D311yescharacteristic
93−1+2⋊Q8—31no—
10#6204D312yes—
11C9⋊SL(2,𝔽3)—21no—
12C9⋊SL(2,𝔽3)—22no—
13C9×SL(2,𝔽3)—21no—
14C9×SL(2,𝔽3)C2×C311yescharacteristic
153−1+2⋊C8—31no—
16C2×C3×3−1+2—41no—
17C3⋊#226—31no—
18C3×#226—93no—
19C3×#216—91no—
20#3688—32no—
21#3674—31no—
22C3×C3⋊Dic3—121no—
23C4×3−1+2—31no—
24C3⋊Dic9—41no—
253−1+2⋊C4—61no—
263−1+2⋊C4—122no—
27C3×3−1+2—41noSylow
28C32×Q8C3⋊D311yescharacteristic
29C3×Dic6—91no—
30C3×SL(2,𝔽3)—21no—
31C3×SL(2,𝔽3)—33no—
32C3×SL(2,𝔽3)C3×D311yescharacteristic
33C3×C3⋊C8—91no—
34Q8×C9—21no—
35Q8×C9C3×D311yescharacteristic
36Dic18—31no—
37Q8⋊C9—22no—
38Q8⋊C9C3×D312yes—
39C9⋊C8—31no—
40C2×C33—41no—
41C2×3−1+2—42no—
42C2×3−1+2—82no—
43C2×3−1+2—81no—
44C2×3−1+2S411yescharacteristic
45C2×C3×C9—41no—
46C2×C3×C9—81no—
47#226—93no—
48C3×Q16—271no—
49#216—91no—
50C32×C4—31no—
51C3⋊Dic3—121no—
52C3×Dic3—181no—
53C3×Dic3—363no—
54C4×C9—31no—
55C4×C9—61no—
56Dic9—61no—
57Dic9—122no—
58C33—41no—
593−1+2—42no—
603−1+2—82no—
613−1+2—81no—
623−1+2#22611yescharacteristic, complemented
63C3×C9—41no—
64C3×C9—81no—
65C3×Q8—31no—
66C3×Q8—271no—
67C3×Q8C3×C3⋊D311yescharacteristic
68Dic6—91no—
69SL(2,𝔽3)—61no—
70SL(2,𝔽3)AGL(1,ℤ/9)13yes—
71C3×C8—271no—
72C3⋊C8—91no—
73C2×C32—41no—
74C2×C32—81no—
75C2×C32—123no—
76C2×C32C3⋊S411yescharacteristic
77C2×C9—21no—
78C2×C9—42no—
79C2×C9—82no—
80C2×C9C3×S411yescharacteristic
81Q16—271noSylow
82C3×C4—31no—
83C3×C4—91no—
84C3×C4—541no—
85Dic3—181no—
86Dic3—363no—
87C32—41no—
88C32—81no—
89C32—123no—
90C32C3⋊#22611yescharacteristic
91C9—21no—
92C9—42no—
93C9—82no—
94C9C3×#22611yescharacteristic, complemented
95Q8—271no—
96Q8C3⋊AGL(1,ℤ/9)11yescharacteristic
97C8—271no—
98C2×C3—31no—
99C2×C3—43no—
100C2×C3—241no—
101C2×C3C3×C3⋊S411yescharacteristic
102C4—31no—
103C4—541no—
104C3—31no—
105C3—43no—
106C3—241no—
107C3C3×C3⋊#22611yescharacteristic
108C2A4⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
center, hypercenterC2A4⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9×SL(2,𝔽3)C2×C3
Frattini, socleC2×C3C3×C3⋊S4
FittingQ8×3−1+2D3
prime radicalC32×SL(2,𝔽3)D3
LCSG ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
3. C9⋊#226C3semidirect
62. 3−1+2#226semidirect
94. C9C3×#226semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×3−1+22
C3×C3⋊#2263
C9⋊#2263
#3688⋊C33
#3674⋊C33
C3⋊(3−1+2⋊C4)4