Finite Group Databasebeta

← #10593639

Subgroups — #10593639

Order 576.

160 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 31 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#64266C211yesmaximal, characteristic
2C3×#131⋊C3C211yesmaximal, characteristic, complemented
3C3⋊GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#5405—33nomaximal
5#5145—31nomaximal
6C3×#41⋊C3C2211yescharacteristic
7C3⋊GL(2,𝔽3)C411yescharacteristic, complemented
8C3⋊#226C411yescharacteristic, complemented
9#3743—41nomaximal
10#985—33no—
11C3×#131D311yescharacteristic, complemented
12#950—31no—
13#901—31no—
14#131⋊C3D313yescomplemented
15GU(2,𝔽3)—33no—
16#837—31no—
17#825—31no—
18#822—31no—
19#815—31no—
20C4×C3⋊D3—41no—
21C3×SL(2,𝔽3)C2×C411yescharacteristic
22C32×C8—41no—
23C3⋊(C3⋊C8)—41no—
24#444—91noSylow
25C3×#41C2×D311yescharacteristic
26#235—31no—
27#41⋊C3C2×D313yes—
28GL(2,𝔽3)—33no—
29#226—33no—
30C3×M16—31no—
31C2×C3×C8—31no—
32#216—31no—
33#215—31no—
34#214—31no—
35#213—31no—
36C4×Dic3—31no—
37#208—31no—
38C2×C3⋊C8—31no—
39ΓL(1,𝔽25)—61no—
40ΓL(1,𝔽25)—123no—
41D3×C8—61no—
42C2×C3⋊D3—41no—
43C32×C4—41no—
44C3⋊Dic3—41no—
45#135—91no—
46#131—91no—
47#131C3⋊D311yescharacteristic, complemented
48#108—91no—
49#104—91no—
50#104—91no—
51#97—91no—
52C3×Q8C4×D311yescharacteristic
53C3×D4—31no—
54C2×C3×C4—31no—
55#67—61no—
56C2×Dic3—61no—
57D12—31no—
58C4×D3—61no—
59C4×D3—123no—
60Dic6—31no—
61SL(2,𝔽3)C4×D313yes—
62C3×C8—31no—
63C3×C8—43no—
64C3×C8S411yescharacteristic
65C3⋊C8—31no—
66C3⋊C8—31no—
67C3⋊C8—61no—
68C3⋊C8—123no—
69C2×C32—41no—
70C3⋊D3—81no—
71#41—91no—
72#41C2×C3⋊D311yescharacteristic
73Q16—91no—
74SD16—91no—
75SD16—91no—
76D8—91no—
77M16—31no—
78M16—91no—
79M16—181no—
80C2×C8—31no—
81C2×C8—91no—
82C2×C8—181no—
83C42—91no—
84C22×C3—31no—
85C2×D3—61no—
86C2×D3—123no—
87C3×C4—31no—
88C3×C4—43no—
89C3×C4C2×S411yescharacteristic
90Dic3—61no—
91Dic3—123no—
92Dic3—121no—
93C32—41noSylow
94Q8—91no—
95Q8C4×C3⋊D311yescharacteristic
96D4—31no—
97D4—91no—
98D4—181no—
99C2×C4—31no—
100C2×C4—181no—
101C2×C4—181no—
102C8—31no—
103C8—91no—
104C8—91no—
105C8—181no—
106C8C3⋊S411yescharacteristic
107C2×C3—43no—
108C2×C3—61no—
109C2×C3C4×S411yescharacteristic
110D3—121no—
111D3—243no—
112C22—31no—
113C22—181no—
114C4—31no—
115C4—181no—
116C4—361no—
117C4C2×C3⋊S411yescharacteristic
118C3—43no—
119C3#540511yescharacteristic, complemented
120C2—61no—
121C2—361no—
122C2C4×C3⋊S411yescharacteristic
subgroupG/H
center, FrattiniC4C2×C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C4
FittingC3×#131D3
socleC2×C3C4×S4
prime radical#64266C2
cosocleC3×#41⋊C3C22
hypercenterC8C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C3×#131⋊C3C2semidirect
3. C3⋊GU(2,𝔽3)C2semidirect
7. C3⋊GL(2,𝔽3)C4semidirect
8. C3⋊#226C4semidirect
11. C3×#131D3semidirect
14. #131⋊C3D3semidirect
47. #131C3⋊D3semidirect
119. C3#5405semidirect

Maximal subgroups

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

subgroupindex
#642662
C3×#131⋊C32
C3⋊GU(2,𝔽3)2
#54053
#51453
#37434