Finite Group Databasebeta

← #10593191

Subgroups — #10593191

Order 576.

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

normal subgroups only

This group has 169 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
1C2×C3⋊(#41⋊C3)C211yesmaximal, characteristic
2C3⋊#859C211yesmaximal, characteristic, complemented
3C3×#859C211yesmaximal, characteristic, complemented
4#5424—31nomaximal
5#4804—31nomaximal
6C2×C3×SL(2,𝔽3)C2211yescharacteristic
7C3⋊(#41⋊C3)C412yescomplemented
8Dic32—41nomaximal
9C2×#239D311yescharacteristic
10C2×#41⋊C3—31no—
11C2×D3×C8—31no—
12#892—31no—
13#859—31no—
14#859—61no—
15#859D311yescharacteristic, complemented
16C3×#103—31no—
17#835—31no—
18#821—31no—
19#809—31no—
20C2×C3⋊Dic3—41no—
21C2×C3×Dic3—41no—
22C2×C3×Dic3—41no—
23C3×SL(2,𝔽3)C2×C411yescharacteristic
24#416—91noSylow
25C2×C3×Q8C2×D311yescharacteristic
26#239—32no—
27#239Dic312yescomplemented
28C2×D12—31no—
29C2×C4×D3—31no—
30#41⋊C3—32no—
31C2×SL(2,𝔽3)—21no—
32C2×SL(2,𝔽3)C2×D311yescharacteristic
33C2×C3×C8—31no—
34C3×C4⋊C4—31no—
35(C2×D3)⋊C4—61no—
36C4⋊Dic3—31no—
37C4×Dic3—61no—
38C4×Dic3—121no—
39C2×C3⋊C8—31no—
40D3×C8—62no—
41C22×C32—41no—
42C3⋊Dic3—42no—
43C3×Dic3—42no—
44C3×Dic3—42no—
45C2×#41—31no—
46C22×C8—91no—
47#117—91no—
48#103—31no—
49#103—91no—
50#102—91no—
51#102—91no—
52C22×D3—31no—
53C3×Q8—31no—
54C3×Q8C2×Dic311yescharacteristic
55C2×C3×C4—31no—
56C2×C3×C4—61no—
57C2×C3×C4—121no—
58C2×Dic3—41no—
59C2×Dic3—61no—
60C2×Dic3—121no—
61C2×Dic3—241no—
62C2×Dic3S411yescharacteristic
63D12—32no—
64D12—61no—
65C4×D3—32no—
66C4×D3—32no—
67SL(2,𝔽3)—21no—
68SL(2,𝔽3)C4×D311yescharacteristic
69C3×C8—61no—
70C3⋊C8—61no—
71C2×C32—41no—
72C2×C32—41no—
73C2×C32—41no—
74#41—32no—
75#41—92no—
76C2×Q8D3211yescharacteristic
77C2×D4—91no—
78C22×C4—91no—
79C2×C8—31no—
80C2×C8—91no—
81C2×C8—182no—
82C4⋊C4—31no—
83C4⋊C4—91no—
84C22⋊C4—181no—
85C42—181no—
86C22×C3—41no—
87C22×C3—81no—
88C22×C3C2×S411yescharacteristic
89C2×D3—32no—
90C2×D3—61no—
91C2×D3—61no—
92C3×C4—31no—
93C3×C4—31no—
94C3×C4—122no—
95C3×C4—121no—
96Dic3—42no—
97Dic3—122no—
98Dic3—121no—
99Dic3—242no—
100Dic3SL(2,ℤ/4)12yes—
101C32—41noSylow
102C23—91no—
103Q8—31no—
104Q8D3×Dic311yescharacteristic
105D4—92no—
106D4—181no—
107C2×C4—31no—
108C2×C4—31no—
109C2×C4—61no—
110C2×C4—92no—
111C2×C4—92no—
112C2×C4—181no—
113C8—61no—
114C8—181no—
115C2×C3—41no—
116C2×C3—41no—
117C2×C3—41no—
118C2×C3—81no—
119C2×C3—81no—
120C2×C3—81no—
121C2×C3#98511yescharacteristic
122C2×C3#98511yescharacteristic
123C2×C3C2×SL(2,ℤ/4)11yescharacteristic
124D3—62no—
125C22—92no—
126C22—181no—
127C22—181no—
128C22D3×S411yescharacteristic
129C4—31no—
130C4—31no—
131C4—32no—
132C4—121no—
133C4—361no—
134C3—41no—
135C3—81no—
136C3#542411yescharacteristic, complemented
137C2—182no—
138C2#6419711yescharacteristic
139C2#6419811yescharacteristic
140C2D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C4
FittingC2×C3×Q8C2×D3
socleC22×C3C2×S4
prime radicalC2×C3⋊(#41⋊C3)C2
cosocleC2×C3×SL(2,𝔽3)C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3⋊#859C2semidirect
3. C3×#859C2semidirect
7. C3⋊(#41⋊C3)C4semidirect
15. #859D3semidirect
27. #239Dic3semidirect
136. C3#5424semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(#41⋊C3)2
C3⋊#8592
C3×#8592
#54243
#48043
Dic324