Finite Group Databasebeta

← #10592570

Subgroups — #10592570

Order 576.

306 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 87 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#64102C211yesmaximal, characteristic
2#64099C211yesmaximal, characteristic, complemented
3C32×#106C211yesmaximal, characteristic, complemented
4#63648C211yesmaximal, characteristic
5#63641C211yesmaximal, characteristic, complemented
6#63636C211yesmaximal, characteristic
7#63634C211yesmaximal, characteristic, complemented
8#4871—34nomaximal
9C2×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C4×C3⋊D3—21no—
11C2×C32×C8C2211yescharacteristic
12C32×C4⋊C4C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14#3752—21no—
15C4⋊(C3⋊Dic3)—21no—
16C4⋊(C3⋊Dic3)C2211yescharacteristic
17#3750—21no—
18C4×C3⋊Dic3C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20#893—34no—
21#890—34no—
22C3×#106D314yescomplemented
23#821—34no—
24Dic3⋊C8—34no—
25#809—34no—
26#807—34no—
27C22×C3⋊D3—21no—
28C2×C32×C4—21no—
29C2×C32×C4—21no—
30C2×C32×C4C2311yescharacteristic
31C2×C3⋊Dic3—21no—
32C2×C3⋊Dic3D411yescharacteristic
33C2×C3⋊Dic3D411yescharacteristic
34C4⋊(C3⋊D3)—21no—
35C4⋊(C3⋊D3)Q812yes—
36C4×C3⋊D3—42no—
37C32×C8—21no—
38C3⋊(C3⋊C8)—21no—
39#478—91noSylow
40C2×D12—34no—
41C2×C4×D3—64no—
42C2×C3×C8C2×D314yes—
43C3×C4⋊C4C2×D314yes—
44C3×C4⋊C4C2×D314yes—
45(C2×D3)⋊C4—64no—
46C4⋊Dic3—34no—
47C4⋊Dic3—64no—
48Dic3⋊C4—64no—
49C4×Dic3—34no—
50C2×C3⋊C8—34no—
51C22×C32C2×D411yescharacteristic
52C2×C3⋊D3—22no—
53C2×C3⋊D3—41no—
54C2×C3⋊D3—41no—
55C32×C4—41no—
56C32×C4—41no—
57C32×C4C2×Q811yescharacteristic
58C32×C4#4111yescharacteristic
59C3⋊Dic3—21no—
60C3⋊Dic3—21no—
61C3⋊Dic3—41no—
62#125—91no—
63C4×D4—91no—
64#107—91no—
65#106C3⋊D311yescharacteristic, complemented
66C4⋊C8—91no—
67#102—91no—
68#102—91no—
69C22×D3—64no—
70C2×C3×C4—24no—
71C2×C3×C4—24no—
72C2×C3×C4C22×D314yes—
73C2×Dic3—34no—
74C2×Dic3—34no—
75C2×Dic3—64no—
76D12—38no—
77D12—64no—
78C4×D3—128no—
79C3×C8—24no—
80C3⋊C8—64no—
81C2×C32C22⋊Q811yescharacteristic
82C2×C32#13511yescharacteristic
83C2×C32AGL(1,ℤ/8)11yescharacteristic
84C3⋊D3—42no—
85C2×D4—91no—
86C22×C4—181no—
87C2×C8—91no—
88C2×C8C2×C3⋊D311yescharacteristic
89C4⋊C4—91no—
90C4⋊C4—181no—
91C4⋊C4—181no—
92C4⋊C4C2×C3⋊D311yescharacteristic
93C4⋊C4C2×C3⋊D311yescharacteristic
94C22⋊C4—181no—
95C42—91no—
96C22×C3D3×D414yes—
97C2×D3—68no—
98C2×D3—124no—
99C2×D3—124no—
100C3×C4—44no—
101C3×C4—44no—
102C3×C4#23514yes—
103C3×C4D3×Q814yes—
104Dic3—64no—
105Dic3—64no—
106Dic3—124no—
107C32#47811yescharacteristic, complemented, Sylow
108C23—181no—
109D4—92no—
110D4—181no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—91no—
114C2×C4—91no—
115C2×C4—181no—
116C2×C4—362no—
117C2×C4C22×C3⋊D311yescharacteristic
118C8—21no—
119C8—181no—
120C2×C3#89614yes—
121C2×C3#91414yes—
122C2×C3#91614yes—
123D3—128no—
124C22—182no—
125C22—361no—
126C22—361no—
127C22D4×C3⋊D311yescharacteristic
128C4—41no—
129C4—41no—
130C4—181no—
131C4—181no—
132C4—361no—
133C4#382811yescharacteristic
134C4Q8×C3⋊D311yescharacteristic
135C3#487114yescomplemented
136C2—362no—
137C2#6410511yescharacteristic
138C2#6412311yescharacteristic
139C2#6412511yescharacteristic
subgroupG/H
centerC22D4×C3⋊D3
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×#106C2
socleC22×C32C2×D4
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenter#106C3⋊D3
hypocenterC32#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #106

Complements

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

NH ≅ G/Ntype
2. #64099C2semidirect
3. C32×#106C2semidirect
5. #63641C2semidirect
7. #63634C2semidirect
22. C3×#106D3semidirect
65. #106C3⋊D3semidirect
107. C32#478semidirect
135. C3#4871semidirect

Maximal subgroups

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

subgroupindex
#641022
#640992
C32×#1062
#636482
#636412
#636362
#636342
#48713