Finite Group Databasebeta

← #10590344

Subgroups — #10590344

Order 576.

284 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C3×C4×D12C212yesmaximal, complemented
2C2×#3713C211yesmaximal, characteristic
3#63630C211yesmaximal, characteristic, complemented
4#63574C211yesmaximal, characteristic, complemented
5#63562C212yesmaximal, complemented
6#5014—32nomaximal
7C2×C3×D12C2212yes—
8C2×C3×C4×D3—22no—
9C32×C42C2211yescharacteristic
10C2×C3⋊(C3⋊C8)C2211yescharacteristic
11C2×C3⋊(C3⋊C8)C2211yescharacteristic
12C3×(C2×D3)⋊C4—22no—
13C3×C4⋊Dic3C2212yes—
14#3713C414yescomplemented
15C3×C4×D4—32no—
16C2×#213—32no—
17C4×D12D312yescomplemented
18#832—32no—
19#809—32no—
20#808—32no—
21#803—32no—
22#803—61no—
23C22×C3×D3—22no—
24C2×C32×C4D412yes—
25C2×C32×C4C2311yescharacteristic
26C2×C3×Dic3—22no—
27C3×D12—22no—
28C3×D12C2×C414yescomplemented
29C3×C4×D3—44no—
30C3⋊(C3⋊C8)—21no—
31C3⋊(C3⋊C8)C2×C412yescomplemented
32#442—91noSylow
33C2×C3×D4—32no—
34C22×C3×C4—62no—
35C2×D12C2×D312yes—
36C2×C4×D3—22no—
37C3×C4⋊C4—32no—
38C3×C22⋊C4—62no—
39C3×C42—21no—
40C3×C42C2×D312yes—
41#213—38no—
42(C2×D3)⋊C4—22no—
43C4⋊Dic3C2×D312yes—
44C2×C3⋊C8—32no—
45C2×C3⋊C8—32no—
46C2×C3⋊C8—61no—
47C2×C3⋊C8—61no—
48C22×C32C2×D411yescharacteristic
49C2×C3×D3—24no—
50C2×C3×D3—42no—
51C2×C3×D3—42no—
52C32×C4—22no—
53C32×C4C22×C411yescharacteristic
54C32×C4#4111yescharacteristic
55C3×Dic3—42no—
56C2×D8—91no—
57C4×D4—32no—
58#106—91no—
59#102—92no—
60#97—91no—
61C23×C3—62no—
62C22×D3—22no—
63C3×D4—34no—
64C3×D4—62no—
65C2×C3×C4—22no—
66C2×C3×C4—21no—
67C2×C3×C4—62no—
68C2×C3×C4—124no—
69C2×C3×C4#6712yes—
70C2×C3×C4#6712yes—
71C2×C3×C4C22×D312yes—
72C2×Dic3—22no—
73D12—22no—
74D12C4×D314yescomplemented
75C4×D3—44no—
76C3⋊C8—34no—
77C3⋊C8—62no—
78C3⋊C8—62no—
79C3⋊C8—62no—
80C2×C32C4×D411yescharacteristic
81C2×C32AGL(1,ℤ/8)12yes—
82C3×D3—44no—
83C2×D4—32no—
84C22×C4—62no—
85D8—94no—
86C2×C8—91no—
87C2×C8—91no—
88C4⋊C4—32no—
89C22⋊C4—62no—
90C42D3211yescharacteristic
91C22×C3—21no—
92C22×C3—64no—
93C22×C3—122no—
94C22×C3—122no—
95C22×C3C2×#6712yes—
96C2×D3—24no—
97C2×D3—42no—
98C2×D3—42no—
99C3×C4—22no—
100C3×C4—22no—
101C3×C4—21no—
102C3×C4—21no—
103C3×C4—42no—
104C3×C4—122no—
105C3×C4C2×C4×D312yes—
106C3×C4#23512yes—
107Dic3—42no—
108C32#44211yescharacteristic, complemented, Sylow
109C23—62no—
110D4—34no—
111D4—62no—
112C2×C4—62no—
113C2×C4—124no—
114C2×C4#62512yes—
115C2×C4C2×D3211yescharacteristic
116C8—92no—
117C8—181no—
118C2×C3—22no—
119C2×C3—21no—
120C2×C3—124no—
121C2×C3C4×#6712yes—
122C2×C3#93212yes—
123C2×C3#94912yes—
124D3—44no—
125C22—64no—
126C22—122no—
127C22—122no—
128C22C2×#62511yescharacteristic
129C4—22no—
130C4—122no—
131C4#379811yescharacteristic
132C4C4×D3211yescharacteristic
133C3—21no—
134C3#501412yescomplemented
135C2—124no—
136C2#6382312yes—
137C2C4×#62511yescharacteristic
subgroupG/H
centerC22C2×#625
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC32×C42C22
socleC22×C32C2×D4
prime radicalC2×#3713C2
cosocleC2×C32×C4C23
hypercenterC42D32
hypocenterC32#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×D12C2semidirect
3. #63630C2semidirect
4. #63574C2semidirect
5. #63562C2semidirect
14. #3713C4semidirect
17. C4×D12D3semidirect
28. C3×D12C2×C4semidirect
31. C3⋊(C3⋊C8)C2×C4semidirect
74. D12C4×D3semidirect
108. C32#442semidirect
134. C3#5014semidirect

Maximal subgroups

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

subgroupindex
C3×C4×D122
C2×#37132
#636302
#635742
#635622
#50143