Finite Group Databasebeta

← #10590741

Subgroups — #10590741

Order 576.

284 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 72 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#63898C212yesmaximal, complemented
2C2×#3713C211yesmaximal, characteristic
3#63642C211yesmaximal, characteristic, complemented
4C3×#821C212yesmaximal, complemented
5#63574C211yesmaximal, characteristic, complemented
6#4785—32nomaximal
7C2×C3×D12C2212yes—
8C2×D3×Dic3—22no—
9C2×C32×C8C2211yescharacteristic
10C4×C3⋊Dic3C2211yescharacteristic
11C2×C3⋊(C3⋊C8)C2211yescharacteristic
12C3×C4⋊Dic3C2212yes—
13(C2×D3)⋊Dic3—22no—
14#3713C414yescomplemented
15D4×Dic3—32no—
16C2×#213—32no—
17#893—32no—
18C3×#102—32no—
19#821D312yescomplemented
20#815—32no—
21#815—61no—
22#808—32no—
23C22×C3×D3—22no—
24C2×C32×C4C2311yescharacteristic
25C2×C3⋊Dic3D412yes—
26C2×C3×Dic3—22no—
27C3×D12—22no—
28C3×D12C2×C414yescomplemented
29D3×Dic3—44no—
30C32×C8—21no—
31C3⋊(C3⋊C8)C2×C412yescomplemented
32#442—91noSylow
33C2×C3×D4—32no—
34C22×Dic3—62no—
35C2×D12C2×D312yes—
36C2×C4×D3—62no—
37C2×C3×C8—21no—
38C2×C3×C8C2×D312yes—
39C3×C4⋊C4—32no—
40#217—62no—
41#213—38no—
42(C2×D3)⋊C4—62no—
43C4⋊Dic3C2×D312yes—
44C4×Dic3—32no—
45C4×Dic3—61no—
46C2×C3⋊C8—32no—
47C2×C3⋊C8—61no—
48C22×C32C2×D411yescharacteristic
49C2×C3×D3—24no—
50C2×C3×D3—42no—
51C2×C3×D3—42no—
52C32×C4C22×C411yescharacteristic
53C32×C4#4111yescharacteristic
54C3⋊Dic3—22no—
55C3×Dic3—42no—
56C2×D8—91no—
57C4×D4—92no—
58#106—91no—
59#102—32no—
60#97—91no—
61C23×C3—62no—
62C22×D3—22no—
63C3×D4—34no—
64C3×D4—62no—
65C2×C3×C4—21no—
66C2×C3×C4—62no—
67C2×C3×C4C22×D312yes—
68C2×Dic3—22no—
69C2×Dic3—32no—
70C2×Dic3—32no—
71C2×Dic3—62no—
72C2×Dic3—124no—
73D12—22no—
74D12C4×D314yescomplemented
75C4×D3—124no—
76C3×C8—22no—
77C3×C8—22no—
78C3⋊C8—34no—
79C3⋊C8—62no—
80C2×C32C4×D411yescharacteristic
81C2×C32AGL(1,ℤ/8)12yes—
82C3×D3—44no—
83C2×D4—32no—
84C22×C4—182no—
85D8—94no—
86C2×C8—91no—
87C2×C8D3211yescharacteristic
88C4⋊C4—32no—
89C22⋊C4—182no—
90C42—91no—
91C22×C3—21no—
92C22×C3—64no—
93C22×C3—122no—
94C22×C3—122no—
95C22×C3D3×D412yes—
96C2×D3—24no—
97C2×D3—42no—
98C2×D3—42no—
99C3×C4—21no—
100C3×C4—21no—
101C3×C4—122no—
102C3×C4C2×C4×D312yes—
103C3×C4#23712yes—
104Dic3—42no—
105Dic3—62no—
106Dic3—62no—
107Dic3—122no—
108C32#44211yescharacteristic, complemented, Sylow
109C23—62no—
110D4—34no—
111D4—62no—
112C2×C4—62no—
113C2×C4—92no—
114C2×C4—364no—
115C2×C4C2×D3211yescharacteristic
116C8—21no—
117C8—92no—
118C2×C3—22no—
119C2×C3—21no—
120C2×C3—124no—
121C2×C3#67⋊C412yes—
122C2×C3#91112yes—
123C2×C3#91412yes—
124D3—44no—
125C22—64no—
126C22—122no—
127C22—122no—
128C22#380211yescharacteristic
129C4—122no—
130C4—182no—
131C4C4×D3211yescharacteristic
132C4#380511yescharacteristic
133C3—21no—
134C3#478512yescomplemented
135C2—124no—
136C2#6379812yes—
137C2#6390211yescharacteristic
subgroupG/H
centerC22#3802
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC2×#3713C2
cosocleC2×C32×C4C23
hypercenterC2×C8D32
hypocenterC32#442
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #63898C2semidirect
3. #63642C2semidirect
4. C3×#821C2semidirect
5. #63574C2semidirect
14. #3713C4semidirect
19. #821D3semidirect
28. C3×D12C2×C4semidirect
31. C3⋊(C3⋊C8)C2×C4semidirect
74. D12C4×D3semidirect
108. C32#442semidirect
134. C3#4785semidirect

Maximal subgroups

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

subgroupindex
#638982
C2×#37132
#636422
C3×#8212
#635742
#47853