Finite Group Databasebeta

← #10592561

Subgroups — #10592561

Order 576.

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

normal subgroups only

This group has 312 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#64106C211yesmaximal, characteristic
2C4⋊C4×C3⋊D3C211yesmaximal, characteristic
3C32×#106C211yesmaximal, characteristic, complemented
4#63647C211yesmaximal, characteristic
5#63643C211yesmaximal, characteristic, complemented
6#63637C211yesmaximal, characteristic, complemented
7#63635C211yesmaximal, characteristic, complemented
8#4862—34nomaximal
9C2×C4×C3⋊D3—21no—
10C2×C4×C3⋊D3C2211yescharacteristic
11C2×#634C2211yescharacteristic
12C2×C32×C8C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C32×C4⋊C4C2211yescharacteristic
15#3752—21no—
16C4⋊(C3⋊Dic3)—21no—
17C4⋊(C3⋊Dic3)C2211yescharacteristic
18#3750—21no—
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20#897—34no—
21D3×C4⋊C4—34no—
22C3×#106D314yescomplemented
23(C2×D3)⋊C8—34no—
24#816—34no—
25#810—34no—
26#808—34no—
27C22×C3⋊D3D411yescharacteristic
28C2×C32×C4—21no—
29C2×C32×C4—21no—
30C2×C32×C4C2311yescharacteristic
31C2×C3⋊Dic3—21no—
32C2×C3⋊Dic3—21no—
33C2×C3⋊Dic3D411yescharacteristic
34C4×C3⋊D3—21no—
35C4×C3⋊D3—21no—
36C4×C3⋊D3—42no—
37#634—21no—
38#634—21no—
39C32×C8—21no—
40C3⋊(C3⋊C8)—21no—
41#483—91noSylow
42C2×C4×D3—34no—
43C2×C4×D3—64no—
44C2×Dic6—34no—
45C2×C3×C8C2×D314yes—
46C3×C4⋊C4C2×D314yes—
47C3×C4⋊C4C2×D314yes—
48(C2×D3)⋊C4—64no—
49C4⋊Dic3—34no—
50C4⋊Dic3—64no—
51Dic3⋊C4—64no—
52C2×C3⋊C8—34no—
53C22×C32C2×D411yescharacteristic
54C2×C3⋊D3—21no—
55C2×C3⋊D3—21no—
56C2×C3⋊D3SD1612yes—
57C32×C4—41no—
58C32×C4—41no—
59C32×C4#4111yescharacteristic
60C32×C4#4111yescharacteristic
61C3⋊Dic3—21no—
62C3⋊Dic3—41no—
63C3⋊Dic3—41no—
64C22⋊Q8—91no—
65C2×C4⋊C4—91no—
66#106—91no—
67#106C3⋊D311yescharacteristic, complemented
68#103—91no—
69#103—91no—
70C22⋊C8—91no—
71C22×D3—34no—
72C2×C3×C4—24no—
73C2×C3×C4—24no—
74C2×C3×C4C22×D314yes—
75C2×Dic3—34no—
76C2×Dic3—64no—
77C2×Dic3—64no—
78C4×D3—64no—
79C4×D3—64no—
80C4×D3—128no—
81Dic6—64no—
82Dic6—64no—
83C3×C8—24no—
84C3⋊C8—64no—
85C2×C32#12311yescharacteristic
86C2×C32C2×SD1611yescharacteristic
87C2×C32#13711yescharacteristic
88C3⋊D3—22no—
89C2×Q8—91no—
90C22×C4—91no—
91C22×C4—181no—
92C2×C8—91no—
93C2×C8C2×C3⋊D311yescharacteristic
94C4⋊C4—91no—
95C4⋊C4—181no—
96C4⋊C4—181no—
97C4⋊C4C2×C3⋊D311yescharacteristic
98C4⋊C4C2×C3⋊D311yescharacteristic
99C22⋊C4—181no—
100C22×C3D3×D414yes—
101C2×D3—38no—
102C2×D3—64no—
103C2×D3—64no—
104C3×C4—44no—
105C3×C4—44no—
106C3×C4#23514yes—
107C3×C4#23914yes—
108Dic3—64no—
109Dic3—124no—
110Dic3—124no—
111C32#48311yescharacteristic, complemented, Sylow
112C23—91no—
113Q8—181no—
114Q8—181no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—91no—
118C2×C4—181no—
119C2×C4—181no—
120C2×C4—181no—
121C2×C4—181no—
122C2×C4—362no—
123C2×C4C22×C3⋊D311yescharacteristic
124C8—21no—
125C8—181no—
126C2×C3#89414yes—
127C2×C3D3×SD1614yes—
128C2×C3#91514yes—
129D3—68no—
130C22—92no—
131C22—181no—
132C22—181no—
133C22D4×C3⋊D311yescharacteristic
134C4—41no—
135C4—41no—
136C4—181no—
137C4—361no—
138C4—361no—
139C4#382811yescharacteristic
140C4#383211yescharacteristic
141C3#486214yescomplemented
142C2—182no—
143C2#6410311yescharacteristic
144C2SD16×C3⋊D311yescharacteristic
145C2#6412411yescharacteristic
subgroupG/H
centerC22D4×C3⋊D3
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×#106C2
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenter#106C3⋊D3
hypocenterC32#483
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
3. C32×#106C2semidirect
5. #63643C2semidirect
6. #63637C2semidirect
7. #63635C2semidirect
22. C3×#106D3semidirect
67. #106C3⋊D3semidirect
111. C32#483semidirect
141. C3#4862semidirect

Maximal subgroups

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

subgroupindex
#641062
C4⋊C4×C3⋊D32
C32×#1062
#636472
#636432
#636372
#636352
#48623