Finite Group Databasebeta

← #10596171

Subgroups — #10596171

Order 576.

660 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C32×#125C211yesmaximal, characteristic, complemented
2#64105C214yesmaximal, complemented
3#64104C214yesmaximal, complemented
4#64103C214yesmaximal, complemented
5#64084C212yesmaximal, complemented
6#5700—34nomaximal
7C2×C4⋊(C3⋊D3)C2212yes—
8C2×C4⋊(C3⋊D3)C2214yes—
9C2×C4×C3⋊D3C2214yes—
10C2×#634C2212yes—
11C32×C4⋊C4C2212yes—
12C32×C4⋊C4C2214yes—
13C32×C42C2211yescharacteristic
14#3752C2214yes—
15#3752C2214yes—
16#3752C2214yes—
17#3750C2214yes—
18C3×#125D314yescomplemented
19#896—316no—
20#895—316no—
21#894—316no—
22#875—38no—
23C22×C3⋊D3C2314yes—
24C2×C32×C4C2314yes—
25C2×C32×C4C2311yescharacteristic
26C2×C32×C4C2312yes—
27C2×C3⋊Dic3C2314yes—
28C4⋊(C3⋊D3)—42no—
29C4⋊(C3⋊D3)—44no—
30C4×C3⋊D3—44no—
31#634—42no—
32#562—91noSylow
33C2×D12—38no—
34C2×D12—316no—
35C2×C4×D3—316no—
36C2×Dic6—38no—
37C3×C4⋊C4C2×D318yes—
38C3×C4⋊C4C2×D3116yes—
39C3×C42C2×D314yes—
40(C2×D3)⋊C4—316no—
41(C2×D3)⋊C4—316no—
42(C2×D3)⋊C4—316no—
43Dic3⋊C4—316no—
44C22×C32C2411yescharacteristic
45C2×C3⋊D3—24no—
46C2×C3⋊D3—24no—
47C2×C3⋊D3—24no—
48C32×C4—22no—
49C32×C4—24no—
50C32×C4—21no—
51C3⋊Dic3—24no—
52#125C3⋊D311yescharacteristic, complemented
53#124—92no—
54#123—94no—
55C22⋊Q8—94no—
56#121—94no—
57C22×D3—316no—
58C2×C3×C4C22×D3116yes—
59C2×C3×C4C22×D314yes—
60C2×C3×C4C22×D318yes—
61C2×Dic3—316no—
62D12—128no—
63D12—1216no—
64C4×D3—1216no—
65Dic6—128no—
66C2×C32H(5,𝔽2)12yes—
67C2×C322−1+411yescharacteristic
68C3⋊D3—44no—
69C2×Q8—92no—
70C2×D4—94no—
71C2×D4—92no—
72C22×C4—94no—
73C4⋊C4—94no—
74C4⋊C4C2×C3⋊D314yes—
75C4⋊C4C2×C3⋊D312yes—
76C22⋊C4—94no—
77C22⋊C4—94no—
78C22⋊C4—94no—
79C42C2×C3⋊D311yescharacteristic
80C22×C3C23×D314yes—
81C2×D3—616no—
82C2×D3—616no—
83C2×D3—616no—
84C3×C4—24no—
85C3×C4—28no—
86C3×C4—216no—
87Dic3—616no—
88C32#56211yescharacteristic, complemented, Sylow
89C23—94no—
90Q8—362no—
91D4—364no—
92D4—362no—
93C2×C4—94no—
94C2×C4—364no—
95C2×C4C22×C3⋊D311yescharacteristic
96C2×C4C22×C3⋊D312yes—
97C2×C4C22×C3⋊D314yes—
98C2×C3#100714yes—
99C2×C3#100918yes—
100D3—1216no—
101C22—184no—
102C22—184no—
103C22—184no—
104C22C23×C3⋊D311yescharacteristic
105C4—21no—
106C4—24no—
107C4—22no—
108C4—184no—
109C3#570014yescomplemented
110C2—364no—
111C2#6436411yescharacteristic
112C2#6436612yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
hypercenter#125C3⋊D3
hypocenterC32#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. C32×#125C2semidirect
2. #64105C2semidirect
3. #64104C2semidirect
4. #64103C2semidirect
5. #64084C2semidirect
18. C3×#125D3semidirect
52. #125C3⋊D3semidirect
88. C32#562semidirect
109. C3#5700semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641052
#641042
#641032
#640842
#57003