Finite Group Databasebeta

← #10596205

Subgroups — #10596205

Order 576.

708 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 195 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×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156C214yesmaximal, complemented
3#64104C214yesmaximal, complemented
4#64101C214yesmaximal, complemented
5C4×C4⋊(C3⋊D3)C212yesmaximal, complemented
6#5734—34nomaximal
7C2×C32×Q8C2212yes—
8C2×C4⋊(C3⋊D3)C2212yes—
9C2×C4⋊(C3⋊D3)C2214yes—
10C2×C4×C3⋊D3C2214yes—
11C32×C4⋊C4C2214yes—
12C32×C42C2211yescharacteristic
13#3752C2214yes—
14#3752C2218yes—
15C4⋊(C3⋊Dic3)C2212yes—
16C4×C3⋊Dic3C2214yes—
17C3×C4⋊Q8D314yescomplemented
18#947—316no—
19#895—316no—
20#892—316no—
21C4×D12—38no—
22C22×C3⋊D3C2314yes—
23C32×Q8—42no—
24C2×C32×C4C2314yes—
25C2×C32×C4C2311yescharacteristic
26C2×C32×C4C2312yes—
27C2×C3⋊Dic3C2314yes—
28C4⋊(C3⋊D3)—24no—
29C4⋊(C3⋊D3)—44no—
30C4×C3⋊D3—28no—
31#555—91noSylow
32C2×C3×Q8C2×D318yes—
33C2×D12—38no—
34C2×D12—316no—
35C2×C4×D3—316no—
36C3×C4⋊C4C2×D3116yes—
37C3×C42C2×D314yes—
38(C2×D3)⋊C4—316no—
39(C2×D3)⋊C4—332no—
40C4⋊Dic3—38no—
41C4×Dic3—316no—
42C22×C32C2411yescharacteristic
43C2×C3⋊D3—24no—
44C2×C3⋊D3—24no—
45C2×C3⋊D3—24no—
46C32×C4—24no—
47C32×C4—21no—
48C32×C4#4114yes—
49C3⋊Dic3—24no—
50C4⋊Q8C3⋊D311yescharacteristic, complemented
51#124—94no—
52#121—94no—
53C4×D4—92no—
54#117—94no—
55C22×D3—316no—
56C3×Q8—48no—
57C2×C3×C4C22×D3116yes—
58C2×C3×C4C22×D314yes—
59C2×C3×C4C22×D318yes—
60C2×Dic3—316no—
61D12—616no—
62D12—1216no—
63C4×D3—632no—
64C2×C32C2×#4112yes—
65C2×C32H(5,𝔽2)11yescharacteristic
66C3⋊D3—44no—
67C2×Q8C2×C3⋊D312yes—
68C2×D4—92no—
69C2×D4—94no—
70C22×C4—94no—
71C4⋊C4—92no—
72C4⋊C4C2×C3⋊D314yes—
73C22⋊C4—98no—
74C22⋊C4—94no—
75C42—94no—
76C42C2×C3⋊D311yescharacteristic
77C22×C3C23×D314yes—
78C2×D3—616no—
79C2×D3—616no—
80C2×D3—616no—
81C3×C4—216no—
82C3×C4—24no—
83C3×C4#239116yes—
84Dic3—616no—
85C32#55511yescharacteristic, complemented, Sylow
86C23—94no—
87Q8—42no—
88D4—184no—
89D4—364no—
90C2×C4—94no—
91C2×C4—188no—
92C2×C4C22×C3⋊D314yes—
93C2×C4C22×C3⋊D311yescharacteristic
94C2×C4C22×C3⋊D312yes—
95C2×C3#100414yes—
96C2×C3C2×#23918yes—
97D3—1216no—
98C22—184no—
99C22—184no—
100C22—184no—
101C22C23×C3⋊D311yescharacteristic
102C4—24no—
103C4—21no—
104C4—184no—
105C4#383214yes—
106C3#573414yescomplemented
107C2—364no—
108C2#6436111yescharacteristic
109C2C2×#383212yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊Q8C2
hypercenterC4⋊Q8C3⋊D3
hypocenterC32#555
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C32×C4⋊Q8C2semidirect
2. #64156C2semidirect
3. #64104C2semidirect
4. #64101C2semidirect
5. C4×C4⋊(C3⋊D3)C2semidirect
17. C3×C4⋊Q8D3semidirect
50. C4⋊Q8C3⋊D3semidirect
85. C32#555semidirect
106. C3#5734semidirect

Maximal subgroups

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

subgroupindex
C32×C4⋊Q82
#641562
#641042
#641012
C4×C4⋊(C3⋊D3)2
#57343