Finite Group Databasebeta

← #10596168

Subgroups — #10596168

Order 576.

720 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 183 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#64104C214yesmaximal, complemented
3#64104C212yesmaximal, complemented
4#64103C214yesmaximal, complemented
5#64102C212yesmaximal, complemented
6#64083C211yesmaximal, characteristic, complemented
7#64081C211yesmaximal, characteristic, complemented
8#5697—34nomaximal
9C2×C4⋊(C3⋊D3)C2212yes—
10C2×C4⋊(C3⋊D3)C2214yes—
11C2×C4⋊(C3⋊D3)C2214yes—
12C2×C4×C3⋊D3C2211yescharacteristic
13C2×C4×C3⋊D3C2214yes—
14C32×C4⋊C4C2212yes—
15C32×C4⋊C4C2214yes—
16C32×C42C2211yescharacteristic
17#3752C2212yes—
18#3752C2214yes—
19#3752C2214yes—
20#3750C2212yes—
21C4×C3⋊Dic3C2211yescharacteristic
22C3×#125D314yescomplemented
23#895—316no—
24#895—38no—
25#894—316no—
26#893—38no—
27#874—34no—
28#872—34no—
29C22×C3⋊D3C2311yescharacteristic
30C22×C3⋊D3C2314yes—
31C2×C32×C4C2312yes—
32C2×C32×C4C2314yes—
33C2×C32×C4C2311yescharacteristic
34C2×C3⋊Dic3C2312yes—
35C2×C3⋊Dic3C2311yescharacteristic
36C4⋊(C3⋊D3)—24no—
37C4⋊(C3⋊D3)—44no—
38C4⋊(C3⋊D3)—44no—
39C4×C3⋊D3—22no—
40C4×C3⋊D3—44no—
41#540—91noSylow
42C2×D12—38no—
43C2×D12—316no—
44C2×D12—316no—
45C2×C4×D3—34no—
46C2×C4×D3—316no—
47C3×C4⋊C4C2×D318yes—
48C3×C4⋊C4C2×D3116yes—
49C3×C42C2×D314yes—
50(C2×D3)⋊C4—38no—
51(C2×D3)⋊C4—316no—
52(C2×D3)⋊C4—316no—
53Dic3⋊C4—38no—
54C4×Dic3—34no—
55C22×C32C2411yescharacteristic
56C2×C3⋊D3—21no—
57C2×C3⋊D3—24no—
58C2×C3⋊D3—24no—
59C2×C3⋊D3—22no—
60C2×C3⋊D3—24no—
61C32×C4—24no—
62C32×C4—22no—
63C32×C4#4112yes—
64C3⋊Dic3—21no—
65C3⋊Dic3—22no—
66#127—91no—
67#125C3⋊D311yescharacteristic, complemented
68#123—94no—
69#121—94no—
70#121—92no—
71C4×D4—92no—
72#117—91no—
73C22×D3—34no—
74C22×D3—316no—
75C2×C3×C4C22×D318yes—
76C2×C3×C4C22×D3116yes—
77C2×C3×C4C22×D314yes—
78C2×Dic3—38no—
79C2×Dic3—34no—
80D12—616no—
81D12—1216no—
82D12—1216no—
83C4×D3—68no—
84C4×D3—1216no—
85C2×C32C2×#4111yescharacteristic
86C2×C32H(5,𝔽2)12yes—
87C3⋊D3—41no—
88C3⋊D3—44no—
89C2×D4—92no—
90C2×D4—94no—
91C2×D4—94no—
92C22×C4—91no—
93C22×C4—94no—
94C4⋊C4—92no—
95C4⋊C4C2×C3⋊D312yes—
96C4⋊C4C2×C3⋊D314yes—
97C22⋊C4—92no—
98C22⋊C4—94no—
99C22⋊C4—94no—
100C42—91no—
101C42C2×C3⋊D311yescharacteristic
102C22×C3C23×D314yes—
103C2×D3—64no—
104C2×D3—616no—
105C2×D3—616no—
106C2×D3—68no—
107C2×D3—616no—
108C3×C4—216no—
109C3×C4—28no—
110C3×C4#23918yes—
111Dic3—64no—
112Dic3—68no—
113C32#54011yescharacteristic, complemented, Sylow
114C23—91no—
115C23—94no—
116D4—184no—
117D4—364no—
118D4—364no—
119C2×C4—92no—
120C2×C4—91no—
121C2×C4—182no—
122C2×C4—364no—
123C2×C4C22×C3⋊D312yes—
124C2×C4C22×C3⋊D314yes—
125C2×C4C22×C3⋊D311yescharacteristic
126C2×C3C2×#23914yes—
127C2×C3#100918yes—
128D3—124no—
129D3—1216no—
130C22—181no—
131C22—184no—
132C22—184no—
133C22—182no—
134C22—184no—
135C22C23×C3⋊D311yescharacteristic
136C4—24no—
137C4—22no—
138C4—181no—
139C4—182no—
140C4#383212yes—
141C3#569714yescomplemented
142C2—361no—
143C2—364no—
144C2C2×#383211yescharacteristic
145C2#6436612yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#125C2
hypercenter#125C3⋊D3
hypocenterC32#540
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. #64104C2semidirect
3. #64104C2semidirect
4. #64103C2semidirect
5. #64102C2semidirect
6. #64083C2semidirect
7. #64081C2semidirect
22. C3×#125D3semidirect
67. #125C3⋊D3semidirect
113. C32#540semidirect
141. C3#5697semidirect

Maximal subgroups

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

subgroupindex
C32×#1252
#641042
#641032
#641022
#640832
#640812
#56973