Finite Group Databasebeta

← #10592584

Subgroups — #10592584

Order 576.

318 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 85 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#64104C211yesmaximal, characteristic
2#64101C211yesmaximal, characteristic, complemented
3C32×#107C211yesmaximal, characteristic, complemented
4#63648C211yesmaximal, characteristic, complemented
5#63647C211yesmaximal, characteristic, complemented
6#63636C211yesmaximal, characteristic, complemented
7#63635C211yesmaximal, characteristic, complemented
8#4885—34nomaximal
9C2×C4⋊(C3⋊D3)—21no—
10C2×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C4×C3⋊D3C2211yescharacteristic
12C2×C32×C8C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C32×C4⋊C4C2211yescharacteristic
15#3752—21no—
16#3752—21no—
17C4⋊(C3⋊Dic3)C2211yescharacteristic
18C4×C3⋊Dic3—21no—
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20#895—34no—
21#892—34no—
22C3×#107D314yescomplemented
23#821—34no—
24(C2×D3)⋊C8—34no—
25#809—34no—
26#808—34no—
27C22×C3⋊D3—21no—
28C22×C3⋊D3D411yescharacteristic
29C2×C32×C4—21no—
30C2×C32×C4—21no—
31C2×C32×C4C2311yescharacteristic
32C2×C3⋊Dic3—21no—
33C2×C3⋊Dic3D411yescharacteristic
34C4⋊(C3⋊D3)—21no—
35C4⋊(C3⋊D3)—21no—
36C4⋊(C3⋊D3)—42no—
37C4×C3⋊D3—21no—
38C4×C3⋊D3—21no—
39C32×C8—21no—
40C3⋊(C3⋊C8)—21no—
41#482—91noSylow
42C2×D12—34no—
43C2×D12—64no—
44C2×C4×D3—34no—
45C2×C3×C8C2×D314yes—
46C3×C4⋊C4C2×D314yes—
47C3×C4⋊C4C2×D314yes—
48(C2×D3)⋊C4—64no—
49(C2×D3)⋊C4—64no—
50C4⋊Dic3—34no—
51C4×Dic3—64no—
52C2×C3⋊C8—34no—
53C22×C32C2×D411yescharacteristic
54C2×C3⋊D3—21no—
55C2×C3⋊D3—21no—
56C2×C3⋊D3—21no—
57C2×C3⋊D3—41no—
58C2×C3⋊D3—41no—
59C2×C3⋊D3—41no—
60C32×C4—41no—
61C32×C4—41no—
62C32×C4#4111yescharacteristic
63C32×C4#4111yescharacteristic
64C3⋊Dic3—21no—
65C3⋊Dic3—41no—
66#121—91no—
67#117—91no—
68#107C3⋊D311yescharacteristic, complemented
69#106—91no—
70#102—91no—
71#102—91no—
72C22⋊C8—91no—
73C22×D3—34no—
74C22×D3—64no—
75C2×C3×C4—24no—
76C2×C3×C4—24no—
77C2×C3×C4C22×D314yes—
78C2×Dic3—34no—
79C2×Dic3—64no—
80D12—64no—
81D12—64no—
82D12—128no—
83C4×D3—64no—
84C4×D3—64no—
85C3×C8—24no—
86C3⋊C8—64no—
87C2×C32#12311yescharacteristic
88C2×C32#13511yescharacteristic
89C2×C32AGL(1,ℤ/8)11yescharacteristic
90C3⋊D3—41no—
91C3⋊D3—81no—
92C2×D4—91no—
93C2×D4—181no—
94C22×C4—91no—
95C2×C8—91no—
96C2×C8C2×C3⋊D311yescharacteristic
97C4⋊C4—91no—
98C4⋊C4C2×C3⋊D311yescharacteristic
99C4⋊C4C2×C3⋊D311yescharacteristic
100C22⋊C4—181no—
101C22⋊C4—181no—
102C42—181no—
103C22×C3D3×D414yes—
104C2×D3—64no—
105C2×D3—64no—
106C2×D3—64no—
107C2×D3—124no—
108C2×D3—124no—
109C2×D3—124no—
110C3×C4—44no—
111C3×C4—44no—
112C3×C4#23514yes—
113C3×C4#23914yes—
114Dic3—64no—
115Dic3—124no—
116C32#48211yescharacteristic, complemented, Sylow
117C23—91no—
118C23—181no—
119D4—181no—
120D4—181no—
121D4—362no—
122C2×C4—21no—
123C2×C4—21no—
124C2×C4—91no—
125C2×C4—181no—
126C2×C4—181no—
127C2×C4—181no—
128C2×C4C22×C3⋊D311yescharacteristic
129C8—21no—
130C8—181no—
131C2×C3#89414yes—
132C2×C3#91114yes—
133C2×C3#91914yes—
134D3—124no—
135D3—244no—
136C22—181no—
137C22—181no—
138C22—181no—
139C22—361no—
140C22—361no—
141C22—361no—
142C22D4×C3⋊D311yescharacteristic
143C4—41no—
144C4—41no—
145C4—181no—
146C4—361no—
147C4#382811yescharacteristic
148C4#383211yescharacteristic
149C3#488514yescomplemented
150C2—361no—
151C2—721no—
152C2#6410311yescharacteristic
153C2#6412011yescharacteristic
154C2#6412811yescharacteristic
subgroupG/H
centerC22D4×C3⋊D3
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C22×C3⋊D3
FittingC32×#107C2
socleC22×C32C2×D4
prime radical#64104C2
hypercenter#107C3⋊D3
hypocenterC32#482
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
2. #64101C2semidirect
3. C32×#107C2semidirect
4. #63648C2semidirect
5. #63647C2semidirect
6. #63636C2semidirect
7. #63635C2semidirect
22. C3×#107D3semidirect
68. #107C3⋊D3semidirect
116. C32#482semidirect
149. C3#4885semidirect

Maximal subgroups

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

subgroupindex
#641042
#641012
C32×#1072
#636482
#636472
#636362
#636352
#48853