Finite Group Databasebeta

← #10593299

Subgroups — #10593299

Order 576.

168 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 37 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#64223C211yesmaximal, characteristic
2#63805C212yesmaximal, complemented
3C8×#157C211yesmaximal, characteristic, complemented
4#63733C211yesmaximal, characteristic, complemented
5#63731C212yesmaximal, complemented
6#3798C2212yes—
7C4×#157C2211yescharacteristic
8#3787C2211yescharacteristic
9#3776C411yescharacteristic, complemented
10#3775C412yescomplemented
11#3774C411yescharacteristic, complemented
12C8×C3⋊D3C2211yescharacteristic
13C3×ΓL(1,𝔽25)—22no—
14#3712C2212yes—
15D3×C3⋊C8—22no—
16#907—62no—
17C2×#157—21no—
18C4×C3⋊D3C2311yescharacteristic
19C3×C4×D3—22no—
20#627C2×C412yes—
21#626—22no—
22#625C2×C412yes—
23#622C2×C411yescharacteristic
24#622C2×C411yescharacteristic
25C32×C8D411yescharacteristic
26C3⋊(C3⋊C8)D411yescharacteristic
27C3×C3⋊C8—22no—
28#443—91nomaximal, Sylow
29#235—62no—
30C3×M16—62no—
31C2×C3⋊C8—62no—
32ΓL(1,𝔽25)—22no—
33ΓL(1,𝔽25)—62no—
34D3×C8—62no—
35D3×C8—62no—
36C2×C3⋊D3#4111yescharacteristic
37C2×C3×D3—22no—
38#157—22no—
39C32×C4C2×D411yescharacteristic
40C3⋊Dic3C22×C411yescharacteristic
41C3×Dic3—22no—
42#135—91no—
43#131—92no—
44#108—91no—
45#104—92no—
46C4×C8—91no—
47C2×C3×C4—62no—
48#67—62no—
49#67—62no—
50D12—62no—
51C4×D3—22no—
52C4×D3—62no—
53Dic6—62no—
54C3×C8—22no—
55C3×C8—62no—
56C3⋊C8—22no—
57C3⋊C8—62no—
58C2×C32C4×D411yescharacteristic
59C3⋊D3—21no—
60C3×D3—42no—
61#41—92no—
62Q16—91no—
63SD16—92no—
64D8—91no—
65M16—62no—
66M16—92no—
67C2×C8—91no—
68C2×C8—91no—
69C2×C8—182no—
70C42—91no—
71C22×C3—62no—
72C2×D3—22no—
73C2×D3—62no—
74C3×C4—22no—
75C3×C4—62no—
76Dic3—22no—
77Dic3—62no—
78C32#44311yescharacteristic, complemented, Sylow
79Q8—92no—
80D4—92no—
81D4—182no—
82C2×C4—62no—
83C2×C4—91no—
84C2×C4—181no—
85C8—62no—
86C8—91no—
87C8—91no—
88C8—91no—
89C8GO+(4,𝔽2)11yescharacteristic
90C2×C3—22no—
91C2×C3—122no—
92D3—42no—
93D3—122no—
94C22—62no—
95C22—91no—
96C4—62no—
97C4—91no—
98C4—182no—
99C4C2×GO+(4,𝔽2)11yescharacteristic
100C3—22no—
101C2—122no—
102C2—181no—
103C2C4×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC4C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C4
FittingC32×C8D4
socleC2×C32C4×D4
prime radical#64223C2
cosocleC4×C3⋊D3C23
hypercenterC8GO+(4,𝔽2)
hypocenterC32#443
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. #63805C2semidirect
3. C8×#157C2semidirect
4. #63733C2semidirect
5. #63731C2semidirect
9. #3776C4semidirect
10. #3775C4semidirect
11. #3774C4semidirect
78. C32#443semidirect

Maximal subgroups

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

subgroupindex
#642232
#638052
C8×#1572
#637332
#637312
#4439