Finite Group Databasebeta

← #10593336

Subgroups — #10593336

Order 576.

143 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 143 conjugacy classes of subgroups — too many to draw. Only its 29 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
1C2×#3718C211yesmaximal, characteristic, complemented
2#63822C211yesmaximal, characteristic
3#63773C211yesmaximal, characteristic, complemented
4#63736C212yesmaximal, complemented
5#63735C212yesmaximal, complemented
6C3×#235—21no—
7C2×C3×Dic6—21no—
8C2×C3⋊(C3⋊C8)C2211yescharacteristic
9#3718—21no—
10#3718C2211yescharacteristic
11#3718C2212yes—
12#3715—21no—
13#3713C2211yescharacteristic
14#3708C2212yes—
15#950—61no—
16C2×#216—61no—
17C2×C32×C4D411yescharacteristic
18C3×#67—41no—
19C2×C3×Dic3—41no—
20C3×D12—21no—
21C3×C4×D3—41no—
22C3×Dic6—21no—
23C3×Dic6—22no—
24C3×Dic6—41no—
25C3⋊(C3⋊C8)D411yescharacteristic
26C3⋊(C3⋊C8)C2311yescharacteristic
27#510—91nomaximal, Sylow
28C3×#41—61no—
29C2×C3×Q8—61no—
30#235—21no—
31C2×Dic6—21no—
32#216—62no—
33#216—61no—
34#216—62no—
35#215—61no—
36#214—61no—
37#213—61no—
38C2×C3⋊C8—61no—
39C2×C3⋊C8—61no—
40C22×C32D811yescharacteristic
41C2×C3×D3—41no—
42C32×C4D811yescharacteristic
43C32×C4C2×D411yescharacteristic
44C3×Dic3—42no—
45C3×Dic3—41no—
46#135—91no—
47C2×Q16—91no—
48Q32—92no—
49SD32—92no—
50M32—91no—
51C3×Q8—61no—
52C3×Q8—62no—
53C3×Q8—121no—
54C3×D4—61no—
55C3×D4—121no—
56C2×C3×C4—21no—
57C2×C3×C4—21no—
58C2×C3×C4—121no—
59C2×C3×C4—121no—
60#67—41no—
61C2×Dic3—41no—
62D12—21no—
63C4×D3—41no—
64Dic6—21no—
65Dic6—22no—
66Dic6—41no—
67C3⋊C8—61no—
68C3⋊C8—61no—
69C3⋊C8—61no—
70C3⋊C8—61no—
71C2×C32—21no—
72C2×C32C2×D811yescharacteristic
73C3×D3—81no—
74#41—61no—
75C2×Q8—61no—
76Q16—92no—
77Q16—91no—
78Q16—181no—
79SD16—181no—
80D8—91no—
81C2×C8—91no—
82C16—92no—
83C22×C3—21no—
84C22×C3—21no—
85C22×C3—121no—
86C2×D3—41no—
87C3×C4—21no—
88C3×C4—21no—
89C3×C4—21no—
90C3×C4—21no—
91C3×C4—121no—
92C3×C4—122no—
93Dic3—42no—
94Dic3—41no—
95C32#51011yescharacteristic, complemented, Sylow
96Q8—61no—
97Q8—62no—
98Q8—121no—
99D4—61no—
100D4—121no—
101C2×C4—121no—
102C2×C4—121no—
103C2×C4GO+(4,𝔽2)11yescharacteristic
104C8—91no—
105C8—91no—
106C2×C3—22no—
107C2×C3—21no—
108C2×C3—21no—
109C2×C3—41no—
110C2×C3—241no—
111D3—81no—
112C22—121no—
113C22#377411yescharacteristic
114C4—121no—
115C4—122no—
116C4#377411yescharacteristic
117C4C2×GO+(4,𝔽2)11yescharacteristic
118C3—21no—
119C3—21no—
120C2—21no—
121C2—241no—
122C2C2×#377411yescharacteristic
subgroupG/H
centerC2C2×#3774
commutator, cosocleC3⋊(C3⋊C8)C23
FrattiniC4C2×GO+(4,𝔽2)
FittingC2×C32×C4D4
socleC2×C32C2×D8
prime radical#63822C2
hypercenterC2×C4GO+(4,𝔽2)
hypocenterC32#510
LCSG ▷ C3⋊(C3⋊C8) ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3718C2semidirect
3. #63773C2semidirect
4. #63736C2semidirect
5. #63735C2semidirect
95. C32#510semidirect

Maximal subgroups

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

subgroupindex
C2×#37182
#638222
#637732
#637362
#637352
#5109