Finite Group Databasebeta

← #68375

Subgroups — #68375

Order 384.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C3⋊#564C211yesmaximal, characteristic, complemented
2#5067C211yesmaximal, characteristic
3#4713C212yesmaximal
4C3×#330C211yesmaximal, characteristic, complemented
5#4465C212yesmaximal, complemented
6#1658—31nomaximal, Sylow
7C3×#125C2211yescharacteristic
8#890—22no—
9#889—22no—
10#889—22no—
11#874C2211yescharacteristic
12#870C2212yes—
13C3×#97C2212yes—
14#821—22no—
15#821—22no—
16#809—22no—
17#803C2211yescharacteristic
18#564—31no—
19#490—31no—
20#490—32no—
21#330—32no—
22#330D311yescharacteristic, complemented
23C2×D12—21no—
24C2×D12—22no—
25C2×C3×C8—22no—
26C3×C4⋊C4—22no—
27C3×C4⋊C4D412yes—
28C3×C42C2311yescharacteristic
29C4⋊Dic3—22no—
30C4⋊Dic3D412yes—
31C4⋊Dic3D412yes—
32Dic3⋊C4—22no—
33Dic3⋊C4—22no—
34Dic3⋊C4—22no—
35Dic3⋊C4—22no—
36C4×Dic3—22no—
37C2×C3⋊C8—21no—
38#127—31no—
39#125—32no—
40#125—62no—
41#125—62no—
42#125—62no—
43#125C2×D311yescharacteristic
44#102—62no—
45#102—62no—
46#102—62no—
47#97—31no—
48#97C2×D312yes—
49C22×D3—41no—
50C2×C3×C4—22no—
51C2×C3×C4C2×D411yescharacteristic
52C2×C3×C4C2×D412yes—
53C2×Dic3—22no—
54C2×Dic3—22no—
55D12—42no—
56D12—44no—
57C3×C8—42no—
58C3⋊C8—41no—
59C2×D4—61no—
60C2×D4—62no—
61C2×C8—22no—
62C2×C8—61no—
63C4⋊C4—22no—
64C4⋊C4—32no—
65C4⋊C4—32no—
66C4⋊C4—62no—
67C4⋊C4—62no—
68C4⋊C4—62no—
69C4⋊C4—62no—
70C4⋊C4—62no—
71C4⋊C4D1212yes—
72C42—62no—
73C42C22×D311yescharacteristic
74C22×C3#12011yescharacteristic
75C2×D3—81no—
76C2×D3—82no—
77C3×C4—21no—
78C3×C4—22no—
79C3×C4—42no—
80Dic3—42no—
81Dic3—42no—
82C23—121no—
83D4—122no—
84D4—124no—
85C2×C4—22no—
86C2×C4—62no—
87C2×C4—62no—
88C2×C4C2×D1211yescharacteristic
89C2×C4D3×D412yes—
90C8—42no—
91C8—121no—
92C2×C3#45411yescharacteristic
93C2×C3#45412yes—
94D3—161no—
95C22—241no—
96C22—242no—
97C22#88211yescharacteristic
98C4—21no—
99C4—22no—
100C4—42no—
101C4—122no—
102C4—122no—
103C3#165811yescharacteristic, complemented, Sylow
104C2—481no—
105C2#475311yescharacteristic
106C2#482712yes—
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#330C2
socleC22×C3#120
prime radical#874C22
hypercenter#330D3
hypocenterC3#1658
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. C3⋊#564C2semidirect
4. C3×#330C2semidirect
5. #4465C2semidirect
22. #330D3semidirect
103. C3#1658semidirect

Maximal subgroups

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

subgroupindex
C3⋊#5642
#50672
#47132
C3×#3302
#44652
#16583