Finite Group Databasebeta

← #88672

Subgroups — #88672

Order 432.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1084C211yesmaximal, characteristic, complemented
2#638⋊C3C212yesmaximal, complemented
3#6198C211yesmaximal, characteristic, complemented
4((C3⋊D3)⋊C3)⋊C4C212yesmaximal, complemented
5#6196C211yesmaximal, characteristic, complemented
6#3804—32nomaximal
7C22×H(3,𝔽3)C2211yescharacteristic
8C2×(C3⋊D3)⋊C3C2212yes—
9#1084C2211yescharacteristic
10#1084C2211yescharacteristic
11(C3⋊Dic3)⋊C3C2212yes—
12#638D312yescomplemented
13C3×#67—32no—
14C2×C3×Dic3—32no—
15C2×C3×Dic3—61no—
16#627—32no—
17#626—32no—
18#624—32no—
19D3×Dic3—32no—
20C2×H(3,𝔽3)—21no—
21C2×H(3,𝔽3)C2311yescharacteristic
22(C3⋊D3)⋊C3—22no—
23#237—92no—
24#235—91no—
25C22×C32—21no—
26C22×C32C2×D312yes—
27C2×C3⋊D3C2×D312yes—
28C2×C3×D3—32no—
29C3⋊Dic3C2×D312yes—
30C3×Dic3—32no—
31C3×Dic3—32no—
32C3×Dic3—32no—
33C3×Dic3—61no—
34C3×Dic3—61no—
35H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
36C3×D4—92no—
37C2×C3×C4—91no—
38#67—32no—
39#67—92no—
40#67—92no—
41C2×Dic3—32no—
42C2×Dic3—61no—
43C2×Dic3—92no—
44D12—91no—
45C4×D3—92no—
46C4×D3—92no—
47Dic6—91no—
48Dic6—92no—
49C2×C32—22no—
50C2×C32—22no—
51C2×C32—21no—
52C2×C32C22×D312yes—
53C3⋊D3—22no—
54C3×D3—62no—
55#41—271noSylow
56C22×C3—32no—
57C22×C3—61no—
58C22×C3—92no—
59C22×C3D3211yescharacteristic
60C2×D3—32no—
61C2×D3—92no—
62C3×C4—92no—
63C3×C4—91no—
64C3×C4—91no—
65Dic3—32no—
66Dic3—32no—
67Dic3—32no—
68Dic3—61no—
69Dic3—61no—
70Dic3—92no—
71C32—21no—
72C32#23712yescomplemented
73Q8—271no—
74D4—92no—
75D4—271no—
76C2×C4—91no—
77C2×C4—272no—
78C2×C3—21no—
79C2×C3—32no—
80C2×C3—62no—
81C2×C3—62no—
82C2×C3—61no—
83C2×C3—182no—
84C2×C3C2×D3211yescharacteristic
85D3—62no—
86D3—182no—
87C22—92no—
88C22((C3⋊D3)⋊C3)⋊C211yescharacteristic
89C4—91no—
90C4—91no—
91C4—92no—
92C3—32no—
93C3—61no—
94C3#380511yescharacteristic
95C2—21no—
96C2—182no—
97C2C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C23
Frattini, socleC2×C3C2×D32
FittingC22×H(3,𝔽3)C22
hypercenterC22((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#1084C2semidirect
2. #638⋊C3C2semidirect
3. #6198C2semidirect
4. ((C3⋊D3)⋊C3)⋊C4C2semidirect
5. #6196C2semidirect
12. #638D3semidirect
35. H(3,𝔽3)#41semidirect
72. C32#237semidirect

Maximal subgroups

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

subgroupindex
C2×#10842
#638⋊C32
#61982
((C3⋊D3)⋊C3)⋊C42
#61962
#38043