Finite Group Databasebeta

← C2×(C3⋊S4)⋊C3 #88888

Subgroups — C2×(C3⋊S4)⋊C3 #88888

Order 432.

134 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 22 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2(C3⋊S4)⋊C3C212yesmaximal, direct product factor
3C2×C3⋊S4C311yesmaximal, characteristic, complemented
4C2×C3×#67—31nomaximal
5C2×(C3⋊D3)⋊C3—41nomaximal
6C22⋊H(3,𝔽3)C2211yescharacteristic, complemented
7C23×C32D311yescharacteristic, complemented
8C22×C3×D3—31no—
9C2×C3×A4—21no—
10C2×C3×A4C2×C311yescharacteristic, complemented
11C3⋊S4C2×C312yescomplemented
12C3×#67—32no—
13C3×#67—32no—
14C2×C3×Dic3—31no—
15C2×H(3,𝔽3)—41no—
16(C3⋊D3)⋊C3—42no—
17C2×S4—91no—
18C2×C3×D4—91no—
19C2×#67—31no—
20C22×C32—31no—
21C22×C32—31no—
22C22×C32C2×D311yescharacteristic, complemented
23C2×C3⋊D3—41no—
24C2×C3×D3—32no—
25C2×C3×D3—61no—
26C2×C3×D3—61no—
27C3×A4—21no—
28C3×A4C22×C311yescharacteristic, complemented
29C3×Dic3—32no—
30H(3,𝔽3)—41noSylow
31C23×C3—31no—
32C23×C3—91no—
33C23×C3C3×D311yescharacteristic
34C22×D3—31no—
35C2×A4—31no—
36C2×A4—61no—
37S4—92no—
38C3×D4—92no—
39C3×D4—92no—
40C2×C3×C4—91no—
41#67—32no—
42#67—32no—
43C2×Dic3—31no—
44C2×C32—31no—
45C2×C32—31no—
46C2×C32—41no—
47C2×C32—81no—
48C2×C32S411yescharacteristic, complemented
49C3⋊D3—42no—
50C3×D3—62no—
51C2×D4—91noSylow
52C22×C3—31no—
53C22×C3—31no—
54C22×C3—31no—
55C22×C3—31no—
56C22×C3—31no—
57C22×C3—61no—
58C22×C3—61no—
59C22×C3—92no—
60C22×C3—181no—
61C22×C3—181no—
62C22×C3C2×C3×D311yescharacteristic
63C2×D3—32no—
64C2×D3—61no—
65C2×D3—61no—
66C2×D3—361no—
67A4—31no—
68A4—61no—
69C3×C4—92no—
70Dic3—32no—
71C32—41no—
72C32—81no—
73C32C2×S411yescharacteristic, complemented
74C23—91no—
75C23(C3⋊D3)⋊C311yescharacteristic, complemented
76D4—92no—
77D4—92no—
78C2×C4—91no—
79C2×C3—31no—
80C2×C3—31no—
81C2×C3—31no—
82C2×C3—31no—
83C2×C3—31no—
84C2×C3—61no—
85C2×C3—61no—
86C2×C3—121no—
87C2×C3—182no—
88C2×C3—241no—
89C2×C3C3×S411yescharacteristic
90D3—62no—
91D3—362no—
92C22—31no—
93C22—31no—
94C22—92no—
95C22—181no—
96C22—181no—
97C22C2×(C3⋊D3)⋊C311yescharacteristic, complemented
98C4—92no—
99C3—31no—
100C3—121no—
101C3—241no—
102C3C2×C3×S411yescharacteristic
103C2—31no—
104C2—31no—
105C2—182no—
106C2(C3⋊S4)⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2(C3⋊S4)⋊C3
commutator, cosocle, hypocenterC3×A4C22×C3
FrattiniC3C2×C3×S4
FittingC23×C32D3
socleC23×C3C3×D3
LCSG ▷ C3×A4
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊H(3,𝔽3)C2semidirect
2. (C3⋊S4)⋊C3C2direct (N × H)
3. C2×C3⋊S4C3semidirect
6. C22⋊H(3,𝔽3)C22semidirect
7. C23×C32D3semidirect
10. C2×C3×A4C2×C3semidirect
11. C3⋊S4C2×C3semidirect
22. C22×C32C2×D3semidirect
28. C3×A4C22×C3semidirect
48. C2×C32S4semidirect
73. C32C2×S4semidirect
75. C23(C3⋊D3)⋊C3semidirect
97. C22C2×(C3⋊D3)⋊C3semidirect
106. C2(C3⋊S4)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊H(3,𝔽3)2
(C3⋊S4)⋊C32
C2×C3⋊S43
C2×C3×#673
C2×(C3⋊D3)⋊C34