Finite Group Databasebeta

← C42×S4 #85286

Subgroups — C42×S4 #85286

Order 384.

497 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 84 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×C4×S4C213yesmaximal
2C42×A4C211yesmaximal, characteristic, complemented
3C4×SL(2,ℤ/4)C213yesmaximal, complemented
4C42×D4—31nomaximal, Sylow
5C22×S4C2211yescharacteristic
6C2×C4×A4C2213yes—
7C2×SL(2,ℤ/4)C2213yes—
8C4×S4C4112yesdirect product factor
9C42×D3—41nomaximal
10C2×C4×D4—33no—
11C22×C42—31no—
12C22×C42D311yescharacteristic, complemented
13C4×C4⋊C4—33no—
14C4×C22⋊C4—33no—
15C4×C22⋊C4—33no—
16C43—31no—
17C22×A4C2311yescharacteristic
18C2×S4C2×C416yes—
19C2×C4×D3—43no—
20C4×A4C2×C416yescomplemented
21SL(2,ℤ/4)C2×C416yescomplemented
22C3×C42—41no—
23C4×Dic3—43no—
24C22×D4—31no—
25C23×C4—33no—
26C23×C4C2×D313yes—
27C4×D4—312no—
28C4×D4—312no—
29C2×C4⋊C4—33no—
30C2×C22⋊C4—33no—
31C2×C22⋊C4—33no—
32C2×C42—33no—
33C2×C42—31no—
34C2×C42—33no—
35C2×C42—33no—
36C2×C42—33no—
37C2×C42—63no—
38C2×C42—61no—
39#95—36no—
40C22×D3—41no—
41C2×A4C22×C413yes—
42S4C4214yesdirect product factor
43C2×C3×C4—43no—
44C2×Dic3—43no—
45C4×D3—412no—
46C24—31no—
47C24C22×D311yescharacteristic
48C2×D4—36no—
49C2×D4—36no—
50C22×C4—31no—
51C22×C4—33no—
52C22×C4—33no—
53C22×C4—36no—
54C22×C4—33no—
55C22×C4—33no—
56C22×C4—312no—
57C22×C4—63no—
58C22×C4—66no—
59C22×C4—63no—
60C22×C4C4×D316yescomplemented
61C4⋊C4—312no—
62C22⋊C4—312no—
63C22⋊C4—36no—
64C22⋊C4—36no—
65C42—312no—
66C42—33no—
67C42—61no—
68C42—66no—
69C42—63no—
70C42—63no—
71C42S411yescharacteristic, direct product factor
72C22×C3—41no—
73C2×D3—46no—
74A4C2×C4211yescharacteristic, complemented
75C3×C4—46no—
76Dic3—46no—
77C23—31no—
78C23—33no—
79C23—36no—
80C23—63no—
81C23—61no—
82C23C2×C4×D313yes—
83D4—312no—
84D4—34no—
85C2×C4—36no—
86C2×C4—33no—
87C2×C4—36no—
88C2×C4—36no—
89C2×C4—36no—
90C2×C4—36no—
91C2×C4—36no—
92C2×C4—63no—
93C2×C4—612no—
94C2×C4—66no—
95C2×C4—63no—
96C2×C4—63no—
97C2×C4—612no—
98C2×C4C2×S413yes—
99C2×C3—43no—
100D3—44no—
101C22—33no—
102C22—33no—
103C22—33no—
104C22—34no—
105C22—61no—
106C22—66no—
107C22—66no—
108C22C42×D311yescharacteristic, complemented
109C22C22×S411yescharacteristic
110C4—34no—
111C4—36no—
112C4—66no—
113C4—66no—
114C4C4×S416yesdirect product factor
115C3—41noSylow
116C2—31no—
117C2—33no—
118C2—64no—
119C2C2×C4×S413yes—
subgroupG/H
center, hypercenterC42S4
commutator, hypocenterA4C2×C42
FrattiniC22C22×S4
FittingC22×C42D3
socleC24C22×D3
prime radicalC22×S4C22
cosocleC22×A4C23
LCSG ▷ A4
UCSC1 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C42×A4C2semidirect
3. C4×SL(2,ℤ/4)C2semidirect
8. C4×S4C4direct (N × H)
12. C22×C42D3semidirect
20. C4×A4C2×C4semidirect
21. SL(2,ℤ/4)C2×C4semidirect
42. S4C42direct (N × H)
60. C22×C4C4×D3semidirect
71. C42S4direct (N × H)
74. A4C2×C42semidirect
108. C22C42×D3semidirect
114. C4C4×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×S42
C42×A42
C4×SL(2,ℤ/4)2
C42×D43
C42×D34