Finite Group Databasebeta

← C2×C4×S4 #5910

Subgroups — C2×C4×S4 #5910

Order 192.

233 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 233 conjugacy classes of subgroups — too many to draw. Only its 43 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
1C22×S4C211yesmaximal, characteristic
2C2×C4×A4C211yesmaximal, characteristic, complemented
3C2×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C4×S4C214yesmaximal, direct product factor
5C2×C4×D4—31nomaximal, Sylow
6C22×A4C2211yescharacteristic
7C2×S4C414yesdirect product factor
8C2×S4C2212yes—
9C2×C4×D3—41nomaximal
10C4×A4C2212yescomplemented
11SL(2,ℤ/4)C2212yescomplemented
12C22×D4—31no—
13C23×C4—31no—
14C23×C4D311yescharacteristic, complemented
15C4×D4—34no—
16C4×D4—34no—
17C2×C4⋊C4—31no—
18C2×C22⋊C4—31no—
19C2×C22⋊C4—31no—
20C2×C42—31no—
21C22×D3—41no—
22C2×A4C2×C412yescomplemented
23C2×A4C2311yescharacteristic
24S4C2×C414yesdirect product factor
25C2×C3×C4—41no—
26C2×Dic3—41no—
27C4×D3—44no—
28C24—31no—
29C24C2×D311yescharacteristic
30C2×D4—32no—
31C2×D4—34no—
32C2×D4—34no—
33C2×D4—32no—
34C22×C4—31no—
35C22×C4—31no—
36C22×C4—31no—
37C22×C4—32no—
38C22×C4—31no—
39C22×C4—31no—
40C22×C4—34no—
41C22×C4—61no—
42C22×C4—62no—
43C22×C4—61no—
44C22×C4C2×D312yescomplemented
45C4⋊C4—34no—
46C22⋊C4—34no—
47C22⋊C4—32no—
48C22⋊C4—32no—
49C42—34no—
50C22×C3—41no—
51C2×D3—42no—
52C2×D3—44no—
53A4C22×C411yescharacteristic, complemented
54C3×C4—42no—
55Dic3—42no—
56C23—31no—
57C23—31no—
58C23—32no—
59C23—32no—
60C23—34no—
61C23—61no—
62C23—62no—
63C23—61no—
64C23C4×D312yescomplemented
65C23C22×D311yescharacteristic
66D4—38no—
67D4—34no—
68D4—34no—
69C2×C4—32no—
70C2×C4—34no—
71C2×C4—31no—
72C2×C4—32no—
73C2×C4—32no—
74C2×C4—32no—
75C2×C4—32no—
76C2×C4—32no—
77C2×C4—61no—
78C2×C4—64no—
79C2×C4—62no—
80C2×C4—61no—
81C2×C4—61no—
82C2×C4—64no—
83C2×C4S411yescharacteristic, direct product factor
84C2×C3—42no—
85C2×C3—41no—
86D3—44no—
87C22—32no—
88C22—31no—
89C22—32no—
90C22—32no—
91C22—31no—
92C22—31no—
93C22—34no—
94C22—61no—
95C22—62no—
96C22—64no—
97C22—64no—
98C22—62no—
99C22C2×C4×D311yescharacteristic, complemented
100C22C2×S411yescharacteristic
101C4—34no—
102C4—32no—
103C4—62no—
104C4—62no—
105C4C2×S412yesdirect product factor
106C3—41noSylow
107C2—31no—
108C2—31no—
109C2—32no—
110C2—64no—
111C2C4×S412yesdirect product factor
112C2C22×S411yescharacteristic
subgroupG/H
center, hypercenterC2×C4S4
commutator, hypocenterA4C22×C4
FrattiniC2C22×S4
FittingC23×C4D3
socleC24C2×D3
prime radicalC22×S4C2
cosocleC2×A4C23
LCSG ▷ A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×A4C2semidirect
3. C2×SL(2,ℤ/4)C2semidirect
4. C4×S4C2direct (N × H)
7. C2×S4C4direct (N × H)
10. C4×A4C22semidirect
11. SL(2,ℤ/4)C22semidirect
14. C23×C4D3semidirect
22. C2×A4C2×C4semidirect
24. S4C2×C4direct (N × H)
44. C22×C4C2×D3semidirect
53. A4C22×C4semidirect
64. C23C4×D3semidirect
83. C2×C4S4direct (N × H)
99. C22C2×C4×D3semidirect
105. C4C2×S4direct (N × H)
111. C2C4×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×S42
C2×C4×A42
C2×SL(2,ℤ/4)2
C4×S42
C2×C4×D43
C2×C4×D34