Finite Group Databasebeta

← C23×C3×S4 #10596786

Subgroups — C23×C3×S4 #10596786

Order 576.

1434 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1434 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C23×C3×A4C211yesmaximal, characteristic, complemented
2C22×C3×S4C2114yesmaximal, direct product factor
3C23×S4C311yesmaximal, characteristic, direct product factor
4C23×C3×D4—31nomaximal
5C23×C3×D3—41nomaximal
6C22×C3×A4C2217yescomplemented
7C2×C3×S4C22128yesdirect product factor
8C25×C3—31no—
9C25×C3D311yescharacteristic, complemented
10C23×A4—21no—
11C23×A4C2×C311yescharacteristic, complemented
12C22×S4C2×C3114yesdirect product factor
13C22×C3×D4—314no—
14C22×C3×D4—314no—
15C23×C3×C4—31no—
16C23×C32—41no—
17C22×C3×D3—414no—
18C2×C3×A4C2317yescomplemented
19C3×S4C2318yesdirect product factor
20C23×D4—31noSylow
21C24×C3—31no—
22C24×C3—37no—
23C24×C3—314no—
24C24×C3—67no—
25C24×C3—61no—
26C24×C3C2×D317yescomplemented
27C23×D3—41no—
28C22×A4—27no—
29C22×A4C22×C317yescomplemented
30C2×S4C22×C3128yesdirect product factor
31C2×C3×D4—384no—
32C2×C3×D4—328no—
33C22×C3×C4—314no—
34C22×C32—47no—
35C2×C3×D3—428no—
36C3×A4C2411yescharacteristic, complemented
37C25—31no—
38C25C3×D311yescharacteristic, complemented
39C22×D4—314no—
40C22×D4—314no—
41C23×C4—31no—
42C23×C3—37no—
43C23×C3—37no—
44C23×C3—321no—
45C23×C3—328no—
46C23×C3—41no—
47C23×C3—67no—
48C23×C3—642no—
49C23×C3—614no—
50C23×C3—81no—
51C23×C3S411yescharacteristic, direct product factor
52C23×C3C22×D317yescomplemented
53C22×D3—414no—
54C2×A4—27no—
55C2×A4C23×C317yescomplemented
56S4C23×C318yesdirect product factor
57C3×D4—356no—
58C3×D4—38no—
59C2×C3×C4—328no—
60C2×C32—47no—
61C3×D3—48no—
62C24—31no—
63C24—37no—
64C24—314no—
65C24—67no—
66C24—61no—
67C24C2×C3×D317yescomplemented
68C2×D4—384no—
69C2×D4—328no—
70C22×C4—314no—
71C22×C3—37no—
72C22×C3—37no—
73C22×C3—321no—
74C22×C3—38no—
75C22×C3—47no—
76C22×C3—67no—
77C22×C3—628no—
78C22×C3—628no—
79C22×C3—87no—
80C22×C3C2×S417yesdirect product factor
81C22×C3C23×D311yescharacteristic, complemented
82C2×D3—428no—
83A4—21no—
84A4C24×C311yescharacteristic, complemented
85C3×C4—38no—
86C32—41noSylow
87C23—37no—
88C23—37no—
89C23—321no—
90C23—328no—
91C23—67no—
92C23—642no—
93C23—614no—
94C23C3×S411yescharacteristic, direct product factor
95C23C22×C3×D317yescomplemented
96D4—356no—
97D4—38no—
98C2×C4—328no—
99C2×C3—31no—
100C2×C3—37no—
101C2×C3—47no—
102C2×C3—68no—
103C2×C3—87no—
104C2×C3C22×S417yesdirect product factor
105D3—48no—
106C22—37no—
107C22—37no—
108C22—321no—
109C22—38no—
110C22—67no—
111C22—628no—
112C22—628no—
113C22C2×C3×S417yesdirect product factor
114C22C23×C3×D311yescharacteristic, complemented
115C4—38no—
116C3—41no—
117C3—81no—
118C3C23×S411yescharacteristic, direct product factor
119C2—31no—
120C2—37no—
121C2—68no—
122C2C22×C3×S417yesdirect product factor
subgroupG/H
center, hypercenterC23×C3S4
commutator, cosocle, hypocenterA4C24×C3
FrattiniC1G
Fitting, socleC25×C3D3
LCSG ▷ A4
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×A4C2semidirect
2. C22×C3×S4C2direct (N × H)
3. C23×S4C3direct (N × H)
6. C22×C3×A4C22semidirect
7. C2×C3×S4C22direct (N × H)
9. C25×C3D3semidirect
11. C23×A4C2×C3semidirect
12. C22×S4C2×C3direct (N × H)
18. C2×C3×A4C23semidirect
19. C3×S4C23direct (N × H)
26. C24×C3C2×D3semidirect
29. C22×A4C22×C3semidirect
30. C2×S4C22×C3direct (N × H)
36. C3×A4C24semidirect
38. C25C3×D3semidirect
51. C23×C3S4direct (N × H)
52. C23×C3C22×D3semidirect
55. C2×A4C23×C3semidirect
56. S4C23×C3direct (N × H)
67. C24C2×C3×D3semidirect
80. C22×C3C2×S4direct (N × H)
81. C22×C3C23×D3semidirect
84. A4C24×C3semidirect
94. C23C3×S4direct (N × H)
95. C23C22×C3×D3semidirect
104. C2×C3C22×S4direct (N × H)
113. C22C2×C3×S4direct (N × H)
114. C22C23×C3×D3semidirect
118. C3C23×S4direct (N × H)
122. C2C22×C3×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×A42
C22×C3×S42
C23×S43
C23×C3×D43
C23×C3×D34