Finite Group Databasebeta

← C24×#1113 #13181867

Subgroups — C24×#1113 #13181867

Order 1728.

24978 conjugacy classes of subgroups; only the 4215 normal subgroups are shown.

normal subgroups only

This group has 24978 conjugacy classes of subgroups — too many to draw. Only its 4215 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
1C24×C3×C3⋊D3C213yesmaximal, complemented
2C23×#1113C2160yesmaximal, direct product factor
3C24×D32—33nomaximal
4C24×C33C2211yescharacteristic, complemented
5C23×C3×C3⋊D3C22190yescomplemented
6C22×#1113C221560yesdirect product factor
7C24×C3⋊D3D313yescomplemented
8C24×C3×D3—36no—
9C24×C3×D3—63no—
10C23×D32—3180no—
11C23×C33C23115yescomplemented
12C22×C3×C3⋊D3C231420yescomplemented
13C2×#1113C231960yesdirect product factor
14C25×D3—93no—
15C24×C32—23no—
16C24×C32—41no—
17C24×C32C2×D313yescomplemented
18C23×C3⋊D3C2×D3190yescomplemented
19C23×C3×D3—3180no—
20C23×C3×D3—690no—
21C22×D32—31680no—
22C22×C33C24135yescomplemented
23C2×C3×C3⋊D3C241360yescomplemented
24#1113C241256yesdirect product factor
25C25×C3—93no—
26C24×D3—36no—
27C24×D3—63no—
28C24×D3—9180no—
29C23×C32—245no—
30C23×C32—415no—
31C23×C32C22×D3145yescomplemented
32C22×C3⋊D3C22×D31420yescomplemented
33C22×C3×D3—3840no—
34C22×C3×D3—6420no—
35C2×D32—32880no—
36C26—271noSylow
37C2×C33C25115yescomplemented
38C3×C3⋊D3C25148yescomplemented
39C24×C3—23no—
40C24×C3—41no—
41C24×C3—990no—
42C24×C3D3213yescomplemented
43C23×D3—3180no—
44C23×D3—690no—
45C23×D3—91680no—
46C22×C32—2105no—
47C22×C32—435no—
48C22×C32C23×D31105yescomplemented
49C2×C3⋊D3C23×D31360yescomplemented
50C2×C3×D3—3720no—
51C2×C3×D3—6360no—
52D32—3768no—
53C25—93no—
54C25—2760no—
55C33C2611yescharacteristic, complemented, Sylow
56C23×C3—245no—
57C23×C3—415no—
58C23×C3—9420no—
59C23×C3C2×D32145yescomplemented
60C22×D3—3840no—
61C22×D3—6420no—
62C22×D3—92880no—
63C2×C32—245no—
64C2×C32—415no—
65C2×C32C24×D3145yescomplemented
66C3⋊D3C24×D3148yescomplemented
67C3×D3—396no—
68C3×D3—648no—
69C24—990no—
70C24—27560no—
71C24#111311yescharacteristic, direct product factor
72C22×C3—2105no—
73C22×C3—435no—
74C22×C3—9360no—
75C22×C3C22×D321105yescomplemented
76C2×D3—3720no—
77C2×D3—6360no—
78C2×D3—9768no—
79C32—23no—
80C32—41no—
81C32C25×D313yescomplemented
82C23—9420no—
83C23—27960no—
84C23C2×#1113115yesdirect product factor
85C2×C3—245no—
86C2×C3—415no—
87C2×C3—948no—
88C2×C3C23×D32145yescomplemented
89D3—396no—
90D3—648no—
91C22—9360no—
92C22—27256no—
93C22C22×#1113135yesdirect product factor
94C3—23no—
95C3—41no—
96C3C24×D3213yescomplemented
97C2—948no—
98C2C23×#1113115yesdirect product factor
subgroupG/H
center, hypercenterC24#1113
commutator, cosocle, hypocenterC33C26
FrattiniC1G
Fitting, socleC24×C33C22
LCSG ▷ C33
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C24×C3×C3⋊D3C2semidirect
2. C23×#1113C2direct (N × H)
4. C24×C33C22semidirect
5. C23×C3×C3⋊D3C22semidirect
6. C22×#1113C22direct (N × H)
7. C24×C3⋊D3D3semidirect
11. C23×C33C23semidirect
12. C22×C3×C3⋊D3C23semidirect
13. C2×#1113C23direct (N × H)
17. C24×C32C2×D3semidirect
18. C23×C3⋊D3C2×D3semidirect
22. C22×C33C24semidirect
23. C2×C3×C3⋊D3C24semidirect
24. #1113C24direct (N × H)
31. C23×C32C22×D3semidirect
32. C22×C3⋊D3C22×D3semidirect
37. C2×C33C25semidirect
38. C3×C3⋊D3C25semidirect
42. C24×C3D32semidirect
48. C22×C32C23×D3semidirect
49. C2×C3⋊D3C23×D3semidirect
55. C33C26semidirect
59. C23×C3C2×D32semidirect
65. C2×C32C24×D3semidirect
66. C3⋊D3C24×D3semidirect
71. C24#1113direct (N × H)
75. C22×C3C22×D32semidirect
81. C32C25×D3semidirect
84. C23C2×#1113direct (N × H)
88. C2×C3C23×D32semidirect
93. C22C22×#1113direct (N × H)
96. C3C24×D32semidirect
98. C2C23×#1113direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C23×#11132
C24×D323