Finite Group Databasebeta

← C24×C32×A4 #13181870

Subgroups — C24×C32×A4 #13181870

Order 1728.

9134 conjugacy classes of subgroups; only the 2278 normal subgroups are shown.

normal subgroups only

This group has 9134 conjugacy classes of subgroups — too many to draw. Only its 2278 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×C32×A4C2115yesmaximal, direct product factor
2C26×C32C311yesmaximal, characteristic, complemented
3C24×C3×A4C3112yesmaximal, direct product factor
4C24×C33—41nomaximal
5C22×C32×A4C22135yesdirect product factor
6C25×C32—315no—
7C25×C32—31no—
8C25×C32C2×C3115yescomplemented
9C23×C3×A4C2×C31180yesdirect product factor
10C23×C33—415no—
11C2×C32×A4C23115yesdirect product factor
12C26×C3C3214yescomplemented
13C24×A4C3219yesdirect product factor
14C24×C32—370no—
15C24×C32—315no—
16C24×C32—315no—
17C24×C32—3105no—
18C24×C32—412no—
19C24×C32A411yescharacteristic, direct product factor
20C24×C32C22×C3135yescomplemented
21C22×C3×A4C22×C31420yesdirect product factor
22C22×C33—435no—
23C32×A4C2411yescharacteristic, direct product factor
24C25×C3—360no—
25C25×C3—34no—
26C25×C3C2×C32160yescomplemented
27C23×A4C2×C321135yesdirect product factor
28C23×C32—3105no—
29C23×C32—3210no—
30C23×C32—3105no—
31C23×C32—335no—
32C23×C32—4180no—
33C23×C32C2×A4115yesdirect product factor
34C23×C32C23×C3115yescomplemented
35C2×C3×A4C23×C31180yesdirect product factor
36C26C3311yescharacteristic, complemented, Sylow
37C2×C33—415no—
38C24×C3—3420no—
39C24×C3—3280no—
40C24×C3—360no—
41C24×C3—360no—
42C24×C3—49no—
43C24×C3C3×A414yesdirect product factor
44C24×C3C22×C321140yescomplemented
45C22×A4C22×C321315yesdirect product factor
46C22×C32—315no—
47C22×C32—370no—
48C22×C32—3105no—
49C22×C32—315no—
50C22×C32—4420no—
51C22×C32C22×A4135yesdirect product factor
52C22×C32C24×C311yescharacteristic, complemented
53C3×A4C24×C3112yesdirect product factor
54C25—315no—
55C25—31no—
56C25C2×C33115yescomplemented
57C33—41noSylow
58C23×C3—3420no—
59C23×C3—3840no—
60C23×C3—3420no—
61C23×C3—3140no—
62C23×C3—4135no—
63C23×C3C2×C3×A4160yesdirect product factor
64C23×C3C23×C32160yescomplemented
65C2×A4C23×C321135yesdirect product factor
66C2×C32—315no—
67C2×C32—31no—
68C2×C32—4180no—
69C2×C32C23×A4115yesdirect product factor
70C24—3105no—
71C24—370no—
72C24—315no—
73C24—315no—
74C24C32×A411yescharacteristic, direct product factor
75C24C22×C33135yescomplemented
76C22×C3—360no—
77C22×C3—3280no—
78C22×C3—360no—
79C22×C3—3420no—
80C22×C3—4315no—
81C22×C3C22×C3×A41140yesdirect product factor
82C22×C3C24×C3214yescomplemented
83A4C24×C3219yesdirect product factor
84C32—412no—
85C32C24×A411yescharacteristic, direct product factor
86C23—3105no—
87C23—3210no—
88C23—335no—
89C23—3105no—
90C23C2×C32×A4115yesdirect product factor
91C23C23×C33115yescomplemented
92C2×C3—34no—
93C2×C3—360no—
94C2×C3—4135no—
95C2×C3C23×C3×A4160yesdirect product factor
96C22—315no—
97C22—370no—
98C22—315no—
99C22—3105no—
100C22C22×C32×A4135yesdirect product factor
101C22C24×C3311yescharacteristic, complemented
102C3—49no—
103C3C24×C3×A414yesdirect product factor
104C2—31no—
105C2—315no—
106C2C23×C32×A4115yesdirect product factor
subgroupG/H
center, hypercenterC24×C32A4
commutator, cosocle, hypocenterC22C24×C33
FrattiniC1G
Fitting, socleC26×C32C3
LCSG ▷ C22
UCSC1 ◁ C24×C32

Complements

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

NH ≅ G/Ntype
1. C23×C32×A4C2direct (N × H)
2. C26×C32C3semidirect
3. C24×C3×A4C3direct (N × H)
5. C22×C32×A4C22direct (N × H)
8. C25×C32C2×C3semidirect
9. C23×C3×A4C2×C3direct (N × H)
11. C2×C32×A4C23direct (N × H)
12. C26×C3C32semidirect
13. C24×A4C32direct (N × H)
19. C24×C32A4direct (N × H)
20. C24×C32C22×C3semidirect
21. C22×C3×A4C22×C3direct (N × H)
23. C32×A4C24direct (N × H)
26. C25×C3C2×C32semidirect
27. C23×A4C2×C32direct (N × H)
33. C23×C32C2×A4direct (N × H)
34. C23×C32C23×C3semidirect
35. C2×C3×A4C23×C3direct (N × H)
36. C26C33semidirect
43. C24×C3C3×A4direct (N × H)
44. C24×C3C22×C32semidirect
45. C22×A4C22×C32direct (N × H)
51. C22×C32C22×A4direct (N × H)
52. C22×C32C24×C3semidirect
53. C3×A4C24×C3direct (N × H)
56. C25C2×C33semidirect
63. C23×C3C2×C3×A4direct (N × H)
64. C23×C3C23×C32semidirect
65. C2×A4C23×C32direct (N × H)
69. C2×C32C23×A4direct (N × H)
74. C24C32×A4direct (N × H)
75. C24C22×C33semidirect
81. C22×C3C22×C3×A4direct (N × H)
82. C22×C3C24×C32semidirect
83. A4C24×C32direct (N × H)
85. C32C24×A4direct (N × H)
90. C23C2×C32×A4direct (N × H)
91. C23C23×C33semidirect
95. C2×C3C23×C3×A4direct (N × H)
100. C22C22×C32×A4direct (N × H)
101. C22C24×C33semidirect
103. C3C24×C3×A4direct (N × H)
106. C2C23×C32×A4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C26×C323
C24×C3×A43
C24×C334