Finite Group Databasebeta

← C23×C32×SL(2,𝔽3) #13181698

Subgroups — C23×C32×SL(2,𝔽3) #13181698

Order 1728.

3008 conjugacy classes of subgroups; only the 850 normal subgroups are shown.

normal subgroups only

This group has 3008 conjugacy classes of subgroups — too many to draw. Only its 850 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×C32×SL(2,𝔽3)C217yesmaximal, direct product factor
2C23×C32×Q8C311yesmaximal, characteristic, complemented
3C23×C3×SL(2,𝔽3)C3112yesmaximal, direct product factor
4C24×C33—41nomaximal
5C2×C32×SL(2,𝔽3)C2217yesdirect product factor
6C22×C32×Q8—37no—
7C22×C32×Q8C2×C317yescomplemented
8C23×C32×C4—31no—
9C22×C3×SL(2,𝔽3)C2×C3184yesdirect product factor
10C23×C33—48no—
11C23×C33—47no—
12C32×SL(2,𝔽3)C2311yescharacteristic, direct product factor
13C23×C3×Q8C3214yescomplemented
14C23×SL(2,𝔽3)C3219yesdirect product factor
15C24×C32—412no—
16C24×C32A411yescharacteristic
17C2×C32×Q8—314no—
18C2×C32×Q8—321no—
19C2×C32×Q8C22×C317yescomplemented
20C22×C32×C4—37no—
21C22×C32×C4—37no—
22C2×C3×SL(2,𝔽3)C22×C3184yesdirect product factor
23C22×C33—428no—
24C22×C33—47no—
25C22×C3×Q8—328no—
26C22×C3×Q8C2×C32128yescomplemented
27C23×C3×C4—34no—
28C22×SL(2,𝔽3)C2×C32163yesdirect product factor
29C23×C32—496no—
30C23×C32—484no—
31C23×C32SL(2,𝔽3)18yesdirect product factor
32C23×C32C2×A417yes—
33C32×Q8—37no—
34C32×Q8—314no—
35C32×Q8C23×C311yescharacteristic, complemented
36C2×C32×C4—321no—
37C2×C32×C4—37no—
38C3×SL(2,𝔽3)C23×C3112yesdirect product factor
39C23×Q8C3311yescharacteristic, complemented, Sylow
40C2×C33—414no—
41C2×C33—41no—
42C24×C3—49no—
43C24×C3C3×A414yes—
44C2×C3×Q8—356no—
45C2×C3×Q8—384no—
46C2×C3×Q8C22×C32128yescomplemented
47C22×C3×C4—328no—
48C22×C3×C4—328no—
49C2×SL(2,𝔽3)C22×C32163yesdirect product factor
50C22×C32—484no—
51C22×C32—4336no—
52C22×C32C2×SL(2,𝔽3)128yesdirect product factor
53C22×C32C22×A417yes—
54C32×C4—37no—
55C32×C4—31no—
56C22×Q8—37no—
57C22×Q8C2×C3317yescomplemented
58C23×C4—31no—
59C33—41noSylow
60C23×C3—472no—
61C23×C3—463no—
62C23×C3C3×SL(2,𝔽3)132yesdirect product factor
63C23×C3C2×C3×A4128yes—
64C3×Q8—328no—
65C3×Q8—356no—
66C3×Q8C23×C3214yescomplemented
67C2×C3×C4—384no—
68C2×C3×C4—328no—
69SL(2,𝔽3)C23×C3219yesdirect product factor
70C2×C32—4168no—
71C2×C32—412no—
72C2×C32C22×SL(2,𝔽3)114yesdirect product factor
73C2×C32C23×A411yescharacteristic
74C24C32×A411yescharacteristic
75C2×Q8—314no—
76C2×Q8—321no—
77C2×Q8C22×C3317yescomplemented
78C22×C4—37no—
79C22×C4—37no—
80C22×C3—463no—
81C22×C3—4252no—
82C22×C3C2×C3×SL(2,𝔽3)1112yesdirect product factor
83C22×C3C22×C3×A4128yes—
84C3×C4—34no—
85C3×C4—328no—
86C32—412no—
87C32C23×SL(2,𝔽3)11yescharacteristic, direct product factor
88C23C32×SL(2,𝔽3)18yesdirect product factor
89C23C2×C32×A417yes—
90Q8—37no—
91Q8—314no—
92Q8C23×C3311yescharacteristic, complemented
93C2×C4—37no—
94C2×C4—321no—
95C2×C3—4126no—
96C2×C3—49no—
97C2×C3C22×C3×SL(2,𝔽3)156yesdirect product factor
98C2×C3C23×C3×A414yes—
99C22C2×C32×SL(2,𝔽3)128yesdirect product factor
100C22C22×C32×A417yes—
101C4—31no—
102C4—37no—
103C3—49no—
104C3C23×C3×SL(2,𝔽3)14yesdirect product factor
105C2C22×C32×SL(2,𝔽3)114yesdirect product factor
106C2C23×C32×A411yescharacteristic
subgroupG/H
center, socle, hypercenterC24×C32A4
commutator, cosocle, hypocenterQ8C23×C33
FrattiniC2C23×C32×A4
FittingC23×C32×Q8C3
LCSG ▷ Q8
UCSC1 ◁ C24×C32

Complements

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

NH ≅ G/Ntype
1. C22×C32×SL(2,𝔽3)C2direct (N × H)
2. C23×C32×Q8C3semidirect
3. C23×C3×SL(2,𝔽3)C3direct (N × H)
5. C2×C32×SL(2,𝔽3)C22direct (N × H)
7. C22×C32×Q8C2×C3semidirect
9. C22×C3×SL(2,𝔽3)C2×C3direct (N × H)
12. C32×SL(2,𝔽3)C23direct (N × H)
13. C23×C3×Q8C32semidirect
14. C23×SL(2,𝔽3)C32direct (N × H)
19. C2×C32×Q8C22×C3semidirect
22. C2×C3×SL(2,𝔽3)C22×C3direct (N × H)
26. C22×C3×Q8C2×C32semidirect
28. C22×SL(2,𝔽3)C2×C32direct (N × H)
31. C23×C32SL(2,𝔽3)direct (N × H)
35. C32×Q8C23×C3semidirect
38. C3×SL(2,𝔽3)C23×C3direct (N × H)
39. C23×Q8C33semidirect
46. C2×C3×Q8C22×C32semidirect
49. C2×SL(2,𝔽3)C22×C32direct (N × H)
52. C22×C32C2×SL(2,𝔽3)direct (N × H)
57. C22×Q8C2×C33semidirect
62. C23×C3C3×SL(2,𝔽3)direct (N × H)
66. C3×Q8C23×C32semidirect
69. SL(2,𝔽3)C23×C32direct (N × H)
72. C2×C32C22×SL(2,𝔽3)direct (N × H)
77. C2×Q8C22×C33semidirect
82. C22×C3C2×C3×SL(2,𝔽3)direct (N × H)
87. C32C23×SL(2,𝔽3)direct (N × H)
88. C23C32×SL(2,𝔽3)direct (N × H)
92. Q8C23×C33semidirect
97. C2×C3C22×C3×SL(2,𝔽3)direct (N × H)
99. C22C2×C32×SL(2,𝔽3)direct (N × H)
104. C3C23×C3×SL(2,𝔽3)direct (N × H)
105. C2C22×C32×SL(2,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C23×C32×Q83
C23×C3×SL(2,𝔽3)3
C24×C334