Finite Group Databasebeta

← H(3,𝔽3)×3−1+2×C43 #44531832

Subgroups — H(3,𝔽3)×3−1+2×C43 #44531832

Order 31347.

1244 conjugacy classes of subgroups; only the 518 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 518 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
1C32×3−1+2×C43C314yesmaximal, complemented
2C32×H(3,𝔽3)×C43C311yesmaximal, characteristic
3C43×C9⋊H(3,𝔽3)C3124yesmaximal, complemented
4C43×#6891C318yesmaximal, complemented
5C9×H(3,𝔽3)×C43C313yesmaximal, complemented
6C34×C43C3214yes—
7C3×3−1+2×C43—336no—
8C3×3−1+2×C43C3218yes—
9C3×3−1+2×C43C3211yescharacteristic
10C3×3−1+2×C43C32124yescomplemented
11C3×H(3,𝔽3)×C43—39no—
12C3×H(3,𝔽3)×C43C3218yes—
13C3×H(3,𝔽3)×C43C3211yescharacteristic
14C32×C9×C43C32112yescomplemented
15C43×C32⋊C9C32124yescomplemented
16C43×C32⋊C9C32148yescomplemented
17C33×C43—312no—
18C33×C43—312no—
19C33×C43—98no—
20C33×C43C3318yes—
21C33×C43C3311yescharacteristic
22C33×C43C3314yes—
233−1+2×C43—324no—
243−1+2×C43—372no—
253−1+2×C43H(3,𝔽3)19yesdirect product factor
26H(3,𝔽3)×C43—324no—
27H(3,𝔽3)×C433−1+219yesdirect product factor
28C3×C9×C43—336no—
29C3×C9×C43C33124yes—
30C3×C9×C43C3313yes—
31H(3,𝔽3)×3−1+2C4311yesmaximal, characteristic, direct product factor, Sylow
32C32×C43—34no—
33C32×C43—31no—
34C32×C43—32no—
35C32×C43—38no—
36C32×C43—38no—
37C32×C43—38no—
38C32×C43—316no—
39C32×C43—936no—
40C32×C43C3×H(3,𝔽3)13yes—
41C32×C43C3×3−1+2112yescomplemented
42C32×C43C3411yescharacteristic
43C9×C43—324no—
44C9×C43C3×H(3,𝔽3)19yescomplemented
45C32×3−1+2C3×C4314yescomplemented
46C32×H(3,𝔽3)C3×C4311yescharacteristic
47C9⋊H(3,𝔽3)C3×C43124yescomplemented
48#6891C3×C4318yescomplemented
49C9×H(3,𝔽3)C3×C4313yescomplemented
50C3×C43—33no—
51C3×C43—312no—
52C3×C43—98no—
53C3×C43C32×H(3,𝔽3)11yescharacteristic
54C3×C43C32×3−1+211yescharacteristic
55C3×C433−1+412yes—
56C34C32×C4314yes—
57C3×3−1+2—336no—
58C3×3−1+2C32×C4318yes—
59C3×3−1+2C32×C4311yescharacteristic
60C3×3−1+2C32×C43124yescomplemented
61C3×H(3,𝔽3)—39no—
62C3×H(3,𝔽3)C32×C4318yes—
63C3×H(3,𝔽3)C32×C4311yescharacteristic
64C32×C9C32×C43112yescomplemented
65C32⋊C9C32×C43124yescomplemented
66C32⋊C9C32×C43148yescomplemented
67C43H(3,𝔽3)×3−1+211yescharacteristic, direct product factor, Sylow
68C33—312no—
69C33—312no—
70C33—98no—
71C33C33×C4318yes—
72C33C33×C4311yescharacteristic
73C33C33×C4314yes—
743−1+2—324no—
753−1+2—372no—
763−1+2H(3,𝔽3)×C4319yesdirect product factor
77H(3,𝔽3)—324no—
78H(3,𝔽3)3−1+2×C4319yesdirect product factor
79C3×C9—336no—
80C3×C9C33×C43124yes—
81C3×C9C33×C4313yes—
82C32—34no—
83C32—31no—
84C32—32no—
85C32—38no—
86C32—38no—
87C32—38no—
88C32—316no—
89C32—936no—
90C32C3×H(3,𝔽3)×C4313yes—
91C32C3×3−1+2×C43112yescomplemented
92C32C34×C4311yescharacteristic
93C9—324no—
94C9C3×H(3,𝔽3)×C4319yescomplemented
95C3—33no—
96C3—312no—
97C3—98no—
98C3C32×H(3,𝔽3)×C4311yescharacteristic
99C3C32×3−1+2×C4311yescharacteristic
100C3C43×3−1+412yes—
subgroupG/H
center, socleC32×C43C34
commutator, Frattini, cosocleC32C34×C43
prime radicalC32×H(3,𝔽3)×C43C3
LCSG ▷ C32 ▷ C1
UCSC1 ◁ C32×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C32×3−1+2×C43C3semidirect
3. C43×C9⋊H(3,𝔽3)C3semidirect
4. C43×#6891C3semidirect
5. C9×H(3,𝔽3)×C43C3semidirect
10. C3×3−1+2×C43C32semidirect
14. C32×C9×C43C32semidirect
15. C43×C32⋊C9C32semidirect
16. C43×C32⋊C9C32semidirect
25. 3−1+2×C43H(3,𝔽3)direct (N × H)
27. H(3,𝔽3)×C433−1+2direct (N × H)
31. H(3,𝔽3)×3−1+2C43direct (N × H)
41. C32×C43C3×3−1+2semidirect
44. C9×C43C3×H(3,𝔽3)semidirect
45. C32×3−1+2C3×C43semidirect
47. C9⋊H(3,𝔽3)C3×C43semidirect
48. #6891C3×C43semidirect
49. C9×H(3,𝔽3)C3×C43semidirect
60. C3×3−1+2C32×C43semidirect
64. C32×C9C32×C43semidirect
65. C32⋊C9C32×C43semidirect
66. C32⋊C9C32×C43semidirect
67. C43H(3,𝔽3)×3−1+2direct (N × H)
76. 3−1+2H(3,𝔽3)×C43direct (N × H)
78. H(3,𝔽3)3−1+2×C43direct (N × H)
91. C32C3×3−1+2×C43semidirect
94. C9C3×H(3,𝔽3)×C43semidirect

Maximal subgroups

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

subgroupindex
C32×3−1+2×C433
C32×H(3,𝔽3)×C433
C43×C9⋊H(3,𝔽3)3
C43×#68913
C9×H(3,𝔽3)×C433
H(3,𝔽3)×3−1+243