Finite Group Databasebeta

← H(3,𝔽3)×C32⋊(C43⋊C3) #44534943

Subgroups — H(3,𝔽3)×C32⋊(C43⋊C3) #44534943

Order 31347.

2102 conjugacy classes of subgroups; only the 316 normal subgroups are shown.

normal subgroups only

This group has 2102 conjugacy classes of subgroups — too many to draw. Only its 316 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×C32⋊(C43⋊C3)C314yesmaximal, complemented
2C3×H(3,𝔽3)×C43⋊C3C313yesmaximal, complemented
3#23782794C318yesmaximal, complemented
4(C3×C43⋊C3)⋊H(3,𝔽3)C3124yesmaximal, complemented
5C32×H(3,𝔽3)×C43C311yesmaximal, characteristic, complemented
6C33×C43⋊C3C32112yescomplemented
7C3×C32⋊(C43⋊C3)—336no—
8C3×C32⋊(C43⋊C3)C3211yescharacteristic
9C3×C32⋊(C43⋊C3)C32124yescomplemented
10C3×C32⋊(C43⋊C3)C32124yescomplemented
11C3×C32⋊(C43⋊C3)C3218yescomplemented
12C3×C32⋊(C43⋊C3)C32148yescomplemented
13H(3,𝔽3)×C43⋊C3—327no—
14C34×C43C3214yescomplemented
15C3×H(3,𝔽3)×C43—39no—
16C3×H(3,𝔽3)×C43C3211yescharacteristic
17C3×H(3,𝔽3)×C43C3218yescomplemented
18C32×C43⋊C3—336no—
19C32×C43⋊C3—336no—
20C32×C43⋊C3—924no—
21C32×C43⋊C3C33124yes—
22C32×C43⋊C3C3313yes—
23C32⋊(C43⋊C3)—372no—
24C32⋊(C43⋊C3)—324no—
25C32⋊(C43⋊C3)—372no—
26C32⋊(C43⋊C3)—948no—
27C32⋊(C43⋊C3)H(3,𝔽3)19yesdirect product factor
28C33×C43—312no—
29C33×C43—312no—
30C33×C43—98no—
31C33×C43C3318yes—
32C33×C43C3314yes—
33C33×C43C3311yescharacteristic
34H(3,𝔽3)×C43—324no—
35H(3,𝔽3)×C43H(3,𝔽3)19yescomplemented
36H(3,𝔽3)2—431nomaximal, Sylow
37C32×C43—31no—
38C32×C43—34no—
39C32×C43—32no—
40C32×C43—38no—
41C32×C43—38no—
42C32×C43—38no—
43C32×C43—316no—
44C32×C43—936no—
45C32×C43C3×H(3,𝔽3)13yescomplemented
46C32×C43C3×H(3,𝔽3)112yescomplemented
47C32×C43C3411yescharacteristic
48C3×C43⋊C3—33no—
49C3×C43⋊C3—36no—
50C3×C43⋊C3—324no—
51C3×C43⋊C3—324no—
52C3×C43⋊C3—348no—
53C3×C43⋊C3—9108no—
54C3×C43⋊C3C3×H(3,𝔽3)19yescomplemented
55C32×H(3,𝔽3)—434no—
56C32×H(3,𝔽3)—433no—
57C32×H(3,𝔽3)C43⋊C311yescharacteristic, complemented
58C32⋊H(3,𝔽3)—438no—
59C32⋊H(3,𝔽3)—4324no—
60C3×C43—312no—
61C3×C43—33no—
62C3×C43—98no—
63C3×C43C32×H(3,𝔽3)11yescharacteristic
64C3×C43C32×H(3,𝔽3)11yescharacteristic
65C3×C43H(5,𝔽3)12yes—
66C43⋊C3—39no—
67C43⋊C3—924no—
68C34—4312no—
69C34C3×C43⋊C314yescomplemented
70C3×H(3,𝔽3)—39no—
71C3×H(3,𝔽3)—431no—
72C3×H(3,𝔽3)—4324no—
73C3×H(3,𝔽3)—438no—
74C3×H(3,𝔽3)—4324no—
75C3×H(3,𝔽3)—4348no—
76C3×H(3,𝔽3)—12936no—
77C3×H(3,𝔽3)—12927no—
78C3×H(3,𝔽3)C3×C43⋊C311yescharacteristic
79C3×H(3,𝔽3)C3×C43⋊C318yescomplemented
80C43H(3,𝔽3)211yescharacteristic, complemented, Sylow
81C33—312no—
82C33—312no—
83C33—98no—
84C33—433no—
85C33—4324no—
86C33—12936no—
87C33—12936no—
88C33—38724no—
89C33C32×C43⋊C318yes—
90C33C32×C43⋊C311yescharacteristic
91C33C32×C43⋊C314yes—
92H(3,𝔽3)—324no—
93H(3,𝔽3)—439no—
94H(3,𝔽3)—12972no—
95H(3,𝔽3)—12924no—
96H(3,𝔽3)—12972no—
97H(3,𝔽3)—38748no—
98H(3,𝔽3)C32⋊(C43⋊C3)19yesdirect product factor
99C32—32no—
100C32—38no—
101C32—34no—
102C32—31no—
103C32—316no—
104C32—38no—
105C32—38no—
106C32—936no—
107C32—439no—
108C32—1296no—
109C32—1293no—
110C32—12948no—
111C32—12924no—
112C32—12924no—
113C32—387108no—
114C32H(3,𝔽3)×C43⋊C313yescomplemented
115C32C3×C32⋊(C43⋊C3)112yescomplemented
116C32C33×C43⋊C311yescharacteristic
117C3—33no—
118C3—312no—
119C3—98no—
120C3—1299no—
121C3—38724no—
122C3C3×H(3,𝔽3)×C43⋊C311yescharacteristic
123C3C32×C32⋊(C43⋊C3)11yescharacteristic
124C3C43⋊H(5,𝔽3)12yes—
subgroupG/H
center, FrattiniC32C33×C43⋊C3
commutator, socle, cosocleC32×C43C34
FittingC32×H(3,𝔽3)×C43C3
hypercenterC32×H(3,𝔽3)C43⋊C3
hypocenterC43H(3,𝔽3)2
LCSG ▷ C32×C43 ▷ C43
UCSC1 ◁ C32 ◁ C32×H(3,𝔽3)

Complements

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

NH ≅ G/Ntype
1. C32×C32⋊(C43⋊C3)C3semidirect
2. C3×H(3,𝔽3)×C43⋊C3C3semidirect
3. #23782794C3semidirect
4. (C3×C43⋊C3)⋊H(3,𝔽3)C3semidirect
5. C32×H(3,𝔽3)×C43C3semidirect
6. C33×C43⋊C3C32semidirect
9. C3×C32⋊(C43⋊C3)C32semidirect
10. C3×C32⋊(C43⋊C3)C32semidirect
11. C3×C32⋊(C43⋊C3)C32semidirect
12. C3×C32⋊(C43⋊C3)C32semidirect
14. C34×C43C32semidirect
17. C3×H(3,𝔽3)×C43C32semidirect
27. C32⋊(C43⋊C3)H(3,𝔽3)direct (N × H)
35. H(3,𝔽3)×C43H(3,𝔽3)semidirect
45. C32×C43C3×H(3,𝔽3)semidirect
46. C32×C43C3×H(3,𝔽3)semidirect
54. C3×C43⋊C3C3×H(3,𝔽3)semidirect
57. C32×H(3,𝔽3)C43⋊C3semidirect
69. C34C3×C43⋊C3semidirect
79. C3×H(3,𝔽3)C3×C43⋊C3semidirect
80. C43H(3,𝔽3)2semidirect
98. H(3,𝔽3)C32⋊(C43⋊C3)direct (N × H)
114. C32H(3,𝔽3)×C43⋊C3semidirect
115. C32C3×C32⋊(C43⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C32×C32⋊(C43⋊C3)3
C3×H(3,𝔽3)×C43⋊C33
#237827943
(C3×C43⋊C3)⋊H(3,𝔽3)3
C32×H(3,𝔽3)×C433
H(3,𝔽3)243