Finite Group Databasebeta

← #44534942

Subgroups — #44534942

Order 31347.

2102 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 2102 conjugacy classes of subgroups — too many to draw. Only its 298 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)C312yesmaximal, complemented
2C3×H(3,𝔽3)×C43⋊C3C316yesmaximal, complemented
3#23782794C311yesmaximal, characteristic, complemented
4(C3×C43⋊C3)⋊H(3,𝔽3)C3112yesmaximal, complemented
5(C3×C43⋊C3)⋊H(3,𝔽3)C3118yesmaximal, complemented
6C43×C32⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
7C33×C43⋊C3C3219yescomplemented
8C33×C43⋊C3C3216yescomplemented
9C3×C32⋊(C43⋊C3)—318no—
10C3×C32⋊(C43⋊C3)C3212yes—
11C3×C32⋊(C43⋊C3)C32136yescomplemented
12C3×C32⋊(C43⋊C3)C32112yescomplemented
13C3×C32⋊(C43⋊C3)C3214yescomplemented
14C3×C32⋊(C43⋊C3)C3216yescomplemented
15C3×C32⋊(C43⋊C3)C32136yescomplemented
16C3×C32⋊(C43⋊C3)C3216yescomplemented
17H(3,𝔽3)×C43⋊C3—354no—
18C34×C43C3211yescharacteristic, complemented
19C3×H(3,𝔽3)×C43C3216yescomplemented
20C3×H(3,𝔽3)×C43C3216yescomplemented
21C32×C43⋊C3—318no—
22C32×C43⋊C3—354no—
23C32×C43⋊C3—318no—
24C32×C43⋊C3—918no—
25C32×C43⋊C3—912no—
26C32×C43⋊C3C3319yes—
27C32×C43⋊C3C33112yes—
28C32×C43⋊C3C3316yes—
29C32⋊(C43⋊C3)—318no—
30C32⋊(C43⋊C3)—312no—
31C32⋊(C43⋊C3)—3108no—
32C32⋊(C43⋊C3)—336no—
33C32⋊(C43⋊C3)—936no—
34C32⋊(C43⋊C3)—96no—
35C32⋊(C43⋊C3)H(3,𝔽3)118yescomplemented
36C33×C43—36no—
37C33×C43—92no—
38C33×C43C3319yes—
39C33×C43C3312yes—
40C33×C43C3312yes—
41H(3,𝔽3)×C43—318no—
42H(3,𝔽3)×C43—96no—
43H(3,𝔽3)2—431nomaximal, Sylow
44C32×C43—32no—
45C32×C43—34no—
46C32×C43—318no—
47C32×C43—34no—
48C32×C43—32no—
49C32×C43—318no—
50C32×C43—32no—
51C32×C43—99no—
52C32×C43C3×H(3,𝔽3)16yescomplemented
53C32×C43C3411yescharacteristic
54C3×C43⋊C3—36no—
55C3×C43⋊C3—312no—
56C3×C43⋊C3—318no—
57C3×C43⋊C3—312no—
58C3×C43⋊C3—312no—
59C3×C43⋊C3—318no—
60C3×C43⋊C3—324no—
61C3×C43⋊C3—981no—
62C3×C43⋊C3—954no—
63C3×C43⋊C3C3×H(3,𝔽3)118yescomplemented
64C32×H(3,𝔽3)—432no—
65C32×H(3,𝔽3)—436no—
66C32⋊H(3,𝔽3)—431no—
67C32⋊H(3,𝔽3)—4318no—
68C32⋊H(3,𝔽3)—4312no—
69C32⋊H(3,𝔽3)C43⋊C311yescharacteristic, complemented
70C3×C43—36no—
71C3×C43—99no—
72C3×C43—92no—
73C3×C43C32×H(3,𝔽3)12yes—
74C3×C43H(5,𝔽3)12yes—
75C43⋊C3—318no—
76C43⋊C3—99no—
77C43⋊C3—912no—
78C34—439no—
79C34—436no—
80C34C3×C43⋊C311yescharacteristic, complemented
81C3×H(3,𝔽3)—432no—
82C3×H(3,𝔽3)—4312no—
83C3×H(3,𝔽3)—4336no—
84C3×H(3,𝔽3)—434no—
85C3×H(3,𝔽3)—436no—
86C3×H(3,𝔽3)—436no—
87C3×H(3,𝔽3)—4336no—
88C3×H(3,𝔽3)—12954no—
89C3×H(3,𝔽3)—12918no—
90C3×H(3,𝔽3)C3×C43⋊C316yescomplemented
91C3×H(3,𝔽3)C3×C43⋊C316yescomplemented
92C43H(3,𝔽3)211yescharacteristic, complemented, Sylow
93C33—36no—
94C33—92no—
95C33—436no—
96C33—4312no—
97C33—439no—
98C33—12918no—
99C33—12954no—
100C33—12918no—
101C33—38712no—
102C33—38718no—
103C33C32×C43⋊C312yes—
104C33C32×C43⋊C319yes—
105C33C32×C43⋊C312yes—
106H(3,𝔽3)—318no—
107H(3,𝔽3)—96no—
108H(3,𝔽3)—4318no—
109H(3,𝔽3)—129108no—
110H(3,𝔽3)—12936no—
111H(3,𝔽3)—12918no—
112H(3,𝔽3)—12912no—
113H(3,𝔽3)—38736no—
114H(3,𝔽3)—3876no—
115C32—34no—
116C32—32no—
117C32—32no—
118C32—318no—
119C32—32no—
120C32—318no—
121C32—34no—
122C32—99no—
123C32—4318no—
124C32—12912no—
125C32—1296no—
126C32—12924no—
127C32—12918no—
128C32—12912no—
129C32—12912no—
130C32—12918no—
131C32—38754no—
132C32—38781no—
133C32H(3,𝔽3)×C43⋊C316yescomplemented
134C32C33×C43⋊C311yescharacteristic
135C3—36no—
136C3—92no—
137C3—99no—
138C3—12918no—
139C3—38712no—
140C3—3879no—
141C3C3×H(3,𝔽3)×C43⋊C312yes—
142C3C43⋊H(5,𝔽3)12yes—
subgroupG/H
center, FrattiniC32C33×C43⋊C3
commutator, socle, cosocleC32×C43C34
FittingC43×C32⋊H(3,𝔽3)C3
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. (C3×C43⋊C3)⋊H(3,𝔽3)C3semidirect
6. C43×C32⋊H(3,𝔽3)C3semidirect
7. C33×C43⋊C3C32semidirect
8. C33×C43⋊C3C32semidirect
11. C3×C32⋊(C43⋊C3)C32semidirect
12. C3×C32⋊(C43⋊C3)C32semidirect
13. C3×C32⋊(C43⋊C3)C32semidirect
14. C3×C32⋊(C43⋊C3)C32semidirect
15. C3×C32⋊(C43⋊C3)C32semidirect
16. C3×C32⋊(C43⋊C3)C32semidirect
18. C34×C43C32semidirect
19. C3×H(3,𝔽3)×C43C32semidirect
20. C3×H(3,𝔽3)×C43C32semidirect
35. C32⋊(C43⋊C3)H(3,𝔽3)semidirect
52. C32×C43C3×H(3,𝔽3)semidirect
63. C3×C43⋊C3C3×H(3,𝔽3)semidirect
69. C32⋊H(3,𝔽3)C43⋊C3semidirect
80. C34C3×C43⋊C3semidirect
90. C3×H(3,𝔽3)C3×C43⋊C3semidirect
91. C3×H(3,𝔽3)C3×C43⋊C3semidirect
92. C43H(3,𝔽3)2semidirect
133. C32H(3,𝔽3)×C43⋊C3semidirect

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
C43×C32⋊H(3,𝔽3)3
H(3,𝔽3)243