Finite Group Databasebeta

← C23×C3×(C3⋊D3)⋊C3 #13049346

Subgroups — C23×C3×(C3⋊D3)⋊C3 #13049346

Order 1296.

1628 conjugacy classes of subgroups; only the 530 normal subgroups are shown.

normal subgroups only

This group has 1628 conjugacy classes of subgroups — too many to draw. Only its 530 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×C3×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C3×(C3⋊D3)⋊C3C2114yesmaximal, direct product factor
3C23×C3×C3⋊D3C311yesmaximal, characteristic, complemented
4C23×C32×D3—31nomaximal
5C23×(C3⋊D3)⋊C3C313yesmaximal, direct product factor
6C22×C3×H(3,𝔽3)C2217yescomplemented
7C2×C3×(C3⋊D3)⋊C3C22128yesdirect product factor
8C23×C33—21no—
9C23×C33D311yescharacteristic, complemented
10C23×C33C2×C311yescharacteristic, complemented
11C22×C3×C3⋊D3C2×C3114yescomplemented
12C22×C32×D3—314no—
13C23×H(3,𝔽3)—23no—
14C23×H(3,𝔽3)C2×C313yescomplemented
15C22×(C3⋊D3)⋊C3C2×C3142yesdirect product factor
16C2×C3×H(3,𝔽3)C2317yescomplemented
17C3×(C3⋊D3)⋊C3C2318yesdirect product factor
18C24×C32—91no—
19C23×C3⋊D3C3211yescharacteristic, complemented
20C23×C3×D3—31no—
21C23×C3×D3—33no—
22C23×C3×D3—91no—
23C22×C33—27no—
24C22×C33C2×D317yescomplemented
25C22×C33C22×C317yescomplemented
26C2×C3×C3⋊D3C22×C3128yescomplemented
27C2×C32×D3—328no—
28C22×H(3,𝔽3)—221no—
29C22×H(3,𝔽3)C22×C3121yescomplemented
30C2×(C3⋊D3)⋊C3C22×C3184yesdirect product factor
31C3×H(3,𝔽3)C2411yescharacteristic, complemented, Sylow
32C23×C32—21no—
33C23×C32—23no—
34C23×C32—31no—
35C23×C32—31no—
36C23×C32—61no—
37C23×C32—61no—
38C23×C32—62no—
39C23×C32—61no—
40C23×C32—914no—
41C23×C32C3×D311yescharacteristic
42C23×C32C3×D313yescomplemented
43C23×C32C2×C3211yescharacteristic, complemented
44C22×C3⋊D3C2×C32114yescomplemented
45C22×C3×D3—314no—
46C22×C3×D3—342no—
47C22×C3×D3—914no—
48C2×C33—27no—
49C2×C33C22×D317yescomplemented
50C2×C33C23×C317yescomplemented
51C3×C3⋊D3C23×C318yescomplemented
52C32×D3—38no—
53C2×H(3,𝔽3)—221no—
54C2×H(3,𝔽3)C23×C3121yescomplemented
55(C3⋊D3)⋊C3C23×C3124yesdirect product factor
56C24×C3—93no—
57C24×C3—91no—
58C23×D3—31no—
59C23×D3—91no—
60C22×C32—27no—
61C22×C32—221no—
62C22×C32—37no—
63C22×C32—37no—
64C22×C32—67no—
65C22×C32—67no—
66C22×C32—614no—
67C22×C32—67no—
68C22×C32—928no—
69C22×C32C2×C3×D317yes—
70C22×C32C2×C3×D3121yescomplemented
71C22×C32C22×C3217yescomplemented
72C2×C3⋊D3C22×C32128yescomplemented
73C2×C3×D3—328no—
74C2×C3×D3—384no—
75C2×C3×D3—928no—
76C33—21no—
77C33C23×D311yescharacteristic, complemented
78C33C24×C311yescharacteristic, complemented
79H(3,𝔽3)—23no—
80H(3,𝔽3)C24×C313yescomplemented
81C23×C3—21no—
82C23×C3—33no—
83C23×C3—31no—
84C23×C3—61no—
85C23×C3—63no—
86C23×C3—942no—
87C23×C3—914no—
88C23×C3(C3⋊D3)⋊C311yescharacteristic, direct product factor
89C23×C3C32×D311yescharacteristic
90C22×D3—314no—
91C22×D3—914no—
92C2×C32—27no—
93C2×C32—221no—
94C2×C32—37no—
95C2×C32—37no—
96C2×C32—67no—
97C2×C32—67no—
98C2×C32—614no—
99C2×C32—67no—
100C2×C32—98no—
101C2×C32C22×C3×D317yes—
102C2×C32C22×C3×D3121yescomplemented
103C2×C32C23×C3217yescomplemented
104C3⋊D3C23×C3218yescomplemented
105C3×D3—38no—
106C3×D3—324no—
107C3×D3—98no—
108C24—91noSylow
109C22×C3—27no—
110C22×C3—37no—
111C22×C3—321no—
112C22×C3—67no—
113C22×C3—621no—
114C22×C3—984no—
115C22×C3—928no—
116C22×C3C2×(C3⋊D3)⋊C317yesdirect product factor
117C22×C3C2×C32×D317yes—
118C2×D3—328no—
119C2×D3—928no—
120C32—21no—
121C32—23no—
122C32—31no—
123C32—31no—
124C32—61no—
125C32—61no—
126C32—62no—
127C32—61no—
128C32C23×C3×D311yescharacteristic
129C32C23×C3×D313yescomplemented
130C32C24×C3211yescharacteristic, complemented
131C23—914no—
132C23C3×(C3⋊D3)⋊C311yescharacteristic, direct product factor
133C2×C3—27no—
134C2×C3—37no—
135C2×C3—321no—
136C2×C3—67no—
137C2×C3—621no—
138C2×C3—924no—
139C2×C3—98no—
140C2×C3C22×(C3⋊D3)⋊C317yesdirect product factor
141C2×C3C22×C32×D317yes—
142D3—38no—
143D3—98no—
144C22—928no—
145C22C2×C3×(C3⋊D3)⋊C317yesdirect product factor
146C3—21no—
147C3—31no—
148C3—33no—
149C3—61no—
150C3—63no—
151C3C23×(C3⋊D3)⋊C311yescharacteristic, direct product factor
152C3C23×C32×D311yescharacteristic
153C2—98no—
154C2C22×C3×(C3⋊D3)⋊C317yesdirect product factor
subgroupG/H
center, hypercenterC23×C3(C3⋊D3)⋊C3
commutator, cosocle, hypocenterC32C24×C32
FrattiniC3C23×C32×D3
FittingC23×C3×H(3,𝔽3)C2
socleC23×C32C3×D3
LCSG ▷ C32
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×H(3,𝔽3)C2semidirect
2. C22×C3×(C3⋊D3)⋊C3C2direct (N × H)
3. C23×C3×C3⋊D3C3semidirect
5. C23×(C3⋊D3)⋊C3C3direct (N × H)
6. C22×C3×H(3,𝔽3)C22semidirect
7. C2×C3×(C3⋊D3)⋊C3C22direct (N × H)
9. C23×C33D3semidirect
10. C23×C33C2×C3semidirect
11. C22×C3×C3⋊D3C2×C3semidirect
14. C23×H(3,𝔽3)C2×C3semidirect
15. C22×(C3⋊D3)⋊C3C2×C3direct (N × H)
16. C2×C3×H(3,𝔽3)C23semidirect
17. C3×(C3⋊D3)⋊C3C23direct (N × H)
19. C23×C3⋊D3C32semidirect
24. C22×C33C2×D3semidirect
25. C22×C33C22×C3semidirect
26. C2×C3×C3⋊D3C22×C3semidirect
29. C22×H(3,𝔽3)C22×C3semidirect
30. C2×(C3⋊D3)⋊C3C22×C3direct (N × H)
31. C3×H(3,𝔽3)C24semidirect
42. C23×C32C3×D3semidirect
43. C23×C32C2×C32semidirect
44. C22×C3⋊D3C2×C32semidirect
49. C2×C33C22×D3semidirect
50. C2×C33C23×C3semidirect
51. C3×C3⋊D3C23×C3semidirect
54. C2×H(3,𝔽3)C23×C3semidirect
55. (C3⋊D3)⋊C3C23×C3direct (N × H)
70. C22×C32C2×C3×D3semidirect
71. C22×C32C22×C32semidirect
72. C2×C3⋊D3C22×C32semidirect
77. C33C23×D3semidirect
78. C33C24×C3semidirect
80. H(3,𝔽3)C24×C3semidirect
88. C23×C3(C3⋊D3)⋊C3direct (N × H)
102. C2×C32C22×C3×D3semidirect
103. C2×C32C23×C32semidirect
104. C3⋊D3C23×C32semidirect
116. C22×C3C2×(C3⋊D3)⋊C3direct (N × H)
129. C32C23×C3×D3semidirect
130. C32C24×C32semidirect
132. C23C3×(C3⋊D3)⋊C3direct (N × H)
140. C2×C3C22×(C3⋊D3)⋊C3direct (N × H)
145. C22C2×C3×(C3⋊D3)⋊C3direct (N × H)
151. C3C23×(C3⋊D3)⋊C3direct (N × H)
154. C2C22×C3×(C3⋊D3)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×H(3,𝔽3)2
C22×C3×(C3⋊D3)⋊C32
C23×C3×C3⋊D33
C23×C32×D33
C23×(C3⋊D3)⋊C33