Finite Group Databasebeta

← C22×#92391 #14519368

Subgroups — C22×#92391 #14519368

Order 1944.

1421 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1421 conjugacy classes of subgroups — too many to draw. Only its 251 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×H(5,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#92391C216yesmaximal, direct product factor
3C22×C3⋊((C3⋊D3)⋊C3)C314yesmaximal, complemented
4C22×C3×(C3⋊D3)⋊C3—34nomaximal
5C2×H(5,𝔽3)C2213yescomplemented
6#92391C2214yesdirect product factor
7C22×C3×H(3,𝔽3)—24no—
8C22×C3×H(3,𝔽3)—212no—
9C22×C3×H(3,𝔽3)D314yescomplemented
10C22×C3×H(3,𝔽3)C2×C314yescomplemented
11C2×C3⋊((C3⋊D3)⋊C3)C2×C3124yescomplemented
12C2×C3×(C3⋊D3)⋊C3—324no—
13H(5,𝔽3)C2311yescharacteristic, complemented, Sylow
14C22×C3⋊(C3⋊D3)C3211yescharacteristic, complemented
15C22×C3×C3⋊D3—34no—
16C22×C32×D3—91no—
17C22×(C3⋊D3)⋊C3—312no—
18C2×C3×H(3,𝔽3)—212no—
19C2×C3×H(3,𝔽3)—236no—
20C2×C3×H(3,𝔽3)C2×D3112yescomplemented
21C2×C3×H(3,𝔽3)C22×C3112yescomplemented
22C3⋊((C3⋊D3)⋊C3)C22×C3116yescomplemented
23C3×(C3⋊D3)⋊C3—316no—
24C22×C33—26no—
25C22×C33—23no—
26C22×C33—24no—
27C22×C33—24no—
28C22×C33C3×D314yescomplemented
29C22×C33C3⋊D311yescharacteristic, complemented
30C22×C33C2×C3211yescharacteristic, complemented
31C2×C3⋊(C3⋊D3)C2×C3216yescomplemented
32C2×C3×C3⋊D3—324no—
33C2×C32×D3—96no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)—28no—
36C22×H(3,𝔽3)—26no—
37C22×H(3,𝔽3)—212no—
38C22×H(3,𝔽3)—212no—
39C22×H(3,𝔽3)C3×D3112yescomplemented
40C2×(C3⋊D3)⋊C3—372no—
41C3×H(3,𝔽3)—24no—
42C3×H(3,𝔽3)—212no—
43C3×H(3,𝔽3)C22×D314yescomplemented
44C3×H(3,𝔽3)C23×C314yescomplemented
45C23×C32—271no—
46C22×C3⋊D3—34no—
47C22×C3⋊D3—271no—
48C22×C3×D3—94no—
49C22×C3×D3—274no—
50C2×C33—218no—
51C2×C33—29no—
52C2×C33—212no—
53C2×C33—212no—
54C2×C33C2×C3×D3112yescomplemented
55C2×C33C2×C3⋊D313yescomplemented
56C2×C33C22×C3213yescomplemented
57C3⋊(C3⋊D3)C22×C3214yescomplemented
58C3×C3⋊D3—316no—
59C32×D3—94no—
60C2×H(3,𝔽3)—23no—
61C2×H(3,𝔽3)—224no—
62C2×H(3,𝔽3)—218no—
63C2×H(3,𝔽3)—236no—
64C2×H(3,𝔽3)—236no—
65C2×H(3,𝔽3)C2×C3×D3136yescomplemented
66(C3⋊D3)⋊C3—348no—
67C22×C32—24no—
68C22×C32—212no—
69C22×C32—91no—
70C22×C32—91no—
71C22×C32—94no—
72C22×C32—184no—
73C22×C32—184no—
74C22×C32—186no—
75C22×C32—183no—
76C22×C32—276no—
77C22×C32C32×D314yes—
78C22×C32C3×C3⋊D314yes—
79C2×C3⋊D3—324no—
80C2×C3⋊D3—276no—
81C2×C3×D3—924no—
82C2×C3×D3—2724no—
83C33—26no—
84C33—23no—
85C33—24no—
86C33—24no—
87C33C22×C3×D314yescomplemented
88C33C22×C3⋊D311yescharacteristic, complemented
89C33C23×C3211yescharacteristic, complemented
90H(3,𝔽3)—21no—
91H(3,𝔽3)—28no—
92H(3,𝔽3)—26no—
93H(3,𝔽3)—212no—
94H(3,𝔽3)—212no—
95H(3,𝔽3)C22×C3×D3112yescomplemented
96C23×C3—274no—
97C22×D3—91no—
98C22×D3—274no—
99C2×C32—212no—
100C2×C32—236no—
101C2×C32—93no—
102C2×C32—93no—
103C2×C32—912no—
104C2×C32—1812no—
105C2×C32—1812no—
106C2×C32—1818no—
107C2×C32—189no—
108C2×C32—274no—
109C2×C32C2×C32×D3112yes—
110C2×C32C2×C3×C3⋊D3112yes—
111C3⋊D3—316no—
112C3⋊D3—274no—
113C3×D3—916no—
114C3×D3—2716no—
115C22×C3—34no—
116C22×C3—34no—
117C22×C3—64no—
118C22×C3—612no—
119C22×C3—2724no—
120C22×C3C32×C3⋊D311yescharacteristic
121C2×D3—96no—
122C2×D3—2724no—
123C32—24no—
124C32—212no—
125C32—91no—
126C32—91no—
127C32—94no—
128C32—184no—
129C32—184no—
130C32—186no—
131C32—183no—
132C32C22×C32×D314yes—
133C32C22×C3×C3⋊D314yes—
134C23—271noSylow
135C2×C3—312no—
136C2×C3—312no—
137C2×C3—612no—
138C2×C3—636no—
139C2×C3—2716no—
140C2×C3C2×C32×C3⋊D313yes—
141D3—94no—
142D3—2716no—
143C22—276no—
144C22#9239111yescharacteristic, direct product factor
145C3—34no—
146C3—34no—
147C3—64no—
148C3—612no—
149C3C22×C32×C3⋊D311yescharacteristic
150C2—274no—
151C2C2×#9239113yesdirect product factor
subgroupG/H
center, hypercenterC22#92391
commutator, cosocle, hypocenterC33C23×C32
FrattiniC3C22×C32×C3⋊D3
FittingC22×H(5,𝔽3)C2
socleC22×C3C32×C3⋊D3
LCSG ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽3)C2semidirect
2. C2×#92391C2direct (N × H)
3. C22×C3⋊((C3⋊D3)⋊C3)C3semidirect
5. C2×H(5,𝔽3)C22semidirect
6. #92391C22direct (N × H)
9. C22×C3×H(3,𝔽3)D3semidirect
10. C22×C3×H(3,𝔽3)C2×C3semidirect
11. C2×C3⋊((C3⋊D3)⋊C3)C2×C3semidirect
13. H(5,𝔽3)C23semidirect
14. C22×C3⋊(C3⋊D3)C32semidirect
20. C2×C3×H(3,𝔽3)C2×D3semidirect
21. C2×C3×H(3,𝔽3)C22×C3semidirect
22. C3⋊((C3⋊D3)⋊C3)C22×C3semidirect
28. C22×C33C3×D3semidirect
29. C22×C33C3⋊D3semidirect
30. C22×C33C2×C32semidirect
31. C2×C3⋊(C3⋊D3)C2×C32semidirect
39. C22×H(3,𝔽3)C3×D3semidirect
43. C3×H(3,𝔽3)C22×D3semidirect
44. C3×H(3,𝔽3)C23×C3semidirect
54. C2×C33C2×C3×D3semidirect
55. C2×C33C2×C3⋊D3semidirect
56. C2×C33C22×C32semidirect
57. C3⋊(C3⋊D3)C22×C32semidirect
65. C2×H(3,𝔽3)C2×C3×D3semidirect
87. C33C22×C3×D3semidirect
88. C33C22×C3⋊D3semidirect
89. C33C23×C32semidirect
95. H(3,𝔽3)C22×C3×D3semidirect
144. C22#92391direct (N × H)
151. C2C2×#92391direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽3)2
C2×#923912
C22×C3⋊((C3⋊D3)⋊C3)3
C22×C3×(C3⋊D3)⋊C33