Finite Group Databasebeta

← C22×C33×D8 #13180024

Subgroups — C22×C33×D8 #13180024

Order 1728.

4732 conjugacy classes of subgroups; only the 2492 normal subgroups are shown.

normal subgroups only

This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 2492 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×C33×D4C212yesmaximal, complemented
2C2×C33×D8C2112yesmaximal, direct product factor
3C22×C33×C8C211yesmaximal, characteristic, complemented
4C22×C32×D8C3113yesmaximal, direct product factor
5C24×C33—22no—
6C2×C33×D4—26no—
7C2×C33×D4C22112yescomplemented
8C22×C33×C4C2211yescharacteristic
9C33×D8C22116yesdirect product factor
10C2×C33×C8C2216yescomplemented
11C22×C32×D4C2×C3126yescomplemented
12C2×C32×D8C2×C31156yesdirect product factor
13C22×C32×C8C2×C3113yescomplemented
14C23×C33—212no—
15C23×C33—48no—
16C23×C33D411yescharacteristic
17C33×D4—212no—
18C33×D4C2318yescomplemented
19C2×C33×C4D413yes—
20C2×C33×C4C2313yes—
21C33×C8C2314yescomplemented
22C22×C3×D8C32113yesdirect product factor
23C24×C32—226no—
24C2×C32×D4—278no—
25C2×C32×D4C22×C31156yescomplemented
26C22×C32×C4C22×C3113yes—
27C32×D8C22×C31208yesdirect product factor
28C2×C32×C8C22×C3178yescomplemented
29C22×C33—28no—
30C22×C33—424no—
31C22×C33D814yesdirect product factor
32C22×C33C2×D413yes—
33C33×C4C2×D413yes—
34C33×C4C2411yescharacteristic
35C22×C3×D4C2×C32126yescomplemented
36C2×C3×D8C2×C321156yesdirect product factor
37C22×C3×C8C2×C32113yescomplemented
38C23×C32—2156no—
39C23×C32—4104no—
40C23×C32C3×D4113yes—
41C32×D4—2156no—
42C32×D4C23×C31104yescomplemented
43C2×C32×C4C3×D4139yes—
44C2×C32×C4C23×C3139yes—
45C32×C8C23×C3152yescomplemented
46C22×D8C3311yescharacteristic, direct product factor, Sylow
47C2×C33—48no—
48C2×C33C2×D816yesdirect product factor
49C2×C33C22×D411yescharacteristic
50C24×C3—226no—
51C2×C3×D4—278no—
52C2×C3×D4C22×C321156yescomplemented
53C22×C3×C4C22×C32113yes—
54C3×D8C22×C321208yesdirect product factor
55C2×C3×C8C22×C32178yescomplemented
56C22×C32—2104no—
57C22×C32—4312no—
58C22×C32C3×D8152yesdirect product factor
59C22×C32C2×C3×D4139yes—
60C32×C4C2×C3×D4139yes—
61C32×C4C24×C3113yes—
62C22×D4C2×C3312yescomplemented
63C2×D8C2×C33112yesdirect product factor
64C22×C8C2×C3311yescharacteristic, complemented
65C33C22×D811yescharacteristic, direct product factor, Sylow
66C23×C3—2156no—
67C23×C3—4104no—
68C23×C3C32×D4113yes—
69C3×D4—2156no—
70C3×D4C23×C321104yescomplemented
71C2×C3×C4C32×D4139yes—
72C2×C3×C4C23×C32139yes—
73C3×C8C23×C32152yescomplemented
74C2×C32—4104no—
75C2×C32C2×C3×D8178yesdirect product factor
76C2×C32C22×C3×D4113yes—
77C24—22no—
78C2×D4—26no—
79C2×D4C22×C33112yescomplemented
80C22×C4C22×C3311yescharacteristic
81D8C22×C33116yesdirect product factor
82C2×C8C22×C3316yescomplemented
83C22×C3—2104no—
84C22×C3—4312no—
85C22×C3C32×D8152yesdirect product factor
86C22×C3C2×C32×D4139yes—
87C3×C4C2×C32×D4139yes—
88C3×C4C24×C32113yes—
89C32C22×C3×D8113yesdirect product factor
90C23—212no—
91C23—48no—
92C23C33×D411yescharacteristic
93D4—212no—
94D4C23×C3318yescomplemented
95C2×C4C33×D413yes—
96C2×C4C23×C3313yes—
97C8C23×C3314yescomplemented
98C2×C3—4104no—
99C2×C3C2×C32×D8178yesdirect product factor
100C2×C3C22×C32×D4113yes—
101C22—28no—
102C22—424no—
103C22C33×D814yesdirect product factor
104C22C2×C33×D413yes—
105C4C2×C33×D413yes—
106C4C24×C3311yescharacteristic
107C3C22×C32×D8113yesdirect product factor
108C2—48no—
109C2C2×C33×D816yesdirect product factor
110C2C22×C33×D411yescharacteristic
subgroupG/H
center, socleC23×C33D4
commutator, Frattini, cosocleC4C24×C33
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C33 ◁ C22×C33×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C33×D4C2semidirect
2. C2×C33×D8C2direct (N × H)
3. C22×C33×C8C2semidirect
4. C22×C32×D8C3direct (N × H)
7. C2×C33×D4C22semidirect
9. C33×D8C22direct (N × H)
10. C2×C33×C8C22semidirect
11. C22×C32×D4C2×C3semidirect
12. C2×C32×D8C2×C3direct (N × H)
13. C22×C32×C8C2×C3semidirect
18. C33×D4C23semidirect
21. C33×C8C23semidirect
22. C22×C3×D8C32direct (N × H)
25. C2×C32×D4C22×C3semidirect
27. C32×D8C22×C3direct (N × H)
28. C2×C32×C8C22×C3semidirect
31. C22×C33D8direct (N × H)
35. C22×C3×D4C2×C32semidirect
36. C2×C3×D8C2×C32direct (N × H)
37. C22×C3×C8C2×C32semidirect
42. C32×D4C23×C3semidirect
45. C32×C8C23×C3semidirect
46. C22×D8C33direct (N × H)
48. C2×C33C2×D8direct (N × H)
52. C2×C3×D4C22×C32semidirect
54. C3×D8C22×C32direct (N × H)
55. C2×C3×C8C22×C32semidirect
58. C22×C32C3×D8direct (N × H)
62. C22×D4C2×C33semidirect
63. C2×D8C2×C33direct (N × H)
64. C22×C8C2×C33semidirect
65. C33C22×D8direct (N × H)
70. C3×D4C23×C32semidirect
73. C3×C8C23×C32semidirect
75. C2×C32C2×C3×D8direct (N × H)
79. C2×D4C22×C33semidirect
81. D8C22×C33direct (N × H)
82. C2×C8C22×C33semidirect
85. C22×C3C32×D8direct (N × H)
89. C32C22×C3×D8direct (N × H)
94. D4C23×C33semidirect
97. C8C23×C33semidirect
99. C2×C3C2×C32×D8direct (N × H)
103. C22C33×D8direct (N × H)
107. C3C22×C32×D8direct (N × H)
109. C2C2×C33×D8direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×D42
C2×C33×D82
C22×C33×C82
C22×C32×D83