Finite Group Databasebeta

← C22×C33×SD16 #13180025

Subgroups — C22×C33×SD16 #13180025

Order 1728.

4172 conjugacy classes of subgroups; only the 2492 normal subgroups are shown.

normal subgroups only

This group has 4172 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×Q8C211yesmaximal, characteristic, complemented
2C22×C33×D4C211yesmaximal, characteristic
3C2×C33×SD16C2112yesmaximal, direct product factor
4C22×C33×C8C211yesmaximal, characteristic, complemented
5C22×C32×SD16C3113yesmaximal, direct product factor
6C24×C33—21no—
7C2×C33×Q8—23no—
8C2×C33×Q8C2216yescomplemented
9C2×C33×D4—23no—
10C2×C33×D4C2216yes—
11C22×C33×C4—21no—
12C22×C33×C4C2211yescharacteristic
13C33×SD16C22116yesdirect product factor
14C2×C33×C8C2216yescomplemented
15C22×C32×Q8C2×C3113yescomplemented
16C22×C32×D4C2×C3113yes—
17C2×C32×SD16C2×C31156yesdirect product factor
18C22×C32×C8C2×C3113yescomplemented
19C23×C33—26no—
20C23×C33—44no—
21C23×C33D411yescharacteristic
22C33×Q8—26no—
23C33×Q8C2314yescomplemented
24C33×D4—26no—
25C33×D4C2314yes—
26C2×C33×C4—26no—
27C2×C33×C4D413yes—
28C2×C33×C4C2313yes—
29C33×C8C2314yescomplemented
30C22×C3×SD16C32113yesdirect product factor
31C24×C32—213no—
32C2×C32×Q8—239no—
33C2×C32×Q8C22×C3178yescomplemented
34C2×C32×D4—239no—
35C2×C32×D4C22×C3178yes—
36C22×C32×C4—213no—
37C22×C32×C4C22×C3113yes—
38C32×SD16C22×C31208yesdirect product factor
39C2×C32×C8C22×C3178yescomplemented
40C22×C33—24no—
41C22×C33—412no—
42C22×C33SD1614yesdirect product factor
43C22×C33C2×D413yes—
44C33×C4—24no—
45C33×C4C2×D413yes—
46C33×C4C2411yescharacteristic
47C22×C3×Q8C2×C32113yescomplemented
48C22×C3×D4C2×C32113yes—
49C2×C3×SD16C2×C321156yesdirect product factor
50C22×C3×C8C2×C32113yescomplemented
51C23×C32—278no—
52C23×C32—452no—
53C23×C32C3×D4113yes—
54C32×Q8—278no—
55C32×Q8C23×C3152yescomplemented
56C32×D4—278no—
57C32×D4C23×C3152yes—
58C2×C32×C4—278no—
59C2×C32×C4C3×D4139yes—
60C2×C32×C4C23×C3139yes—
61C32×C8C23×C3152yescomplemented
62C22×SD16C3311yescharacteristic, direct product factor, Sylow
63C2×C33—44no—
64C2×C33C2×SD1616yesdirect product factor
65C2×C33C22×D411yescharacteristic
66C24×C3—213no—
67C2×C3×Q8—239no—
68C2×C3×Q8C22×C32178yescomplemented
69C2×C3×D4—239no—
70C2×C3×D4C22×C32178yes—
71C22×C3×C4—213no—
72C22×C3×C4C22×C32113yes—
73C3×SD16C22×C321208yesdirect product factor
74C2×C3×C8C22×C32178yescomplemented
75C22×C32—252no—
76C22×C32—4156no—
77C22×C32C3×SD16152yesdirect product factor
78C22×C32C2×C3×D4139yes—
79C32×C4—252no—
80C32×C4C2×C3×D4139yes—
81C32×C4C24×C3113yes—
82C22×Q8C2×C3311yescharacteristic, complemented
83C22×D4C2×C3311yescharacteristic
84C2×SD16C2×C33112yesdirect product factor
85C22×C8C2×C3311yescharacteristic, complemented
86C33C22×SD1611yescharacteristic, direct product factor, Sylow
87C23×C3—278no—
88C23×C3—452no—
89C23×C3C32×D4113yes—
90C3×Q8—278no—
91C3×Q8C23×C32152yescomplemented
92C3×D4—278no—
93C3×D4C23×C32152yes—
94C2×C3×C4—278no—
95C2×C3×C4C32×D4139yes—
96C2×C3×C4C23×C32139yes—
97C3×C8C23×C32152yescomplemented
98C2×C32—452no—
99C2×C32C2×C3×SD16178yesdirect product factor
100C2×C32C22×C3×D4113yes—
101C24—21no—
102C2×Q8—23no—
103C2×Q8C22×C3316yescomplemented
104C2×D4—23no—
105C2×D4C22×C3316yes—
106C22×C4—21no—
107C22×C4C22×C3311yescharacteristic
108SD16C22×C33116yesdirect product factor
109C2×C8C22×C3316yescomplemented
110C22×C3—252no—
111C22×C3—4156no—
112C22×C3C32×SD16152yesdirect product factor
113C22×C3C2×C32×D4139yes—
114C3×C4—252no—
115C3×C4C2×C32×D4139yes—
116C3×C4C24×C32113yes—
117C32C22×C3×SD16113yesdirect product factor
118C23—26no—
119C23—44no—
120C23C33×D411yescharacteristic
121Q8—26no—
122Q8C23×C3314yescomplemented
123D4—26no—
124D4C23×C3314yes—
125C2×C4—26no—
126C2×C4C33×D413yes—
127C2×C4C23×C3313yes—
128C8C23×C3314yescomplemented
129C2×C3—452no—
130C2×C3C2×C32×SD16178yesdirect product factor
131C2×C3C22×C32×D4113yes—
132C22—24no—
133C22—412no—
134C22C33×SD1614yesdirect product factor
135C22C2×C33×D413yes—
136C4—24no—
137C4C2×C33×D413yes—
138C4C24×C3311yescharacteristic
139C3C22×C32×SD16113yesdirect product factor
140C2—44no—
141C2C2×C33×SD1616yesdirect product factor
142C2C22×C33×D411yescharacteristic
subgroupG/H
center, socleC23×C33D4
commutator, Frattini, cosocleC4C24×C33
prime radicalC22×C33×D4C2
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×Q8C2semidirect
3. C2×C33×SD16C2direct (N × H)
4. C22×C33×C8C2semidirect
5. C22×C32×SD16C3direct (N × H)
8. C2×C33×Q8C22semidirect
13. C33×SD16C22direct (N × H)
14. C2×C33×C8C22semidirect
15. C22×C32×Q8C2×C3semidirect
17. C2×C32×SD16C2×C3direct (N × H)
18. C22×C32×C8C2×C3semidirect
23. C33×Q8C23semidirect
29. C33×C8C23semidirect
30. C22×C3×SD16C32direct (N × H)
33. C2×C32×Q8C22×C3semidirect
38. C32×SD16C22×C3direct (N × H)
39. C2×C32×C8C22×C3semidirect
42. C22×C33SD16direct (N × H)
47. C22×C3×Q8C2×C32semidirect
49. C2×C3×SD16C2×C32direct (N × H)
50. C22×C3×C8C2×C32semidirect
55. C32×Q8C23×C3semidirect
61. C32×C8C23×C3semidirect
62. C22×SD16C33direct (N × H)
64. C2×C33C2×SD16direct (N × H)
68. C2×C3×Q8C22×C32semidirect
73. C3×SD16C22×C32direct (N × H)
74. C2×C3×C8C22×C32semidirect
77. C22×C32C3×SD16direct (N × H)
82. C22×Q8C2×C33semidirect
84. C2×SD16C2×C33direct (N × H)
85. C22×C8C2×C33semidirect
86. C33C22×SD16direct (N × H)
91. C3×Q8C23×C32semidirect
97. C3×C8C23×C32semidirect
99. C2×C32C2×C3×SD16direct (N × H)
103. C2×Q8C22×C33semidirect
108. SD16C22×C33direct (N × H)
109. C2×C8C22×C33semidirect
112. C22×C3C32×SD16direct (N × H)
117. C32C22×C3×SD16direct (N × H)
122. Q8C23×C33semidirect
128. C8C23×C33semidirect
130. C2×C3C2×C32×SD16direct (N × H)
134. C22C33×SD16direct (N × H)
139. C3C22×C32×SD16direct (N × H)
141. C2C2×C33×SD16direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C33×Q82
C22×C33×D42
C2×C33×SD162
C22×C33×C82
C22×C32×SD163