Finite Group Databasebeta

← C23×C3×C3⋊(C3⋊C8) #13179531

Subgroups — C23×C3×C3⋊(C3⋊C8) #13179531

Order 1728.

2972 conjugacy classes of subgroups; only the 1518 normal subgroups are shown.

normal subgroups only

This group has 2972 conjugacy classes of subgroups — too many to draw. Only its 1518 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×C33×C4C211yesmaximal, characteristic
2C22×C3×C3⋊(C3⋊C8)C2114yesmaximal, direct product factor
3C23×C3⋊(C3⋊C8)C311yesmaximal, characteristic, direct product factor
4C23×C3×C3⋊C8—34nomaximal
5C24×C33C411yescharacteristic
6C22×C33×C4C417yes—
7C22×C33×C4C2217yes—
8C2×C3×C3⋊(C3⋊C8)C22128yesdirect product factor
9C23×C32×C4—24no—
10C23×C32×C4D314yes—
11C23×C32×C4C2×C311yescharacteristic
12C22×C3⋊(C3⋊C8)C2×C3114yesdirect product factor
13C22×C3×C3⋊C8—356no—
14C23×C33C818yescomplemented
15C23×C33C2×C417yes—
16C2×C33×C4C2×C4121yes—
17C2×C33×C4C2317yes—
18C3×C3⋊(C3⋊C8)C2318yesdirect product factor
19C23×C3×C8—91no—
20C23×C3⋊C8—34no—
21C24×C32—24no—
22C24×C32Dic314yes—
23C24×C32C3×C411yescharacteristic
24C22×C32×C4—228no—
25C22×C32×C4—228no—
26C22×C32×C4Dic3128yes—
27C22×C32×C4C3×C417yes—
28C22×C32×C4C2×D3128yes—
29C22×C32×C4C22×C317yes—
30C2×C3⋊(C3⋊C8)C22×C3128yesdirect product factor
31C2×C3×C3⋊C8—3112no—
32C22×C33C2×C8128yescomplemented
33C22×C33C22×C417yes—
34C33×C4C22×C417yes—
35C33×C4C2411yescharacteristic
36C23×C3×C4—24no—
37C23×C3×C4C3×D314yes—
38C23×C3×C4C3⋊D311yescharacteristic
39C22×C3×C8—914no—
40C22×C3⋊C8—356no—
41C23×C32—232no—
42C23×C32—228no—
43C23×C32C3⋊C8132yescomplemented
44C23×C32C3×C818yescomplemented
45C23×C32C2×Dic3128yes—
46C23×C32C2×C3×C417yes—
47C2×C32×C4—284no—
48C2×C32×C4—228no—
49C2×C32×C4C2×Dic3184yes—
50C2×C32×C4C2×C3×C4121yes—
51C2×C32×C4C22×D3128yes—
52C2×C32×C4C23×C317yes—
53C3⋊(C3⋊C8)C23×C318yesdirect product factor
54C3×C3⋊C8—332no—
55C23×C8—91noSylow
56C2×C33C22×C8114yescomplemented
57C2×C33C23×C411yescharacteristic
58C24×C3—24no—
59C24×C3C3×Dic314yes—
60C24×C3C3⋊Dic311yescharacteristic
61C22×C3×C4—228no—
62C22×C3×C4—228no—
63C22×C3×C4C3×Dic3128yes—
64C22×C3×C4C3⋊Dic317yes—
65C22×C3×C4C2×C3×D3128yes—
66C22×C3×C4C2×C3⋊D317yes—
67C2×C3×C8—928no—
68C2×C3⋊C8—3112no—
69C22×C32—2112no—
70C22×C32—228no—
71C22×C32C2×C3⋊C81112yescomplemented
72C22×C32C2×C3×C8128yescomplemented
73C22×C32C22×Dic3128yes—
74C22×C32C22×C3×C417yes—
75C32×C4—24no—
76C32×C4—228no—
77C32×C4C22×Dic3128yes—
78C32×C4C22×C3×C417yes—
79C32×C4C23×D314yes—
80C32×C4C24×C311yescharacteristic
81C23×C4C3×C3⋊D311yescharacteristic
82C22×C8—914no—
83C33C23×C811yescharacteristic, complemented, Sylow
84C23×C3—232no—
85C23×C3—228no—
86C23×C3C3×C3⋊C8132yescomplemented
87C23×C3C3⋊(C3⋊C8)18yesdirect product factor
88C23×C3C2×C3×Dic3128yes—
89C23×C3C2×C3⋊Dic317yes—
90C2×C3×C4—284no—
91C2×C3×C4—228no—
92C2×C3×C4C2×C3×Dic3184yes—
93C2×C3×C4C2×C3⋊Dic3121yes—
94C2×C3×C4C22×C3×D3128yes—
95C2×C3×C4C22×C3⋊D317yes—
96C3×C8—98no—
97C3⋊C8—332no—
98C2×C32—256no—
99C2×C32—24no—
100C2×C32C22×C3⋊C8156yescomplemented
101C2×C32C22×C3×C8114yescomplemented
102C2×C32C23×Dic314yes—
103C2×C32C23×C3×C411yescharacteristic
104C24C3×C3⋊Dic311yescharacteristic
105C22×C4C3×C3⋊Dic317yes—
106C22×C4C2×C3×C3⋊D317yes—
107C2×C8—928no—
108C22×C3—2112no—
109C22×C3—228no—
110C22×C3C2×C3×C3⋊C81112yescomplemented
111C22×C3C2×C3⋊(C3⋊C8)128yesdirect product factor
112C22×C3C22×C3×Dic3128yes—
113C22×C3C22×C3⋊Dic317yes—
114C3×C4—24no—
115C3×C4—228no—
116C3×C4C22×C3×Dic3128yes—
117C3×C4C22×C3⋊Dic317yes—
118C3×C4C23×C3×D314yes—
119C3×C4C23×C3⋊D311yescharacteristic
120C32—24no—
121C32C23×C3⋊C814yescomplemented
122C32C23×C3×C811yescharacteristic, complemented
123C23C3×C3⋊(C3⋊C8)18yesdirect product factor
124C23C2×C3×C3⋊Dic317yes—
125C2×C4C2×C3×C3⋊Dic3121yes—
126C2×C4C22×C3×C3⋊D317yes—
127C8—98no—
128C2×C3—256no—
129C2×C3—24no—
130C2×C3C22×C3×C3⋊C8156yescomplemented
131C2×C3C22×C3⋊(C3⋊C8)114yesdirect product factor
132C2×C3C23×C3×Dic314yes—
133C2×C3C23×C3⋊Dic311yescharacteristic
134C22C2×C3×C3⋊(C3⋊C8)128yesdirect product factor
135C22C22×C3×C3⋊Dic317yes—
136C4C22×C3×C3⋊Dic317yes—
137C4C23×C3×C3⋊D311yescharacteristic
138C3—24no—
139C3C23×C3×C3⋊C814yescomplemented
140C3C23×C3⋊(C3⋊C8)11yescharacteristic, direct product factor
141C2C22×C3×C3⋊(C3⋊C8)114yesdirect product factor
142C2C23×C3×C3⋊Dic311yescharacteristic
subgroupG/H
center, hypercenterC23×C3×C4C3⋊D3
commutator, hypocenterC32C23×C3×C8
FrattiniC4C23×C3×C3⋊D3
FittingC23×C33×C4C2
socle, prime radicalC24×C33C4
cosocleC32×C4C24×C3
LCSG ▷ C32
UCSC1 ◁ C23×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊(C3⋊C8)C2direct (N × H)
3. C23×C3⋊(C3⋊C8)C3direct (N × H)
8. C2×C3×C3⋊(C3⋊C8)C22direct (N × H)
12. C22×C3⋊(C3⋊C8)C2×C3direct (N × H)
14. C23×C33C8semidirect
18. C3×C3⋊(C3⋊C8)C23direct (N × H)
30. C2×C3⋊(C3⋊C8)C22×C3direct (N × H)
32. C22×C33C2×C8semidirect
43. C23×C32C3⋊C8semidirect
44. C23×C32C3×C8semidirect
53. C3⋊(C3⋊C8)C23×C3direct (N × H)
56. C2×C33C22×C8semidirect
71. C22×C32C2×C3⋊C8semidirect
72. C22×C32C2×C3×C8semidirect
83. C33C23×C8semidirect
86. C23×C3C3×C3⋊C8semidirect
87. C23×C3C3⋊(C3⋊C8)direct (N × H)
100. C2×C32C22×C3⋊C8semidirect
101. C2×C32C22×C3×C8semidirect
110. C22×C3C2×C3×C3⋊C8semidirect
111. C22×C3C2×C3⋊(C3⋊C8)direct (N × H)
121. C32C23×C3⋊C8semidirect
122. C32C23×C3×C8semidirect
123. C23C3×C3⋊(C3⋊C8)direct (N × H)
130. C2×C3C22×C3×C3⋊C8semidirect
131. C2×C3C22×C3⋊(C3⋊C8)direct (N × H)
134. C22C2×C3×C3⋊(C3⋊C8)direct (N × H)
139. C3C23×C3×C3⋊C8semidirect
140. C3C23×C3⋊(C3⋊C8)direct (N × H)
141. C2C22×C3×C3⋊(C3⋊C8)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C33×C42
C22×C3×C3⋊(C3⋊C8)2
C23×C3⋊(C3⋊C8)3
C23×C3×C3⋊C83