Finite Group Databasebeta

← #13141192

Subgroups — #13141192

Order 1728.

156 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 36 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
1C2×C9⋊(C42⋊C3)C211yesmaximal, characteristic
2C3×#384⋊C3C311yesmaximal, characteristic
3C9×#384C311yesmaximal, characteristic, complemented
4#384⋊C9C312yesmaximal, complemented
5C4×C9⋊A4—41nomaximal
6C9⋊(C42⋊C3)C411yescharacteristic, complemented
7C2×C3×C42⋊C3C2×C311yescharacteristic
8C2×C9×C4⋊C4—31no—
9C2×C9×C4⋊C4—31no—
10C2×C42×C9C2×C311yescharacteristic
11C2×C42⋊C9C2×C312yes—
12C2×C9⋊A4—41no—
13C3×#384C3211yescharacteristic
14#384⋊C3—31no—
15C3×C4×A4—41no—
16C3×C42⋊C3C3×C411yescharacteristic
17C22×C4×C9—31no—
18C22×C4×C9—31no—
19C22×C4×C9A411yescharacteristic
20C4×C22⋊C9—42no—
21C9×C4⋊C4—62no—
22C9×C4⋊C4—62no—
23C42×C9—31no—
24C42×C9C3×C411yescharacteristic, complemented
25C42⋊C9C3×C412yescomplemented
26C9⋊A4—41no—
27C4×3−1+2—161no—
28C2×C3×C4⋊C4—31no—
29C2×C3×C4⋊C4—31no—
30C2×C3×C42C2×C3211yescharacteristic
31C2×C42⋊C3—31no—
32C2×C3×A4—41no—
33C23×C9C2×A411yescharacteristic
34C2×C22⋊C9—42no—
35C2×C4×C9—32no—
36C2×C4×C9—32no—
37C2×C4×C9—31no—
38C2×C4×C9—31no—
39C2×C4×C9—61no—
40C2×C4×C9—61no—
41C2×C4×C9—62no—
42#3843−1+211yescharacteristic, complemented, Sylow
43C2×3−1+2—161no—
44C22×C3×C4—31no—
45C22×C3×C4—31no—
46C22×C3×C4C3×A411yescharacteristic
47C4×A4—121no—
48C3×C4⋊C4—62no—
49C3×C4⋊C4—62no—
50C3×C42—31no—
51C3×C42C32×C411yescharacteristic
52C42⋊C3—31no—
53C3×A4—41no—
54C32×C4—161no—
55C22×C9—31no—
56C22×C9—31no—
57C22×C9C4×A411yescharacteristic
58C22⋊C9—42no—
59C4×C9—32no—
60C4×C9—32no—
61C4×C9—41no—
62C4×C9—121no—
63C4×C9—162no—
64C2×C4⋊C4—31no—
65C2×C4⋊C4—31no—
66C2×C42C2×3−1+211yescharacteristic
673−1+2—161noSylow
68C23×C3C2×C3×A411yescharacteristic
69C2×A4—121no—
70C2×C3×C4—32no—
71C2×C3×C4—32no—
72C2×C3×C4—31no—
73C2×C3×C4—31no—
74C2×C3×C4—61no—
75C2×C3×C4—61no—
76C2×C3×C4—62no—
77C2×C32—161no—
78C2×C9—31no—
79C2×C9—31no—
80C2×C9—162no—
81C2×C9#127⋊C311yescharacteristic
82C22×C4—31no—
83C22×C4—31no—
84C22×C4C9⋊A411yescharacteristic
85C4⋊C4—62no—
86C4⋊C4—62no—
87C42—31no—
88C42C4×3−1+211yescharacteristic, complemented
89C22×C3—31no—
90C22×C3—31no—
91C22×C3C3×C4×A411yescharacteristic
92A4—121no—
93C3×C4—32no—
94C3×C4—32no—
95C3×C4—41no—
96C3×C4—121no—
97C3×C4—481no—
98C32—161no—
99C9—162no—
100C9#384⋊C311yescharacteristic, complemented
101C23C2×C9⋊A411yescharacteristic
102C2×C4—32no—
103C2×C4—32no—
104C2×C4—31no—
105C2×C4—31no—
106C2×C4—61no—
107C2×C4—61no—
108C2×C4—62no—
109C2×C3—31no—
110C2×C3—31no—
111C2×C3—481no—
112C2×C3C3×#127⋊C311yescharacteristic
113C22—31no—
114C22—31no—
115C22C4×C9⋊A411yescharacteristic
116C4—32no—
117C4—32no—
118C4—41no—
119C4—121no—
120C3—481no—
121C3C3×#384⋊C311yescharacteristic
122C2—31no—
123C2—31no—
124C2#1172133511yescharacteristic
subgroupG/H
centerC2×C3C3×#127⋊C3
commutatorC3×C42C32×C4
Frattini, socleC23×C3C2×C3×A4
FittingC9×#384C3
prime radicalC2×C3×C42⋊C3C2×C3
cosocleC2×C3×C42C2×C32
hypercenterC2×C9#127⋊C3
hypocenterC42C4×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C9

Complements

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

NH ≅ G/Ntype
3. C9×#384C3semidirect
4. #384⋊C9C3semidirect
6. C9⋊(C42⋊C3)C4semidirect
24. C42×C9C3×C4semidirect
25. C42⋊C9C3×C4semidirect
42. #3843−1+2semidirect
88. C42C4×3−1+2semidirect
100. C9#384⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C9⋊(C42⋊C3)2
C3×#384⋊C33
C9×#3843
#384⋊C93
C4×C9⋊A44