Finite Group Databasebeta

← #13136804

Subgroups — #13136804

Order 1728.

163 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 23 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
1C9×#95⋊C3C211yesmaximal, characteristic, complemented
2C3⋊(#399⋊C3)—31nomaximal
3#10588357C311yesmaximal, characteristic, complemented
4C2×A4×D9—41nomaximal
5C3×#95⋊C3D311yescharacteristic
6C2×(C2×D9)⋊C4—31no—
7C2×C4⋊Dic9—31no—
8C9×#95C2×C311yescharacteristic, complemented
9#95⋊C9—21no—
10C2×C9×A4—41no—
11A4×D9—42no—
12#4677—31no—
13#399⋊C3—91no—
14C2×D3×A4—121no—
15C23×D9A411yescharacteristic
16C22×C4×C9—31no—
17C22×Dic9—31no—
18(C2×D9)⋊C4—62no—
19C4⋊Dic9—62no—
20C2×C3×D9—161no—
21C9×A4—41no—
22C2×(C2×D3)⋊C4—91no—
23C2×C4⋊Dic3—91no—
24C3×#95C3×D311yescharacteristic
25#95⋊C3—21no—
26#95⋊C3D911yescharacteristic, complemented
27C2×C3×A4—41no—
28D3×A4—122no—
29C23×C9C2×A411yescharacteristic
30C22×D9—61no—
31C22×D9—61no—
32C22×D9SL(2,𝔽3)12yes—
33C2×C22⋊C9—81no—
34C2×C4×C9—32no—
35C2×C4×C9—62no—
36C2×Dic9—62no—
37C2×Dic9—61no—
38#399—91noSylow
39C2×C3×C9—161no—
40C3×D9—162no—
41C23×D3—31no—
42C22×A4—361no—
43C22×C3×C4—31no—
44C22×Dic3—91no—
45(C2×D3)⋊C4—182no—
46C4⋊Dic3—182no—
47C2×C3×D3—481no—
48C3×A4—41no—
49C22×C9—31no—
50C22×C9—31no—
51C22×C9C2×SL(2,𝔽3)11yescharacteristic
52C2×D9—41no—
53C2×D9—62no—
54C2×D9—121no—
55C22⋊C9—81no—
56C4×C9—62no—
57Dic9—121no—
58C2×C4⋊C4—271no—
59C2×C22⋊C4—271no—
60#95C3×D911yescharacteristic, complemented
61C3×C9—161noSylow
62C23×C3D3×A411yescharacteristic
63C22×D3—32no—
64C22×D3—181no—
65C22×D3—181no—
66C2×A4—41no—
67C2×A4—81no—
68C2×A4—362no—
69C2×C3×C4—32no—
70C2×C3×C4—62no—
71C2×Dic3—182no—
72C2×Dic3—181no—
73C2×C32—161no—
74C3×D3—482no—
75C2×C9—31no—
76C2×C9—31no—
77C2×C9—321no—
78C2×C9#127⋊C311yescharacteristic
79D9—42no—
80C24—91no—
81C22×C4—31no—
82C22×C4—271no—
83C4⋊C4—542no—
84C22⋊C4—542no—
85C22×C3—31no—
86C22×C3—31no—
87C22×C3—1441no—
88C22×C3D3×SL(2,𝔽3)11yescharacteristic
89C2×D3—121no—
90C2×D3—182no—
91C2×D3—361no—
92A4—41no—
93A4—81no—
94C3×C4—62no—
95Dic3—361no—
96C32—161no—
97C9—321no—
98C9#399⋊C311yescharacteristic, complemented
99C23—92no—
100C23—541no—
101C23—541no—
102C23A4×D911yescharacteristic
103C2×C4—32no—
104C2×C4—62no—
105C2×C4—541no—
106C2×C4—542no—
107C2×C3—31no—
108C2×C3—31no—
109C2×C3—161no—
110C2×C3—321no—
111C2×C3—1442no—
112C2×C3C3⋊(#127⋊C3)11yescharacteristic
113D3—122no—
114C22—31no—
115C22—31no—
116C22—361no—
117C22—542no—
118C22—1081no—
119C22D9×SL(2,𝔽3)11yescharacteristic
120C4—62no—
121C4—1081no—
122C3—161no—
123C3—321no—
124C3C3⋊(#399⋊C3)11yescharacteristic
125C2—31no—
126C2—31no—
127C2—362no—
128C2#1172074211yescharacteristic
subgroupG/H
center, hypercenterC2#11720742
commutator, Fitting, cosocle, hypocenterC9×#95C2×C3
Frattini, socleC23×C3D3×A4
LCSG ▷ C9×#95
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C9×#95⋊C3C2semidirect
3. #10588357C3semidirect
8. C9×#95C2×C3semidirect
26. #95⋊C3D9semidirect
60. #95C3×D9semidirect
98. C9#399⋊C3semidirect

Maximal subgroups

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

subgroupindex
C9×#95⋊C32
C3⋊(#399⋊C3)3
#105883573
C2×A4×D94