Finite Group Databasebeta

← #13141201

Subgroups — #13141201

Order 1728.

177 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 177 conjugacy classes of subgroups — too many to draw. Only its 35 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⋊C3)C211yesmaximal, characteristic, complemented
2C3×#396⋊C3C311yesmaximal, characteristic
3C9×#396C311yesmaximal, characteristic, complemented
4#396⋊C9C312yesmaximal, complemented
5C22×C9⋊A4—41nomaximal
6C3×#95⋊C3C2×C311yescharacteristic
7C2×C9×C4⋊C4—31no—
8C2×C9×C22⋊C4—31no—
9C9×#95C2×C311yescharacteristic, complemented
10#95⋊C9C2×C312yescomplemented
11C2×C9⋊A4—42no—
12C2×C9⋊A4—41no—
13C3×#396C3211yescharacteristic
14#396⋊C3—31no—
15C22×C3×A4—41no—
16C24×C9A411yescharacteristic
17C22×C22⋊C9—42no—
18C22×C4×C9—31no—
19C22×C4×C9—31no—
20C9×C4⋊C4—62no—
21C9×C22⋊C4—62no—
22C22×3−1+2—161no—
23C9⋊A4—41no—
24C2×C3×C4⋊C4—31no—
25C2×C3×C22⋊C4—31no—
26C3×#95C2×C3211yescharacteristic
27#95⋊C3—31no—
28C2×C3×A4—42no—
29C2×C3×A4—41no—
30C23×C9—61no—
31C23×C9—61no—
32C23×C9SL(2,𝔽3)12yes—
33C23×C9C2×A411yescharacteristic
34C2×C22⋊C9—44no—
35C2×C22⋊C9—42no—
36C2×C4×C9—32no—
37C2×C4×C9—61no—
38C2×C4×C9—61no—
39C2×C4×C9—61no—
40C2×C4×C9—61no—
41C2×C4×C9—61no—
42#3963−1+211yescharacteristic, complemented, Sylow
43C2×3−1+2—162no—
44C2×3−1+2—161no—
45C24×C3C3×A411yescharacteristic
46C22×A4—121no—
47C22×C3×C4—31no—
48C22×C3×C4—31no—
49C3×C4⋊C4—62no—
50C3×C22⋊C4—62no—
51C22×C32—161no—
52C3×A4—41no—
53C22×C9—31no—
54C22×C9—31no—
55C22×C9—41no—
56C22×C9—62no—
57C22×C9—121no—
58C22×C9—162no—
59C22×C9C2×SL(2,𝔽3)11yescharacteristic
60C22⋊C9—42no—
61C4×C9—62no—
62C4×C9—121no—
63C2×C4⋊C4—31no—
64C2×C22⋊C4—31no—
65#95C2×3−1+211yescharacteristic, complemented
663−1+2—161noSylow
67C23×C3—61no—
68C23×C3—61no—
69C23×C3C3×SL(2,𝔽3)12yes—
70C23×C3C2×C3×A411yescharacteristic
71C2×A4—122no—
72C2×A4—121no—
73C2×C3×C4—32no—
74C2×C3×C4—61no—
75C2×C3×C4—61no—
76C2×C3×C4—61no—
77C2×C3×C4—61no—
78C2×C3×C4—61no—
79C2×C32—162no—
80C2×C32—161no—
81C2×C9—31no—
82C2×C9—31no—
83C2×C9—42no—
84C2×C9—164no—
85C2×C9—162no—
86C2×C9#126⋊C311yescharacteristic
87C24C9⋊A411yescharacteristic
88C22×C4—31no—
89C22×C4—31no—
90C4⋊C4—62no—
91C22⋊C4—62no—
92C22×C3—31no—
93C22×C3—31no—
94C22×C3—41no—
95C22×C3—62no—
96C22×C3—121no—
97C22×C3—481no—
98C22×C3C2×C3×SL(2,𝔽3)11yescharacteristic
99A4—121no—
100C3×C4—62no—
101C3×C4—121no—
102C32—161no—
103C9—162no—
104C9#396⋊C311yescharacteristic, complemented
105C23—61no—
106C23—61no—
107C23C9⋊SL(2,𝔽3)12yes—
108C23C2×C9⋊A411yescharacteristic
109C2×C4—32no—
110C2×C4—61no—
111C2×C4—61no—
112C2×C4—61no—
113C2×C4—61no—
114C2×C4—61no—
115C2×C3—31no—
116C2×C3—31no—
117C2×C3—42no—
118C2×C3—482no—
119C2×C3—481no—
120C2×C3C3×#126⋊C311yescharacteristic
121C22—31no—
122C22—31no—
123C22—41no—
124C22—62no—
125C22—121no—
126C22C2×C9⋊SL(2,𝔽3)11yescharacteristic
127C4—62no—
128C4—121no—
129C3—481no—
130C3C3×#396⋊C311yescharacteristic
131C2—31no—
132C2—31no—
133C2—42no—
134C2#1172133311yescharacteristic
subgroupG/H
centerC2×C3C3×#126⋊C3
commutator, cosocleC3×#95C2×C32
Frattini, socleC23×C3C2×C3×A4
FittingC9×#396C3
prime radicalC3×#396⋊C3C3
hypercenterC2×C9#126⋊C3
hypocenter#95C2×3−1+2
LCSG ▷ C3×#95 ▷ #95
UCSC1 ◁ C2×C3 ◁ C2×C9

Complements

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

NH ≅ G/Ntype
1. C9⋊(#95⋊C3)C2semidirect
3. C9×#396C3semidirect
4. #396⋊C9C3semidirect
9. C9×#95C2×C3semidirect
10. #95⋊C9C2×C3semidirect
42. #3963−1+2semidirect
65. #95C2×3−1+2semidirect
104. C9#396⋊C3semidirect

Maximal subgroups

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

subgroupindex
C9⋊(#95⋊C3)2
C3×#396⋊C33
C9×#3963
#396⋊C93
C22×C9⋊A44