Finite Group Databasebeta

← #13136830

Subgroups — #13136830

Order 1728.

168 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 20 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
1#95⋊H(3,𝔽3)C211yesmaximal, characteristic
2#10592361C311yesmaximal, characteristic, complemented
3C3⋊(#401⋊C3)—31nomaximal
4(C3⋊Dic3)⋊A4—41nomaximal
5C32×#95C2×C311yescharacteristic
6#63658—31no—
7#63658—31no—
8C3×#95⋊C3—21no—
9C3×#95⋊C3D311yescharacteristic
10C2×C22⋊H(3,𝔽3)—41no—
11C3⋊#401—31no—
12C3⋊#401—91no—
13#401⋊C3—91no—
14C22×C32×C4—31no—
15C22×C3⋊Dic3—31no—
16C22×C3⋊Dic3A411yescharacteristic
17Dic3×A4—121no—
18C22⋊H(3,𝔽3)—41no—
19(C3⋊Dic3)⋊C3—161no—
20C3×#95—31no—
21C3×#95C3×D311yescharacteristic
22C3⋊#95—91no—
23C3⋊#95—93no—
24C3⋊#95—93no—
25C3⋊#95—91no—
26#95⋊C3—31no—
27#95⋊C3—61no—
28C23×C32C2×A411yescharacteristic
29C2×C3×A4—41no—
30C2×C3×A4—81no—
31C2×C32×C4—62no—
32C2×C32×C4—61no—
33C2×C3⋊Dic3—61no—
34C2×C3⋊Dic3—61no—
35C2×C3⋊Dic3—61no—
36C2×C3⋊Dic3—61no—
37#401—91noSylow
38C2×H(3,𝔽3)—161no—
39C22×C3×C4—33no—
40C22×C3×C4—31no—
41C22×Dic3—31no—
42C22×Dic3—91no—
43C22×Dic3—93no—
44C22×Dic3—91no—
45C4×A4—361no—
46C22×C32—31no—
47C22×C32—31no—
48C22×C32#41⋊C311yescharacteristic
49C3×A4—41no—
50C3×A4—81no—
51C32×C4—121no—
52C3⋊Dic3—41no—
53C3⋊Dic3—121no—
54C3×Dic3—481no—
55#95—271no—
56#95—271no—
57#95(C3⋊D3)⋊C311yescharacteristic
58H(3,𝔽3)—161noSylow
59C23×C3—31no—
60C23×C3D3×A411yescharacteristic
61C2×A4—121no—
62C2×A4—241no—
63C2×C3×C4—62no—
64C2×C3×C4—63no—
65C2×C3×C4—66no—
66C2×C3×C4—61no—
67C2×Dic3—181no—
68C2×Dic3—183no—
69C2×Dic3—181no—
70C2×Dic3—183no—
71C2×Dic3—183no—
72C2×Dic3—181no—
73C2×Dic3—183no—
74C2×Dic3—181no—
75C2×C32—31no—
76C2×C32—31no—
77C2×C32—161no—
78C2×C32—321no—
79C2×C32#126⋊C311yescharacteristic
80C22×C4—31no—
81C22×C4—91no—
82C22×C4—271no—
83C22×C3—31no—
84C22×C3—33no—
85C22×C3—33no—
86C22×C3—31no—
87C22×C3—31no—
88C22×C3C3⋊(#41⋊C3)11yescharacteristic
89A4—121no—
90A4—241no—
91C3×C4—123no—
92C3×C4—121no—
93C3×C4—1441no—
94Dic3—121no—
95Dic3—361no—
96Dic3—363no—
97Dic3—361no—
98C32—161no—
99C32—321no—
100C32#401⋊C311yescharacteristic, complemented
101C23(C3⋊D3)⋊A411yescharacteristic
102C2×C4—62no—
103C2×C4—61no—
104C2×C4—541no—
105C2×C4—541no—
106C2×C4—541no—
107C2×C4—541no—
108C2×C3—33no—
109C2×C3—33no—
110C2×C3—31no—
111C2×C3—31no—
112C2×C3—31no—
113C2×C3—481no—
114C2×C3—961no—
115C2×C3C3⋊(#126⋊C3)11yescharacteristic
116C22—31no—
117C22—31no—
118C22#8862211yescharacteristic
119C4—121no—
120C4—361no—
121C4—1081no—
122C3—31no—
123C3—481no—
124C3—961no—
125C3C3⋊(#401⋊C3)11yescharacteristic
126C2—31no—
127C2—31no—
128C2#1172075811yescharacteristic
subgroupG/H
center, hypercenterC2#11720758
commutator, Fitting, cosocle, hypocenterC32×#95C2×C3
Frattini, socleC23×C3D3×A4
prime radical#95⋊H(3,𝔽3)C2
LCSG ▷ C32×#95
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #10592361C3semidirect
100. C32#401⋊C3semidirect

Maximal subgroups

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

subgroupindex
#95⋊H(3,𝔽3)2
#105923613
C3⋊(#401⋊C3)3
(C3⋊Dic3)⋊A44