Finite Group Databasebeta

← #13136721

Subgroups — #13136721

Order 1728.

293 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 39 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×#88631C211yesmaximal, characteristic
2#11721298C212yesmaximal
3C22×#1084C2211yescharacteristic
4#88631—22no—
5C2×#6188C412yescomplemented
6#3757⋊C3—22no—
7#88449—22no—
8#88447C414yescomplemented
9#63955—62no—
10C23×H(3,𝔽3)D411yescharacteristic
11C2×#1084C2×C411yescharacteristic
12C2×#1084C2×C412yes—
13C2×#1084D412yes—
14C2×#1084Q811yescharacteristic
15C2×(C3⋊Dic3)⋊C3—24no—
16#6188—22no—
17#4557—91nomaximal
18C22×C3×Dic3—62no—
19#3757—22no—
20C3×#217—62no—
21Dic3⋊Dic3—64no—
22Dic32—64no—
23C22×H(3,𝔽3)—41no—
24C22×H(3,𝔽3)C22⋊C412yes—
25C22×H(3,𝔽3)C4⋊C411yescharacteristic
26#1084C4211yescharacteristic
27#1084C22⋊C411yescharacteristic
28#1084C4⋊C412yes—
29(C3⋊Dic3)⋊C3—44no—
30C2×C3×M16—91no—
31#926—92no—
32#877—182no—
33C23×C32—22no—
34C2×C3⋊Dic3—24no—
35C2×C3×Dic3—62no—
36C2×C3×Dic3—62no—
37C2×C3×Dic3—62no—
38C2×C3×Dic3—62no—
39C2×C3×Dic3—62no—
40C2×C3×Dic3—62no—
41C2×C3×Dic3—64no—
42H(3,ℤ/4)—271noSylow
43C2×H(3,𝔽3)—22no—
44C2×H(3,𝔽3)—21no—
45C2×H(3,𝔽3)#9511yescharacteristic
46C22×C3×C4—91no—
47C22×Dic3—62no—
48C3×M16—182no—
49C2×C3×C8—92no—
50C3×C22⋊C4—182no—
51#217—62no—
52#217—182no—
53C4⋊Dic3—182no—
54Dic3⋊C4—184no—
55C4×Dic3—94no—
56C4×Dic3—184no—
57C22×C32—22no—
58C22×C32—22no—
59C22×C32—22no—
60C22×C32—44no—
61C3⋊Dic3—44no—
62C3×Dic3—62no—
63C3×Dic3—62no—
64C3×Dic3—62no—
65C3×Dic3—62no—
66C3×Dic3—124no—
67C2×M16—91no—
68#117—272no—
69H(3,𝔽3)H(3,ℤ/4)11yescharacteristic, complemented, Sylow
70C23×C3—62no—
71C23×C3GO+(4,𝔽2)11yescharacteristic
72C2×C3×C4—92no—
73C2×C3×C4—92no—
74C2×C3×C4—91no—
75C2×C3×C4—91no—
76C2×C3×C4—184no—
77C2×Dic3—62no—
78C2×Dic3—62no—
79C2×Dic3—62no—
80C2×Dic3—62no—
81C2×Dic3—62no—
82C2×Dic3—62no—
83C2×Dic3—64no—
84C2×Dic3—184no—
85C3×C8—182no—
86C2×C32—24no—
87C2×C32—22no—
88C2×C32—42no—
89C2×C32—42no—
90C22×C4—91no—
91M16—182no—
92C2×C8—92no—
93C4⋊C4—542no—
94C22⋊C4—182no—
95C42—274no—
96C22×C3—41no—
97C22×C3—62no—
98C22×C3—62no—
99C22×C3—62no—
100C22×C3—124no—
101C22×C3D32⋊C412yes—
102C22×C3#377311yescharacteristic
103C3×C4—91no—
104C3×C4—91no—
105C3×C4—92no—
106C3×C4—364no—
107Dic3—62no—
108Dic3—62no—
109Dic3—62no—
110Dic3—62no—
111Dic3—124no—
112Dic3—364no—
113C32—22no—
114C23#1088⋊C211yescharacteristic
115C2×C4—91no—
116C2×C4—91no—
117C2×C4—92no—
118C2×C4—92no—
119C2×C4—184no—
120C8—182no—
121C2×C3—22no—
122C2×C3—21no—
123C2×C3—64no—
124C2×C3—62no—
125C2×C3—122no—
126C2×C3—122no—
127C2×C3#6373711yescharacteristic
128C22—41no—
129C22#8859011yescharacteristic
130C22(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
131C4—91no—
132C4—91no—
133C4—92no—
134C4—364no—
135C3—62no—
136C3#1059007311yescharacteristic
137C2—22no—
138C2—21no—
139C2#1172070711yescharacteristic
subgroupG/H
centerC2#11720707
commutator#1084C42
FrattiniC23×C3GO+(4,𝔽2)
Fitting, prime radicalC23×H(3,𝔽3)D4
socleC2×C3#63737
cosocleC22×#1084C22
hypercenterC23#1088⋊C2
hypocenterH(3,𝔽3)H(3,ℤ/4)
LCSG ▷ #1084 ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
5. C2×#6188C4semidirect
8. #88447C4semidirect
69. H(3,𝔽3)H(3,ℤ/4)semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
#117212982
#45579