Finite Group Databasebeta

← #13146451

Subgroups — #13146451

Order 1728.

141 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 141 conjugacy classes of subgroups — too many to draw. Only its 25 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#11722292C212yesmaximal, complemented
3C22×#1084C411yescharacteristic
4#88631—22no—
5C2×#6188C411yescharacteristic
6C2×#6188C2211yescharacteristic
7#88586—22no—
8C23×H(3,𝔽3)C811yescharacteristic
9C2×#1084—21no—
10C2×#1084—21no—
11C2×#1084C811yescharacteristic
12C2×#1084C2×C411yescharacteristic
13#6188—21no—
14#6188D412yes—
15#4553—91nomaximal
16C22×C3×Dic3—121no—
17C22×H(3,𝔽3)—21no—
18C22×H(3,𝔽3)—41no—
19C22×H(3,𝔽3)C2×C811yescharacteristic
20#1084—21no—
21#1084C22⋊C411yescharacteristic
22#1084M1611yescharacteristic
23C2×C3×M16—91no—
24#812—92no—
25C23×C32—41no—
26C2×C3×Dic3—121no—
27C2×C3×Dic3—121no—
28C2×C3×Dic3—122no—
29C2×C3×Dic3—122no—
30#349—271noSylow
31C2×H(3,𝔽3)—21no—
32C2×H(3,𝔽3)—41no—
33C2×H(3,𝔽3)C22⋊C811yescharacteristic
34C22×C3×C4—91no—
35C22×Dic3—121no—
36C3×M16—182no—
37C2×C3×C8—91no—
38C2×C3×C8—91no—
39C3⋊C16—182no—
40C22×C32—42no—
41C22×C32—44no—
42C22×C32—41no—
43C3×Dic3—121no—
44C3×Dic3—121no—
45C3×Dic3—122no—
46C2×M16—91no—
47M32—272no—
48H(3,𝔽3)#34911yescharacteristic, complemented, Sylow
49C23×C3—121no—
50C23×C3AGL(1,𝔽9)11yescharacteristic
51C2×C3×C4—91no—
52C2×C3×C4—91no—
53C2×C3×C4—181no—
54C2×C3×C4—181no—
55C2×Dic3—121no—
56C2×Dic3—121no—
57C2×Dic3—122no—
58C2×Dic3—122no—
59C3×C8—92no—
60C3×C8—181no—
61C2×C32—42no—
62C2×C32—44no—
63C2×C32—41no—
64C22×C4—91no—
65M16—182no—
66C2×C8—91no—
67C2×C8—91no—
68C16—542no—
69C22×C3—21no—
70C22×C3—41no—
71C22×C3—122no—
72C22×C3—124no—
73C22×C3—121no—
74C22×C3C2×AGL(1,𝔽9)11yescharacteristic
75C3×C4—91no—
76C3×C4—91no—
77C3×C4—181no—
78Dic3—121no—
79Dic3—121no—
80Dic3—122no—
81C32—41no—
82C23#624911yescharacteristic
83C2×C4—91no—
84C2×C4—91no—
85C2×C4—181no—
86C2×C4—181no—
87C8—92no—
88C8—181no—
89C2×C3—21no—
90C2×C3—41no—
91C2×C3—122no—
92C2×C3—124no—
93C2×C3—121no—
94C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
95C22—21no—
96C22—41no—
97C22C2×#624911yescharacteristic
98C4—91no—
99C4—91no—
100C4—181no—
101C3—121no—
102C3#1059327411yescharacteristic
103C2—21no—
104C2—41no—
105C2C22⋊#624911yescharacteristic
subgroupG/H
centerC2C22⋊#6249
commutatorC22×H(3,𝔽3)C2×C8
FrattiniC22×C3C2×AGL(1,𝔽9)
Fitting, prime radicalC23×H(3,𝔽3)C8
socleC2×C3C22⋊AGL(1,𝔽9)
cosocleC2×#6188C22
hypercenterC23#6249
hypocenterH(3,𝔽3)#349
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #11722292C2semidirect
48. H(3,𝔽3)#349semidirect

Maximal subgroups

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

subgroupindex
C2×#886312
#117222922
#45539