Finite Group Databasebeta

← #13146447

Subgroups — #13146447

Order 1728.

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

normal subgroups only

This group has 124 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
1#11722449C211yesmaximal, characteristic
2#11722288C211yesmaximal, characteristic, complemented
3#11722287C211yesmaximal, characteristic, complemented
4#88744C411yescharacteristic
5#88631—21no—
6C2×#6188—21no—
7#88627C411yescharacteristic
8#88626C2211yescharacteristic
9#88586—21no—
10#88586—21no—
11D4×H(3,𝔽3)C811yescharacteristic
12#6235—21no—
13C2×#1084—21no—
14C4×#272C2×C411yescharacteristic
15#6229C811yescharacteristic
16#6188—21no—
17#6188D411yescharacteristic
18#6188D411yescharacteristic
19#4561—91nomaximal
20C3×#237—121no—
21C22×H(3,𝔽3)—21no—
22C2×#272M1611yescharacteristic
23C4×H(3,𝔽3)C2×C811yescharacteristic
24#1084—21no—
25#1084C22⋊C411yescharacteristic
26C3×#131—91no—
27#812—91no—
28C2×C3⋊C16—91no—
29C32×D4—41no—
30C3×#67—122no—
31C2×C3×Dic3—122no—
32C3×C4×D3—121no—
33C3×Dic6—121no—
34#350—271noSylow
35C2×H(3,𝔽3)—41no—
36C2×H(3,𝔽3)C22⋊C811yescharacteristic
37#272—21no—
38C3×#41—91no—
39#237—121no—
40C3×M16—91no—
41C3×M16—181no—
42C2×C3×C8—91no—
43C2×C3×C8—181no—
44C3⋊C16—181no—
45C3⋊C16—181no—
46C22×C32—42no—
47C2×C3×D3—121no—
48C32×C4—41no—
49C3×Dic3—121no—
50C3×Dic3—122no—
51#131—91no—
52M32—271no—
53C2×C16—271no—
54H(3,𝔽3)#35011yescharacteristic, complemented, Sylow
55C3×Q8—91no—
56C3×D4—121no—
57C3×D4—181no—
58C3×D4AGL(1,𝔽9)11yescharacteristic
59C2×C3×C4—91no—
60C2×C3×C4—181no—
61#67—122no—
62C2×Dic3—122no—
63C4×D3—121no—
64Dic6—121no—
65C3×C8—91no—
66C3×C8—91no—
67C3×C8—181no—
68C2×C32—41no—
69C2×C32—82no—
70C3×D3—241no—
71#41—91no—
72M16—91no—
73M16—181no—
74C2×C8—91no—
75C2×C8—181no—
76C16—541no—
77C16—541no—
78C22×C3—21no—
79C22×C3—91no—
80C22×C3—122no—
81C2×D3—121no—
82C3×C4—91no—
83C3×C4—121no—
84C3×C4—181no—
85C3×C4C2×AGL(1,𝔽9)11yescharacteristic
86Dic3—121no—
87Dic3—122no—
88C32—41no—
89Q8—91no—
90D4—181no—
91D4#624911yescharacteristic
92C2×C4—91no—
93C2×C4—181no—
94C8—91no—
95C8—91no—
96C8—181no—
97C2×C3—41no—
98C2×C3—121no—
99C2×C3—181no—
100C2×C3—242no—
101C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
102D3—241no—
103C22—21no—
104C22—91no—
105C4—91no—
106C4—181no—
107C4C2×#624911yescharacteristic
108C3—121no—
109C3#1059326911yescharacteristic
110C2—41no—
111C2—181no—
112C2C22⋊#624911yescharacteristic
subgroupG/H
centerC2C22⋊#6249
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC3×C4C2×AGL(1,𝔽9)
FittingD4×H(3,𝔽3)C8
socleC2×C3C22⋊AGL(1,𝔽9)
prime radical#88744C4
cosocle#88626C22
hypercenterD4#6249
hypocenterH(3,𝔽3)#350
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #11722288C2semidirect
3. #11722287C2semidirect
54. H(3,𝔽3)#350semidirect

Maximal subgroups

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

subgroupindex
#117224492
#117222882
#117222872
#45619