Finite Group Databasebeta

← #13146449

Subgroups — #13146449

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#11722451C211yesmaximal, characteristic
2#11722288C211yesmaximal, characteristic, complemented
3#11722287C211yesmaximal, characteristic, complemented
4#88748C411yescharacteristic
5#88627—21no—
6#88627C411yescharacteristic
7#88626—21no—
8#88626C2211yescharacteristic
9#88586—21no—
10#88586—21no—
11Q8×H(3,𝔽3)C811yescharacteristic
12C4⋊#272—21no—
13C4⋊#272C811yescharacteristic
14C4×#272—21no—
15C4×#272C2×C411yescharacteristic
16#6188—21no—
17#6188D411yescharacteristic
18#6188D411yescharacteristic
19#4561—91nomaximal
20C3×#239—121no—
21C2×#272—21no—
22C2×#272M1611yescharacteristic
23C4×H(3,𝔽3)—21no—
24C4×H(3,𝔽3)C2×C811yescharacteristic
25#1084C22⋊C411yescharacteristic
26C3×#131—91no—
27#812—91no—
28C2×C3⋊C16—91no—
29C32×Q8—41no—
30C3×D12—121no—
31C3×D12—122no—
32C3×C4×D3—121no—
33C3×C4×D3—122no—
34#350—271noSylow
35C2×H(3,𝔽3)C22⋊C811yescharacteristic
36#272—21no—
37#272—41no—
38C3×#41—91no—
39#239—121no—
40C3×M16—91no—
41C3×M16—181no—
42C2×C3×C8—91no—
43C2×C3×C8—181no—
44C3⋊C16—181no—
45C3⋊C16—181no—
46C2×C3×D3—121no—
47C2×C3×D3—122no—
48C32×C4—42no—
49C32×C4—41no—
50C3×Dic3—121no—
51#131—91no—
52M32—271no—
53C2×C16—271no—
54H(3,𝔽3)#35011yescharacteristic, complemented, Sylow
55C3×Q8—121no—
56C3×Q8AGL(1,𝔽9)11yescharacteristic
57C3×D4—91no—
58C3×D4—181no—
59C2×C3×C4—91no—
60C2×C3×C4—181no—
61D12—121no—
62D12—122no—
63C4×D3—121no—
64C4×D3—122no—
65C3×C8—91no—
66C3×C8—91no—
67C3×C8—181no—
68C2×C32—41no—
69C3×D3—241no—
70C3×D3—242no—
71#41—91no—
72M16—91no—
73M16—181no—
74C2×C8—91no—
75C2×C8—181no—
76C16—541no—
77C16—541no—
78C22×C3—91no—
79C22×C3—181no—
80C2×D3—121no—
81C2×D3—122no—
82C3×C4—21no—
83C3×C4—91no—
84C3×C4—122no—
85C3×C4—121no—
86C3×C4C2×AGL(1,𝔽9)11yescharacteristic
87Dic3—121no—
88C32—41no—
89Q8#624911yescharacteristic
90D4—91no—
91D4—181no—
92C2×C4—91no—
93C2×C4—181no—
94C8—91no—
95C8—91no—
96C8—181no—
97C2×C3—121no—
98C2×C3—181no—
99C2×C3—361no—
100C2×C3C22⋊AGL(1,𝔽9)11yescharacteristic
101D3—241no—
102D3—242no—
103C22—91no—
104C22—181no—
105C4—21no—
106C4—91no—
107C4C2×#624911yescharacteristic
108C3—121no—
109C3#1059327111yescharacteristic
110C2—181no—
111C2—361no—
112C2C22⋊#624911yescharacteristic
subgroupG/H
centerC2C22⋊#6249
commutatorC4×H(3,𝔽3)C2×C8
FrattiniC3×C4C2×AGL(1,𝔽9)
FittingQ8×H(3,𝔽3)C8
socleC2×C3C22⋊AGL(1,𝔽9)
prime radical#88748C4
cosocle#88626C22
hypercenterQ8#6249
hypocenterH(3,𝔽3)#350
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ Q8

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
#117224512
#117222882
#117222872
#45619