Finite Group Databasebeta

← #13165122

Subgroups — #13165122

Order 1728.

162 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 31 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×#88943C211yesmaximal, characteristic
2#11724242C211yesmaximal, characteristic
3C3×#63748C211yesmaximal, characteristic
4#10593169—31nomaximal
5C2×#6296—21no—
6C3×#3792—21no—
7C2×C3×#622C2211yescharacteristic
8#88943—21no—
9#88943C412yes—
10C3⋊#3771—21no—
11C2×#3776—31no—
12#64218—31no—
13#63748D311yescharacteristic
14C2×C3×C3⋊Dic3—41no—
15C2×C3×C3⋊Dic3D411yescharacteristic
16#6296—22no—
17#6296—41no—
18C3×#622—41no—
19C3×#622C2×C411yescharacteristic
20C3×#622D411yescharacteristic
21#4496—91nomaximal
22C2×#627—61no—
23C2×#627—121no—
24#3792—21no—
25C2×#622C2×D311yescharacteristic
26#3776—32no—
27#3776—61no—
28#3771—61no—
29C22×C33SD1611yescharacteristic
30C3×C3⋊Dic3—42no—
31C3×C3⋊Dic3C22⋊C411yescharacteristic
32C3×C3⋊Dic3D811yescharacteristic
33C2×Dic12—91no—
34C3×#108—91no—
35#812—91no—
36C2×C3⋊Dic3—41no—
37C2×C3⋊Dic3#6711yescharacteristic
38C2×C3×Dic3—121no—
39C2×C3×Dic3—121no—
40C2×C3×Dic3—121no—
41C2×C3×Dic3—241no—
42#627—62no—
43#627—122no—
44#627—122no—
45#627—121no—
46#622—41no—
47#622C4×D311yescharacteristic
48#622D1211yescharacteristic
49#362—271noSylow
50C2×C33—21no—
51C2×C33#10211yescharacteristic
52C2×Dic6—181no—
53C2×Dic6—361no—
54C3×M16—181no—
55C2×C3×C8—91no—
56Dic12—92no—
57Dic12—181no—
58C3⋊C16—181no—
59C22×C32—41no—
60C22×C32—81no—
61C22×C32#21511yescharacteristic
62C3⋊Dic3—42no—
63C3⋊Dic3(C2×D3)⋊C411yescharacteristic
64C3⋊Dic3#21311yescharacteristic
65C3×Dic3—121no—
66C3×Dic3—121no—
67C3×Dic3—122no—
68C3×Dic3—122no—
69C3×Dic3—242no—
70C2×Q16—271no—
71M32—271no—
72#108—91no—
73C33#36211yescharacteristic, complemented, Sylow
74C2×C3×C4—91no—
75C2×C3×C4—361no—
76C2×Dic3—121no—
77C2×Dic3—121no—
78C2×Dic3—121no—
79C2×Dic3—241no—
80Dic6—182no—
81Dic6—362no—
82Dic6—361no—
83Dic6—362no—
84C3×C8—91no—
85C3×C8—91no—
86C3×C8—361no—
87C2×C32—21no—
88C2×C32—42no—
89C2×C32—41no—
90C2×C32—82no—
91C2×C32—81no—
92C2×C32#80911yescharacteristic
93C2×Q8—541no—
94Q16—272no—
95Q16—541no—
96M16—181no—
97C2×C8—91no—
98C16—541no—
99C22×C3—41no—
100C22×C3—81no—
101C22×C3AΓL(1,𝔽9)11yescharacteristic
102C3×C4—91no—
103C3×C4—91no—
104C3×C4—362no—
105Dic3—121no—
106Dic3—121no—
107Dic3—122no—
108Dic3—122no—
109Dic3—242no—
110C32—41no—
111C32—81no—
112C32#449611yescharacteristic, complemented
113Q8—542no—
114Q8—1081no—
115C2×C4—91no—
116C2×C4—361no—
117C8—91no—
118C8—91no—
119C8—361no—
120C2×C3—21no—
121C2×C3—42no—
122C2×C3—41no—
123C2×C3—82no—
124C2×C3—81no—
125C2×C3GO+(4,𝔽2)⋊C411yescharacteristic
126C22#8909211yescharacteristic
127C4—91no—
128C4—91no—
129C4—362no—
130C3—41no—
131C3—81no—
132C3#1059316911yescharacteristic, complemented
133C2—21no—
134C2#1172410111yescharacteristic
subgroupG/H
centerC2#11724101
commutatorC3×#622C2×C4
Frattini, hypercenterC22#89092
Fitting, prime radicalC22×C33SD16
socleC2×C33#102
cosocleC2×C3×#622C22
hypocenterC33#362
LCSG ▷ C3×#622 ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
73. C33#362semidirect
112. C32#4496semidirect
132. C3#10593169semidirect

Maximal subgroups

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

subgroupindex
C2×#889432
#117242422
C3×#637482
#105931693
#44969