Finite Group Databasebeta

← #13167682

Subgroups — #13167682

Order 1728.

274 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 79 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#11724511C212yesmaximal
2C2×C3×#3788C211yesmaximal, characteristic
3#11722811C214yesmaximal, complemented
4#10593547—31nomaximal
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88993C2214yes—
7#88989—22no—
8#88988—22no—
9C2×C3×#622C2212yes—
10C3×#3788C414yes—
11#64283—32no—
12C2×#3788D311yescharacteristic
13#63748—34no—
14C22×C3×C3⋊D3Q811yescharacteristic
15C2×C33×C4Q811yescharacteristic
16C2×C3×C3⋊Dic3C2311yescharacteristic
17C3×C4×C3⋊D3C2×C412yes—
18C3×C4×C3⋊D3D412yes—
19C3⋊#622—24no—
20C3×#622C2×C414yes—
21#5125—91nomaximal
22C2×C4×C3⋊D3C2×D311yescharacteristic
23C2×C3×C4×D3—121no—
24#3792—34no—
25C2×#622C2×D312yes—
26#3788—62no—
27#3788Dic314yes—
28#3787—62no—
29C22×C33C2×Q811yescharacteristic
30C2×C3×C3⋊D3—41no—
31C2×C3×C3⋊D3C4⋊C412yes—
32C33×C4C4⋊C412yes—
33C3×C3⋊Dic3C22×C411yescharacteristic
34C3×C3⋊Dic3C2×D411yescharacteristic
35C2×C3×M16—91no—
36C2×#208—92no—
37#819—94no—
38C22×C3⋊D3Dic611yescharacteristic
39C22×C3×D3—121no—
40C2×C32×C4—41no—
41C2×C32×C4—81no—
42C2×C32×C4Dic611yescharacteristic
43C2×C3⋊Dic3C22×D311yescharacteristic
44C4×C3⋊D3D1212yes—
45C4×C3⋊D3C2×Dic312yes—
46C2×C3×Dic3—121no—
47C3×C4×D3—122no—
48C3×C4×D3—122no—
49#622—64no—
50#622C2×Dic314yes—
51#430—271noSylow
52C2×C33—21no—
53C2×C33C2×C4⋊C411yescharacteristic
54C3×C3⋊D3—22no—
55C22×C3×C4—91no—
56C2×C4×D3—121no—
57C3×M16—94no—
58C2×C3×C8—92no—
59#208—94no—
60#208—182no—
61C2×C3⋊C8—182no—
62C22×C32—41no—
63C22×C32—81no—
64C22×C32C2×Dic611yescharacteristic
65C2×C3⋊D3—41no—
66C2×C3⋊D3C4⋊Dic312yes—
67C2×C3×D3—122no—
68C2×C3×D3—124no—
69C32×C4—42no—
70C32×C4—82no—
71C32×C4C4⋊Dic312yes—
72C3⋊Dic3C2×D1211yescharacteristic
73C3⋊Dic3C22×Dic311yescharacteristic
74C3×Dic3—121no—
75C3×Dic3—121no—
76C2×M16—91no—
77C2×M16—272no—
78#108—274no—
79C33#43011yescharacteristic, complemented, Sylow
80C23×C3—91no—
81C22×D3—121no—
82C2×C3×C4—41no—
83C2×C3×C4—81no—
84C2×C3×C4—91no—
85C2×C3×C4—92no—
86C2×C3×C4—92no—
87C2×C3×C4PSU(3,𝔽2)11yescharacteristic
88C2×Dic3—121no—
89C4×D3—122no—
90C4×D3—122no—
91C3×C8—94no—
92C3⋊C8—184no—
93C2×C32—21no—
94C2×C32—42no—
95C2×C32—41no—
96C2×C32—82no—
97C2×C32—81no—
98C2×C32C2×C4⋊Dic311yescharacteristic
99C3⋊D3—22no—
100C3×D3—124no—
101C22×C4—91no—
102M16—94no—
103M16—274no—
104M16—542no—
105C2×C8—92no—
106C2×C8—542no—
107C22×C3—41no—
108C22×C3—81no—
109C22×C3—92no—
110C22×C3—361no—
111C22×C3C2×PSU(3,𝔽2)11yescharacteristic
112C2×D3—122no—
113C2×D3—124no—
114C3×C4—42no—
115C3×C4—82no—
116C3×C4—91no—
117C3×C4—91no—
118C3×C4#377712yes—
119Dic3—121no—
120Dic3—121no—
121C32—41no—
122C32—81no—
123C32#512511yescharacteristic, complemented
124C23—91no—
125C2×C4—91no—
126C2×C4—92no—
127C2×C4—92no—
128C2×C4C3⋊PSU(3,𝔽2)11yescharacteristic
129C8—94no—
130C8—544no—
131C2×C3—21no—
132C2×C3—42no—
133C2×C3—41no—
134C2×C3—82no—
135C2×C3—81no—
136C2×C3—182no—
137C2×C3C2×#377711yescharacteristic
138D3—124no—
139C22—92no—
140C22—361no—
141C22C2×C3⋊PSU(3,𝔽2)11yescharacteristic
142C4—91no—
143C4—91no—
144C4#8894612yes—
145C3—41no—
146C3—81no—
147C3#1059354711yescharacteristic, complemented
148C2—21no—
149C2—182no—
150C2C2×#8894611yescharacteristic
subgroupG/H
centerC2C2×#88946
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×C3⋊PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33C2×C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4C3⋊PSU(3,𝔽2)
hypocenterC33#430
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722811C2semidirect
79. C33#430semidirect
123. C32#5125semidirect
147. C3#10593547semidirect

Maximal subgroups

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

subgroupindex
#117245112
C2×C3×#37882
#117228112
#105935473
#51259