Finite Group Databasebeta

← #11723113

Subgroups — #11723113

Order 864.

264 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 29 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×C3×#638C211yesmaximal, characteristic
2#88993C212yesmaximal, complemented
3#63788—31nomaximal
4C23×C33C411yescharacteristic
5C22×C3×C3⋊D3C411yescharacteristic
6C3×#638—22no—
7C2×C3×C3⋊Dic3C2211yescharacteristic
8C3⋊#622—22no—
9C2×#638D311yescharacteristic
10C2×C3×#67—62no—
11#3792—32no—
12C22×C33—21no—
13C22×C33—41no—
14C22×C33C2×C411yescharacteristic
15C2×C3×C3⋊D3—21no—
16C2×C3×C3⋊D3—41no—
17C3×C3⋊Dic3D412yes—
18#833—91nomaximal
19C23×C32—22no—
20C23×C32—42no—
21C23×C32Dic311yescharacteristic
22C22×C3⋊D3Dic311yescharacteristic
23C22×C3×D3—62no—
24#638—22no—
25C2×C3⋊Dic3C2×D311yescharacteristic
26C3×#67—68no—
27C2×C3×Dic3—62no—
28#622—62no—
29C2×C33—21no—
30C2×C33—41no—
31C2×C33C22⋊C411yescharacteristic
32C3×C3⋊D3—41no—
33C2×C3×D4—91no—
34C2×#67—62no—
35#208—92no—
36C22×C32—21no—
37C22×C32—24no—
38C22×C32—22no—
39C22×C32—44no—
40C22×C32—42no—
41C22×C32—41no—
42C22×C32—44no—
43C22×C32—48no—
44C22×C32C2×Dic311yescharacteristic
45C2×C3⋊D3—21no—
46C2×C3⋊D3—41no—
47C2×C3×D3—64no—
48C2×C3×D3—124no—
49C3⋊Dic3#6712yes—
50C3×Dic3—64no—
51#100—271noSylow
52C33#10011yescharacteristic, complemented, Sylow
53C23×C3—22no—
54C23×C3—42no—
55C23×C3—91no—
56C23×C3#15711yescharacteristic
57C22×D3—62no—
58C3×D4—182no—
59C2×C3×C4—91no—
60#67—68no—
61C2×Dic3—62no—
62C3⋊C8—182no—
63C2×C32—21no—
64C2×C32—24no—
65C2×C32—22no—
66C2×C32—44no—
67C2×C32—42no—
68C2×C32—44no—
69C2×C32—48no—
70C2×C32—41no—
71C2×C32#21711yescharacteristic
72C3⋊D3—41no—
73C3×D3—124no—
74C2×D4—91no—
75M16—272no—
76C22×C3—24no—
77C22×C3—21no—
78C22×C3—22no—
79C22×C3—44no—
80C22×C3—41no—
81C22×C3—44no—
82C22×C3—48no—
83C22×C3—42no—
84C22×C3—181no—
85C22×C3—361no—
86C22×C3C2×#15711yescharacteristic
87C2×D3—64no—
88C2×D3—124no—
89C3×C4—92no—
90Dic3—64no—
91C32—22no—
92C32—42no—
93C32#83311yescharacteristic, complemented
94C23—91no—
95C23C3⋊#15711yescharacteristic
96D4—182no—
97C2×C4—91no—
98C8—542no—
99C2×C3—24no—
100C2×C3—21no—
101C2×C3—22no—
102C2×C3—44no—
103C2×C3—44no—
104C2×C3—48no—
105C2×C3—41no—
106C2×C3—42no—
107C2×C3—361no—
108C2×C3C22⋊#15711yescharacteristic
109D3—124no—
110C22—21no—
111C22—41no—
112C22—181no—
113C22—361no—
114C22C2×C3⋊#15711yescharacteristic
115C4—92no—
116C3—22no—
117C3—42no—
118C3#6378811yescharacteristic, complemented
119C2—21no—
120C2—41no—
121C2—361no—
122C2#8899411yescharacteristic
subgroupG/H
centerC2#88994
commutatorC22×C33C2×C4
FrattiniC22C2×C3⋊#157
FittingC23×C33C4
socleC2×C33C22⋊C4
prime radicalC2×C3×#638C2
cosocleC2×C3×C3⋊Dic3C22
hypercenterC23C3⋊#157
hypocenterC33#100
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #88993C2semidirect
52. C33#100semidirect
93. C32#833semidirect
118. C3#63788semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#6382
#889932
#637883
#8339