Finite Group Databasebeta

← #11724351

Subgroups — #11724351

Order 864.

236 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 35 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#89046C212yesmaximal, complemented
2C2×C3×#622C211yesmaximal, characteristic, complemented
3#88943C211yesmaximal, characteristic, complemented
4#88942C212yesmaximal, complemented
5#88941C211yesmaximal, characteristic, complemented
6#64237—31nomaximal
7C3×#638—22no—
8C2×C3×C3⋊Dic3C2211yescharacteristic
9#6296C2212yes—
10#6295C2212yes—
11#6294—22no—
12C3×#622C2211yescharacteristic
13C3×#622C2211yescharacteristic
14#3805—32no—
15#3804—62no—
16C2×#622D311yescharacteristic, complemented
17#3776—31no—
18#3775—32no—
19#3774—31no—
20C22×C33D411yescharacteristic
21C2×C3×C3⋊D3—22no—
22C3×C3⋊Dic3—22no—
23C3×C3⋊Dic3D411yescharacteristic
24C3×C3⋊Dic3C2311yescharacteristic
25#904—91nomaximal
26#638—22no—
27C2×C3⋊Dic3C2×D311yescharacteristic
28C3×#67—62no—
29C3×#67—62no—
30C3×#67—122no—
31C2×C3×Dic3—62no—
32#627—32no—
33#627—62no—
34#626—62no—
35#625—32no—
36#624—62no—
37D3×Dic3—62no—
38D3×Dic3—62no—
39#622C2×D311yescharacteristic
40#622C2×D311yescharacteristic
41C2×C33—21no—
42C2×C33C2×D411yescharacteristic
43C3×C3⋊D3—42no—
44#237—182no—
45#235—92no—
46C2×C3×C8—91no—
47Dic12—91no—
48D24—91no—
49#204—92no—
50C22×C32—22no—
51C22×C32—42no—
52C22×C32D1211yescharacteristic
53C2×C3⋊D3—22no—
54C2×C3×D3—62no—
55C2×C3×D3—62no—
56C2×C3×D3—122no—
57C3⋊Dic3—22no—
58C3⋊Dic3D1211yescharacteristic
59C3⋊Dic3C22×D311yescharacteristic
60C3×Dic3—62no—
61C3×Dic3—62no—
62C3×Dic3—62no—
63C3×Dic3—62no—
64C3×Dic3—122no—
65#135—271noSylow
66C33#13511yescharacteristic, complemented, Sylow
67C3×D4—182no—
68C2×C3×C4—91no—
69#67—62no—
70#67—62no—
71#67—122no—
72#67—182no—
73C2×Dic3—62no—
74C2×Dic3—182no—
75D12—92no—
76C4×D3—182no—
77C4×D3—182no—
78Dic6—92no—
79Dic6—182no—
80C3×C8—91no—
81C3×C8—91no—
82C2×C32—22no—
83C2×C32—21no—
84C2×C32—42no—
85C2×C32—42no—
86C2×C32—82no—
87C2×C32C2×D1211yescharacteristic
88C3⋊D3—42no—
89C3×D3—122no—
90C3×D3—122no—
91C3×D3—242no—
92#41—272no—
93Q16—271no—
94SD16—272no—
95D8—271no—
96C2×C8—91no—
97C22×C3—22no—
98C22×C3—42no—
99C22×C3—182no—
100C22×C3GO+(4,𝔽2)11yescharacteristic
101C2×D3—62no—
102C2×D3—62no—
103C2×D3—122no—
104C3×C4—91no—
105C3×C4—91no—
106C3×C4—182no—
107Dic3—62no—
108Dic3—62no—
109Dic3—62no—
110Dic3—62no—
111Dic3—122no—
112C32—22no—
113C32—42no—
114C32#90411yescharacteristic, complemented
115Q8—272no—
116D4—182no—
117D4—272no—
118C2×C4—91no—
119C2×C4—542no—
120C8—91no—
121C8—91no—
122C2×C3—21no—
123C2×C3—22no—
124C2×C3—42no—
125C2×C3—42no—
126C2×C3—82no—
127C2×C3—362no—
128C2×C3C2×GO+(4,𝔽2)11yescharacteristic
129D3—122no—
130D3—122no—
131D3—242no—
132C22—182no—
133C22#157⋊D311yescharacteristic
134C4—91no—
135C4—91no—
136C4—182no—
137C3—22no—
138C3—42no—
139C3#6423711yescharacteristic, complemented
140C2—21no—
141C2—362no—
142C2C2×#157⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C2×#157⋊D3
commutator, cosocleC3×C3⋊Dic3C23
FittingC22×C33D4
socleC2×C33C2×D4
hypercenterC22#157⋊D3
hypocenterC33#135
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #89046C2semidirect
2. C2×C3×#622C2semidirect
3. #88943C2semidirect
4. #88942C2semidirect
5. #88941C2semidirect
16. C2×#622D3semidirect
66. C33#135semidirect
114. C32#904semidirect
139. C3#64237semidirect

Maximal subgroups

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

subgroupindex
#890462
C2×C3×#6222
#889432
#889422
#889412
#642373
#9049