Finite Group Databasebeta

← #11724152

Subgroups — #11724152

Order 864.

134 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 36 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#88946C211yesmaximal, characteristic
2#88945C211yesmaximal, characteristic
3#88945C211yesmaximal, characteristic
4C3×#3777C211yesmaximal, characteristic
5Dic3⋊#157C211yesmaximal, characteristic
6Dic3⋊#157C211yesmaximal, characteristic
7Dic3×#157C211yesmaximal, characteristic
8#64243—31nomaximal
9C2×C3⋊#157C2211yescharacteristic
10C2×C3⋊#157C2211yescharacteristic
11C2×C3⋊#157C2211yescharacteristic
12C2×C3×#157C2211yescharacteristic
13C2×C3×#157C2211yescharacteristic
14C2×C3×#157C2211yescharacteristic
15Dic3×C3⋊D3C2211yescharacteristic
16C4⋊#157—31no—
17C4⋊#157—31no—
18C4×#157—31no—
19#3777—31no—
20#3777—31no—
21#3777—31no—
22#3777D311yescharacteristic
23C2×C3×C3⋊D3C2311yescharacteristic
24C3⋊#157—21no—
25C3⋊#157—21no—
26C3⋊#157—21no—
27C3×#157—21no—
28C3×#157—21no—
29C3×#157—21no—
30C3⋊(C3⋊Dic3)Q811yescharacteristic
31C32×Dic3Q811yescharacteristic
32#889—91nomaximal
33C2×#157—31no—
34C2×#157—31no—
35C2×#157—31no—
36C2×#157C2×D311yescharacteristic
37C2×#157C2×D311yescharacteristic
38C2×#157C2×D311yescharacteristic
39C4×C3⋊D3—31no—
40D3×Dic3—121no—
41C2×C33C2×Q811yescharacteristic
42C3×C3⋊D3#4111yescharacteristic
43C3×C3⋊D3#4111yescharacteristic
44C3×C4⋊C4—91no—
45C4⋊Dic3—91no—
46Dic3⋊C4—91no—
47Dic3⋊C4—91no—
48Dic3⋊C4—91no—
49Dic3⋊C4—91no—
50C4×Dic3—91no—
51C2×C3⋊D3C22×D311yescharacteristic
52C2×C3×D3—121no—
53#157—21no—
54#157—21no—
55#157—21no—
56#157—61no—
57#157—61no—
58#157—61no—
59C32×C4—31no—
60C3⋊Dic3—31no—
61C3⋊Dic3—121no—
62C3⋊Dic3—241no—
63C3×Dic3—41no—
64#125—271noSylow
65C33#12511yescharacteristic, complemented, Sylow
66C2×C3×C4—91no—
67C2×C3×C4—91no—
68C2×C3×C4—91no—
69C2×Dic3—91no—
70C2×Dic3—91no—
71C2×Dic3—91no—
72C2×Dic3—91no—
73C4×D3—361no—
74C2×C32—41no—
75C2×C32—81no—
76C2×C32D3×Q811yescharacteristic
77C3⋊D3#23511yescharacteristic
78C3⋊D3#23711yescharacteristic
79C3×D3—121no—
80C3×D3—121no—
81C4⋊C4—91no—
82C4⋊C4—271no—
83C4⋊C4—271no—
84C4⋊C4—271no—
85C4⋊C4—271no—
86C4⋊C4—271no—
87C42—271no—
88C22×C3—91no—
89C2×D3—121no—
90C3×C4—121no—
91C3×C4—181no—
92C3×C4—181no—
93C3×C4—181no—
94Dic3—91no—
95Dic3—181no—
96Dic3—181no—
97Dic3—181no—
98Dic3—361no—
99Dic3—721no—
100Dic3PSU(3,𝔽2)11yescharacteristic
101C32—41no—
102C32—81no—
103C32#88911yescharacteristic, complemented
104C2×C4—91no—
105C2×C4—91no—
106C2×C4—91no—
107C2×C4—271no—
108C2×C4—271no—
109C2×C4—271no—
110C2×C4—271no—
111C2×C3—41no—
112C2×C3—81no—
113C2×C3—91no—
114C2×C3—91no—
115C2×C3C2×PSU(3,𝔽2)11yescharacteristic
116D3—121no—
117D3—121no—
118C22—91no—
119C4—31no—
120C4—181no—
121C4—181no—
122C4—181no—
123C4—271no—
124C4—541no—
125C4—541no—
126C4—541no—
127C3—41no—
128C3—81no—
129C3#6424311yescharacteristic, complemented
130C2—91no—
131C2—91no—
132C2D3×PSU(3,𝔽2)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D3×PSU(3,𝔽2)
commutator, prime radical, cosocleC2×C3×C3⋊D3C23
Fitting, socleC2×C33C2×Q8
hypocenterC33#125
LCSG ▷ C2×C3×C3⋊D3 ▷ C33
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
65. C33#125semidirect
103. C32#889semidirect
129. C3#64243semidirect

Maximal subgroups

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

subgroupindex
#889462
#889452
C3×#37772
Dic3⋊#1572
Dic3×#1572
#642433
#8899