Finite Group Databasebeta

← #11720322

Subgroups — #11720322

Order 864.

174 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 40 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#3828⋊C3C211yesmaximal, characteristic
2#88538C211yesmaximal, characteristic, complemented
3C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#63570—31nomaximal
5#63568—31nomaximal
6C2×C4×H(3,𝔽3)C2211yescharacteristic
7#638⋊C3—21no—
8C2×(C3⋊Dic3)⋊C3—21no—
9C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
10C4×(C3⋊D3)⋊C3—21no—
11#634⋊C3C411yescharacteristic, complemented
12#6180—21no—
13#3828D311yescharacteristic
14C3×#235—31no—
15C4×C3⋊Dic3D311yescharacteristic, complemented
16C3×C4×Dic3—31no—
17C3×#208—31no—
18C3×#208—31no—
19C3×#208—61no—
20C22×H(3,𝔽3)D411yescharacteristic
21C2×(C3⋊D3)⋊C3—21no—
22C4×H(3,𝔽3)C2×C411yescharacteristic
23C4×H(3,𝔽3)D411yescharacteristic
24(C3⋊Dic3)⋊C3—22no—
25(C3⋊Dic3)⋊C3—21no—
26#837—91no—
27#825—91no—
28#805—91no—
29C2×C32×C4—21no—
30C2×C32×C4C2×D311yescharacteristic
31C2×C32×C4C2×D311yescharacteristic
32#638—21no—
33C2×C3⋊Dic3—21no—
34C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
35C4×C3⋊D3—21no—
36#634Dic311yescharacteristic, complemented
37C3×#67—61no—
38C2×C3×Dic3—61no—
39C3×D12—31no—
40C3×C4×D3—61no—
41C3×Dic6—31no—
42C3×C3⋊C8—62no—
43C3×C3⋊C8—61no—
44C3×C3⋊C8—61no—
45C2×H(3,𝔽3)—21no—
46C2×H(3,𝔽3)C22⋊C411yescharacteristic
47(C3⋊D3)⋊C3—41no—
48C3×#41—91no—
49#235—31no—
50#235—91no—
51C3×M16—91no—
52C3×C42—91no—
53C4×Dic3—31no—
54C4×Dic3—181no—
55#208—31no—
56#208—31no—
57#208—61no—
58C22×C32—21no—
59C22×C32#6711yescharacteristic
60C22×C32#6711yescharacteristic
61C2×C3⋊D3—21no—
62C2×C3×D3—61no—
63C32×C4—21no—
64C32×C4—21no—
65C32×C4C4×D311yescharacteristic
66C32×C4D1211yescharacteristic
67C32×C4C2×Dic311yescharacteristic
68C32×C4#6711yescharacteristic
69C3⋊Dic3—22no—
70C3⋊Dic3—21no—
71C3×Dic3—62no—
72C3×Dic3—61no—
73#104—271noSylow
74H(3,𝔽3)#10411yescharacteristic, complemented, Sylow
75C3×Q8—91no—
76C3×D4—91no—
77C3×D4—181no—
78C2×C3×C4—31no—
79C2×C3×C4—61no—
80C2×C3×C4—61no—
81C2×C3×C4—181no—
82C2×C3×C4—181no—
83C2×C3×C4D3211yescharacteristic
84#67—61no—
85#67—181no—
86C2×Dic3—61no—
87C2×Dic3—181no—
88D12—31no—
89D12—91no—
90C4×D3—61no—
91C4×D3—181no—
92Dic6—31no—
93Dic6—91no—
94C3×C8—181no—
95C3⋊C8—61no—
96C3⋊C8—61no—
97C3⋊C8—62no—
98C2×C32—21no—
99C2×C32—21no—
100C2×C32—21no—
101C2×C32—41no—
102C2×C32(C2×D3)⋊C411yescharacteristic
103C2×C32#21711yescharacteristic
104C3⋊D3—41no—
105C3×D3—121no—
106#41—91no—
107M16—91no—
108C42—91no—
109C22×C3—31no—
110C22×C3—61no—
111C22×C3—61no—
112C22×C3—181no—
113C22×C3#62511yescharacteristic
114C2×D3—61no—
115C2×D3—181no—
116C3×C4—31no—
117C3×C4—31no—
118C3×C4—61no—
119C3×C4—61no—
120C3×C4—61no—
121C3×C4—61no—
122C3×C4—181no—
123C3×C4—182no—
124C3×C4D3×Dic311yescharacteristic
125C3×C4#62611yescharacteristic
126Dic3—62no—
127Dic3—61no—
128Dic3—181no—
129Dic3—182no—
130C32—21no—
131C32#80511yescharacteristic, complemented
132C32#83711yescharacteristic, complemented
133Q8—91no—
134D4—91no—
135D4—181no—
136C2×C4—181no—
137C2×C4—181no—
138C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
139C8—181no—
140C2×C3—21no—
141C2×C3—31no—
142C2×C3—61no—
143C2×C3—61no—
144C2×C3—61no—
145C2×C3—61no—
146C2×C3—121no—
147C2×C3—361no—
148C2×C3(C2×D3)⋊Dic311yescharacteristic
149D3—121no—
150D3—361no—
151C22—181no—
152C22#619811yescharacteristic
153C4—182no—
154C4—181no—
155C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
156C4#620011yescharacteristic
157C3—31no—
158C3—61no—
159C3—61no—
160C3#6356911yescharacteristic
161C2—21no—
162C2—361no—
163C2#8845011yescharacteristic
subgroupG/H
centerC4#6200
commutatorC4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC2×C3(C2×D3)⋊Dic3
prime radical#3828⋊C3C2
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #88538C2semidirect
3. C4×(C3⋊Dic3)⋊C3C2semidirect
9. C4⋊((C3⋊D3)⋊C3)C4semidirect
11. #634⋊C3C4semidirect
15. C4×C3⋊Dic3D3semidirect
34. C4⋊(C3⋊D3)Dic3semidirect
36. #634Dic3semidirect
74. H(3,𝔽3)#104semidirect
131. C32#805semidirect
132. C32#837semidirect

Maximal subgroups

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

subgroupindex
#3828⋊C32
#885382
C4×(C3⋊Dic3)⋊C32
#635703
#635683