Finite Group Databasebeta

← #11722393

Subgroups — #11722393

Order 864.

200 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 26 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#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#88611C211yesmaximal, characteristic, complemented
3#88610C211yesmaximal, characteristic, complemented
4C3×#985—33nomaximal
5C3×#950—31nomaximal
6C4×#272—41nomaximal
7Q8⋊H(3,𝔽3)C2211yescharacteristic
8C32×#41D311yescharacteristic, complemented
9C3×#235—31no—
10C3×#41⋊C3D313yescomplemented
11C3×GL(2,𝔽3)—33no—
12C3×#226—33no—
13C3×#216—31no—
14C3×#215—31no—
15C3×#214—31no—
16C3×#213—31no—
17C2×C3×C3⋊C8—31no—
18C2×#272—41no—
19C4×H(3,𝔽3)—41no—
20#1084—41no—
21#985—33no—
22C3×#135—91no—
23#950—31no—
24C32×Q8C2×D311yescharacteristic
25C32×D4—31no—
26C2×C32×C4—31no—
27C3×#67—61no—
28C3×D12—31no—
29C3×C4×D3—61no—
30C3×C4×D3—123no—
31C3×Dic6—31no—
32C3×SL(2,𝔽3)C2×D313yes—
33C3×C3⋊C8—31no—
34C3×C3⋊C8—31no—
35C2×H(3,𝔽3)—41no—
36#272—42no—
37C3×#41—31no—
38C3×#41—91no—
39C3×#41C3⋊D311yescharacteristic
40#235—31no—
41#41⋊C3—33no—
42GL(2,𝔽3)—33no—
43#226—33no—
44C3×Q16—91no—
45C3×SD16—91no—
46C3×SD16—91no—
47C3×D8—91no—
48C2×C3×C8—91no—
49#216—31no—
50#215—31no—
51#214—31no—
52#213—31no—
53C2×C3⋊C8—31no—
54C22×C32—31no—
55C2×C3×D3—61no—
56C2×C3×D3—123no—
57C32×C4—31no—
58C32×C4—43no—
59C32×C4S411yescharacteristic
60C3×Dic3—61no—
61C3×Dic3—123no—
62#135—91noSylow
63H(3,𝔽3)—41noSylow
64C3×Q8—31no—
65C3×Q8—91no—
66C3×Q8C2×C3⋊D311yescharacteristic
67C3×D4—31no—
68C3×D4—31no—
69C3×D4—61no—
70C3×D4—91no—
71C3×D4—181no—
72C2×C3×C4—31no—
73C2×C3×C4—31no—
74C2×C3×C4—61no—
75C2×C3×C4—181no—
76#67—61no—
77D12—31no—
78C4×D3—61no—
79C4×D3—123no—
80Dic6—31no—
81SL(2,𝔽3)—33no—
82C3×C8—91no—
83C3×C8—91no—
84C3⋊C8—31no—
85C3⋊C8—31no—
86C2×C32—43no—
87C2×C32—61no—
88C2×C32C2×S411yescharacteristic
89C3×D3—126no—
90C3×D3—121no—
91#41—91no—
92#41#27211yescharacteristic, complemented
93Q16—91no—
94SD16—91no—
95SD16—91no—
96D8—91no—
97C2×C8—91no—
98C22×C3—31no—
99C22×C3—31no—
100C22×C3—61no—
101C22×C3—181no—
102C2×D3—61no—
103C2×D3—123no—
104C3×C4—31no—
105C3×C4—31no—
106C3×C4—31no—
107C3×C4—61no—
108C3×C4—123no—
109C3×C4—181no—
110C3×C4C3⋊S411yescharacteristic
111Dic3—61no—
112Dic3—123no—
113C32—43no—
114C32#98511yescharacteristic, complemented
115Q8—91no—
116Q8C2×#27211yescharacteristic
117D4—31no—
118D4—91no—
119D4—181no—
120C2×C4—31no—
121C2×C4—181no—
122C8—91no—
123C8—91no—
124C2×C3—31no—
125C2×C3—61no—
126C2×C3—61no—
127C2×C3—121no—
128C2×C3—123no—
129C2×C3—361no—
130C2×C3C2×C3⋊S411yescharacteristic
131D3—121no—
132D3—126no—
133C22—31no—
134C22—181no—
135C4—31no—
136C4—181no—
137C4#625811yescharacteristic
138C3—31no—
139C3—123no—
140C3#6426611yescharacteristic
141C2—61no—
142C2—361no—
143C2C2×#625811yescharacteristic
subgroupG/H
center, hypercenterC3×C4C3⋊S4
commutator, cosocle, hypocenterQ8⋊H(3,𝔽3)C22
Frattini, socleC2×C3C2×C3⋊S4
FittingC32×#41D3
LCSG ▷ Q8⋊H(3,𝔽3)
UCSC1 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. #41⋊H(3,𝔽3)C2semidirect
2. #88611C2semidirect
3. #88610C2semidirect
8. C32×#41D3semidirect
10. C3×#41⋊C3D3semidirect
92. #41#272semidirect
114. C32#985semidirect

Maximal subgroups

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

subgroupindex
#41⋊H(3,𝔽3)2
#886112
#886102
C3×#9853
C3×#9503
C4×#2724