Finite Group Databasebeta

← #11720522

Subgroups — #11720522

Order 864.

242 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 54 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#88740C211yesmaximal, characteristic
2C42×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#88538C211yesmaximal, characteristic, complemented
4C3×#805—31nomaximal
5C3×#805—33nomaximal
6C2×C4×H(3,𝔽3)—21no—
7C2×C4×H(3,𝔽3)C2211yescharacteristic
8#6235—21no—
9C4⋊#272C411yescharacteristic, complemented
10C4×#272—21no—
11#6229C411yescharacteristic, complemented
12#6180—21no—
13C3×#235—31no—
14C3×#235—33no—
15C32×C42D311yescharacteristic, complemented
16C32×C42D313yescomplemented
17C3×#208—31no—
18C3×#208—33no—
19C22×H(3,𝔽3)D411yescharacteristic
20C2×#272—21no—
21C4×H(3,𝔽3)—22no—
22C4×H(3,𝔽3)C2×C411yescharacteristic
23C4×H(3,𝔽3)D411yescharacteristic
24#1084—21no—
25C3×#104—91no—
26#805—31no—
27#805—33no—
28C2×C32×C4—21no—
29C2×C32×C4—23no—
30C2×C32×C4C2×D311yescharacteristic
31C2×C32×C4C2×D313yes—
32C3×#67—61no—
33C3×#67—63no—
34C3×D12—31no—
35C3×D12—33no—
36C3×C4×D3—61no—
37C3×C4×D3—63no—
38C3×Dic6—31no—
39C3×Dic6—33no—
40C3×C3⋊C8—61no—
41C3×C3⋊C8—63no—
42C2×H(3,𝔽3)—21no—
43C2×H(3,𝔽3)C22⋊C411yescharacteristic
44#272—41no—
45C3×#41—91no—
46#235—31no—
47#235—33no—
48C3×M16—91no—
49C3×C42—31no—
50C3×C42—63no—
51C3×C42C3⋊D311yescharacteristic
52#208—31no—
53#208—33no—
54C22×C32#6711yescharacteristic
55C22×C32#6713yes—
56C2×C3×D3—61no—
57C2×C3×D3—63no—
58C32×C4—22no—
59C32×C4—26no—
60C32×C4C4×D311yescharacteristic
61C32×C4C4×D313yes—
62C32×C4D1211yescharacteristic
63C32×C4D1213yes—
64C3×Dic3—61no—
65C3×Dic3—63no—
66#104—91noSylow
67H(3,𝔽3)#10411yescharacteristic, complemented, Sylow
68C3×Q8—91no—
69C3×D4—91no—
70C3×D4—181no—
71C2×C3×C4—21no—
72C2×C3×C4—31no—
73C2×C3×C4—61no—
74C2×C3×C4—63no—
75C2×C3×C4—63no—
76C2×C3×C4—181no—
77C2×C3×C4C2×C3⋊D311yescharacteristic
78#67—61no—
79#67—63no—
80D12—31no—
81D12—33no—
82C4×D3—61no—
83C4×D3—63no—
84Dic6—31no—
85Dic6—33no—
86C3×C8—181no—
87C3⋊C8—61no—
88C3⋊C8—63no—
89C2×C32—21no—
90C2×C32—23no—
91C2×C32(C2×D3)⋊C411yescharacteristic
92C2×C32(C2×D3)⋊C413yes—
93C3×D3—121no—
94C3×D3—123no—
95#41—91no—
96M16—91no—
97C42#27211yescharacteristic, complemented
98C22×C3—31no—
99C22×C3—63no—
100C22×C3—181no—
101C22×C3#63811yescharacteristic
102C2×D3—61no—
103C2×D3—63no—
104C3×C4—22no—
105C3×C4—31no—
106C3×C4—31no—
107C3×C4—62no—
108C3×C4—66no—
109C3×C4—63no—
110C3×C4—63no—
111C3×C4—181no—
112C3×C4C4×C3⋊D311yescharacteristic
113C3×C4C4⋊(C3⋊D3)11yescharacteristic
114Dic3—61no—
115Dic3—63no—
116C32#80511yescharacteristic, complemented
117C32#80513yescomplemented
118Q8—91no—
119D4—91no—
120D4—181no—
121C2×C4—21no—
122C2×C4—181no—
123C2×C4C2×#27211yescharacteristic
124C8—181no—
125C2×C3—21no—
126C2×C3—31no—
127C2×C3—61no—
128C2×C3—63no—
129C2×C3—63no—
130C2×C3—361no—
131C2×C3#375211yescharacteristic
132D3—121no—
133D3—123no—
134C22—181no—
135C22#623511yescharacteristic
136C4—22no—
137C4—181no—
138C4C4×#27211yescharacteristic
139C4C4⋊#27211yescharacteristic
140C3—31no—
141C3—63no—
142C3#6363211yescharacteristic
143C2—21no—
144C2—361no—
145C2#8854211yescharacteristic
subgroupG/H
centerC3×C4C4⋊(C3⋊D3)
commutatorC4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4C2×C3⋊D3
FittingC42×H(3,𝔽3)C2
socleC2×C3#3752
prime radical#88740C2
cosocleC2×C4×H(3,𝔽3)C22
hypercenterC3×C42C3⋊D3
hypocenterH(3,𝔽3)#104
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C42×H(3,𝔽3)C2semidirect
3. #88538C2semidirect
9. C4⋊#272C4semidirect
11. #6229C4semidirect
15. C32×C42D3semidirect
16. C32×C42D3semidirect
67. H(3,𝔽3)#104semidirect
97. C42#272semidirect
116. C32#805semidirect
117. C32#805semidirect

Maximal subgroups

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

subgroupindex
#887402
C42×H(3,𝔽3)2
#885382
C3×#8053