Finite Group Databasebeta

← #11720842

Subgroups — #11720842

Order 864.

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

normal subgroups only

This group has 239 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#88650C211yesmaximal, characteristic
2C8×#272C211yesmaximal, characteristic, complemented
3#3743⋊C3C212yesmaximal, complemented
4#88435C211yesmaximal, characteristic, complemented
5((C3⋊D3)⋊C3)⋊C8C212yesmaximal, complemented
6#63804—32nomaximal
7C4×#272C2211yescharacteristic
8C4×(C3⋊D3)⋊C3C2212yes—
9#6200C412yes—
10#6198C411yescharacteristic
11#6196C411yescharacteristic
12C8×H(3,𝔽3)C2211yescharacteristic
13#6180C2211yescharacteristic
14(C3⋊(C3⋊C8))⋊C3C2212yes—
15#3798—32no—
16#3743D312yescomplemented
17C3×ΓL(1,𝔽25)—32no—
18C3×D3×C8—32no—
19C3×D3×C8—61no—
20#3711—32no—
21#3710—32no—
22D3×C3⋊C8—32no—
23C2×#272C2×C411yescharacteristic
24C2×(C3⋊D3)⋊C3C2×C412yes—
25C4×H(3,𝔽3)C2311yescharacteristic
26#1084C2×C411yescharacteristic
27(C3⋊Dic3)⋊C3C2×C412yes—
28#907—92no—
29#901—91no—
30C4×C3⋊D3C2×D312yes—
31C3×C4×D3—32no—
32C3×C4×D3—32no—
33C3×C4×D3—61no—
34#627—32no—
35#626—32no—
36#626—32no—
37#625—32no—
38C32×C8—21no—
39C32×C8C2×D312yes—
40C3⋊(C3⋊C8)C2×D312yes—
41C3×C3⋊C8—32no—
42C3×C3⋊C8—32no—
43C3×C3⋊C8—61no—
44C2×H(3,𝔽3)C22×C411yescharacteristic
45#272—21no—
46(C3⋊D3)⋊C3—22no—
47#235—91no—
48#235—92no—
49C3×M16—92no—
50C2×C3×C8—91no—
51#208—91no—
52C2×C3⋊C8—92no—
53ΓL(1,𝔽25)—32no—
54ΓL(1,𝔽25)—92no—
55ΓL(1,𝔽25)—92no—
56D3×C8—32no—
57D3×C8—61no—
58D3×C8—92no—
59D3×C8—92no—
60C2×C3⋊D3C4×D312yes—
61C2×C3×D3—32no—
62C2×C3×D3—32no—
63C2×C3×D3—61no—
64C32×C4—21no—
65C32×C4C22×D312yes—
66C3⋊Dic3C4×D312yes—
67C3×Dic3—32no—
68C3×Dic3—32no—
69C3×Dic3—61no—
70#131—271noSylow
71H(3,𝔽3)#13111yescharacteristic, complemented, Sylow
72C2×C3×C4—92no—
73C2×C3×C4—91no—
74#67—92no—
75#67—92no—
76#67—92no—
77D12—91no—
78D12—92no—
79C4×D3—32no—
80C4×D3—32no—
81C4×D3—61no—
82C4×D3—92no—
83Dic6—91no—
84Dic6—92no—
85C3×C8—32no—
86C3×C8—61no—
87C3×C8—92no—
88C3×C8—91no—
89C3×C8D3211yescharacteristic
90C3⋊C8—32no—
91C3⋊C8—32no—
92C3⋊C8—61no—
93C3⋊C8—92no—
94C2×C32—21no—
95C2×C32C2×C4×D312yes—
96C3⋊D3—22no—
97C3×D3—62no—
98C3×D3—62no—
99C3×D3—62no—
100#41—271no—
101M16—92no—
102M16—271no—
103C2×C8—91no—
104C2×C8—272no—
105C22×C3—92no—
106C22×C3—91no—
107C2×D3—32no—
108C2×D3—32no—
109C2×D3—61no—
110C2×D3—92no—
111C3×C4—32no—
112C3×C4—61no—
113C3×C4—92no—
114C3×C4—91no—
115C3×C4C2×D3211yescharacteristic
116Dic3—32no—
117Dic3—32no—
118Dic3—61no—
119Dic3—92no—
120C32—21no—
121C32#90712yescomplemented
122Q8—271no—
123D4—271no—
124D4—272no—
125C2×C4—91no—
126C2×C4—92no—
127C8—91no—
128C8—92no—
129C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
130C2×C3—32no—
131C2×C3—61no—
132C2×C3—182no—
133C2×C3—181no—
134C2×C3C4×D3211yescharacteristic
135D3—62no—
136D3—62no—
137D3—62no—
138D3—182no—
139C22—91no—
140C22—92no—
141C4—91no—
142C4—92no—
143C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
144C3—32no—
145C3—61no—
146C3#6380511yescharacteristic
147C2—181no—
148C2—182no—
149C2C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC4C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
FrattiniC3×C4C2×D32
FittingC8×H(3,𝔽3)C22
socleC2×C3C4×D32
prime radical#88650C2
cosocleC4×H(3,𝔽3)C23
hypercenterC8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#131
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
2. C8×#272C2semidirect
3. #3743⋊C3C2semidirect
4. #88435C2semidirect
5. ((C3⋊D3)⋊C3)⋊C8C2semidirect
16. #3743D3semidirect
71. H(3,𝔽3)#131semidirect
121. C32#907semidirect

Maximal subgroups

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

subgroupindex
#886502
C8×#2722
#3743⋊C32
#884352
((C3⋊D3)⋊C3)⋊C82
#638043