Finite Group Databasebeta

← #11720319

Subgroups — #11720319

Order 864.

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

normal subgroups only

This group has 125 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#88538C211yesmaximal, characteristic
2#3748⋊C3C211yesmaximal, characteristic
3C2×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#63578—31nomaximal
5#63577—31nomaximal
6C2×C4×H(3,𝔽3)C2211yescharacteristic
7#6180—21no—
8(C3⋊(C3⋊C8))⋊C3—21no—
9(C3⋊(C3⋊C8))⋊C3C412yes—
10#3748D311yescharacteristic
11C2×C3⋊(C3⋊C8)D311yescharacteristic
12C3×#208—31no—
13C3×#208—31no—
14C3×#208—31no—
15C3×#208—61no—
16C2×C3×C3⋊C8—31no—
17C22×H(3,𝔽3)Q811yescharacteristic
18C4×H(3,𝔽3)C2×C411yescharacteristic
19C4×H(3,𝔽3)D411yescharacteristic
20#822—91no—
21#822—91no—
22#819—91no—
23C2×C32×C4—21no—
24C2×C32×C4C2×D311yescharacteristic
25C2×C32×C4C2×D311yescharacteristic
26C3⋊(C3⋊C8)—21no—
27C3⋊(C3⋊C8)Dic312yes—
28C3×C3⋊C8—32no—
29C3×C3⋊C8—62no—
30C3×C3⋊C8—61no—
31C3×C3⋊C8—61no—
32C3×C3⋊C8—61no—
33C2×H(3,𝔽3)—21no—
34C2×H(3,𝔽3)C4⋊C411yescharacteristic
35C3×M16—91no—
36C3×M16—91no—
37C2×C3×C8—91no—
38#208—31no—
39#208—31no—
40#208—31no—
41#208—61no—
42#208—181no—
43C2×C3⋊C8—31no—
44C2×C3⋊C8—181no—
45C22×C32—21no—
46C22×C32Dic611yescharacteristic
47C22×C32Dic611yescharacteristic
48C32×C4—21no—
49C32×C4—21no—
50C32×C4C4×D311yescharacteristic
51C32×C4D1211yescharacteristic
52C32×C4C2×Dic311yescharacteristic
53C32×C4#6711yescharacteristic
54#108—271noSylow
55H(3,𝔽3)#10811yescharacteristic, complemented, Sylow
56C2×C3×C4—61no—
57C2×C3×C4—61no—
58C2×C3×C4—121no—
59C2×C3×C4D3211yescharacteristic
60C3×C8—92no—
61C3×C8—181no—
62C3×C8—181no—
63C3⋊C8—32no—
64C3⋊C8—61no—
65C3⋊C8—61no—
66C3⋊C8—62no—
67C3⋊C8—61no—
68C3⋊C8—182no—
69C3⋊C8—181no—
70C2×C32—21no—
71C2×C32—21no—
72C2×C32—21no—
73C2×C32—41no—
74C2×C32Dic3⋊C411yescharacteristic
75C2×C32C4⋊Dic311yescharacteristic
76M16—91no—
77M16—91no—
78C2×C8—91no—
79C22×C3—61no—
80C22×C3—61no—
81C22×C3—121no—
82C22×C3#62711yescharacteristic
83C3×C4—61no—
84C3×C4—61no—
85C3×C4—61no—
86C3×C4—61no—
87C3×C4—121no—
88C3×C4—121no—
89C3×C4D3×Dic311yescharacteristic
90C3×C4#62611yescharacteristic
91C32—21no—
92C32#81911yescharacteristic, complemented
93C32#82211yescharacteristic, complemented
94C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
95C8—92no—
96C8—181no—
97C8—181no—
98C2×C3—21no—
99C2×C3—61no—
100C2×C3—61no—
101C2×C3—61no—
102C2×C3—121no—
103C2×C3—121no—
104C2×C3—121no—
105C2×C3Dic3⋊Dic311yescharacteristic
106C22#619611yescharacteristic
107C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
108C4#620011yescharacteristic
109C3—61no—
110C3—61no—
111C3—121no—
112C3#6357711yescharacteristic
113C2—21no—
114C2#8844811yescharacteristic
subgroupG/H
centerC4#6200
commutatorC4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC2×C3Dic3⋊Dic3
prime radicalC22×H(3,𝔽3)Q8
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#108
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
55. H(3,𝔽3)#108semidirect
92. C32#819semidirect
93. C32#822semidirect

Maximal subgroups

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

subgroupindex
#885382
#3748⋊C32
C2×(C3⋊(C3⋊C8))⋊C32
#635783
#635773