Finite Group Databasebeta

← #11720321

Subgroups — #11720321

Order 864.

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

normal subgroups only

This group has 153 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
1C2×#634⋊C3C211yesmaximal, characteristic
2#88538C211yesmaximal, characteristic
3#3748⋊C3C211yesmaximal, characteristic
4#63561—31nomaximal
5#63560—31nomaximal
6C2×C4×H(3,𝔽3)C2211yescharacteristic
7C2×(C3⋊Dic3)⋊C3C412yes—
8#634⋊C3—21no—
9#634⋊C3—21no—
10#6180—21no—
11(C3⋊(C3⋊C8))⋊C3—21no—
12C2×#634D311yescharacteristic
13C2×C3×Dic6—31no—
14#3748D311yescharacteristic
15C3×#208—31no—
16C3×#208—31no—
17C3×#208—31no—
18C3×#208—61no—
19C22×H(3,𝔽3)C2×C411yescharacteristic
20C4×H(3,𝔽3)D411yescharacteristic
21C4×H(3,𝔽3)D411yescharacteristic
22(C3⋊Dic3)⋊C3—22no—
23#836—91no—
24#824—91no—
25#824—91no—
26C2×C32×C4—21no—
27C2×C32×C4C2×D311yescharacteristic
28C2×C32×C4C2×D311yescharacteristic
29C2×C3⋊Dic3Dic312yes—
30#634—21no—
31#634—21no—
32C2×C3×Dic3—32no—
33C3×Dic6—61no—
34C3×Dic6—61no—
35C3⋊(C3⋊C8)—21no—
36C3×C3⋊C8—62no—
37C3×C3⋊C8—61no—
38C3×C3⋊C8—61no—
39C3×C3⋊C8—61no—
40C2×H(3,𝔽3)—21no—
41C2×H(3,𝔽3)C22⋊C411yescharacteristic
42C2×C3×Q8—91no—
43C2×Dic6—31no—
44C2×Dic6—181no—
45C3×M16—91no—
46C3×M16—91no—
47#208—31no—
48#208—31no—
49#208—31no—
50#208—61no—
51#208—181no—
52C22×C32—21no—
53C22×C32C4×D311yescharacteristic
54C22×C32C2×Dic311yescharacteristic
55C32×C4—21no—
56C32×C4—21no—
57C32×C4D1211yescharacteristic
58C32×C4#6711yescharacteristic
59C32×C4#6711yescharacteristic
60C32×C4#6711yescharacteristic
61C3⋊Dic3—22no—
62C3×Dic3—62no—
63#101—271noSylow
64H(3,𝔽3)#10111yescharacteristic, complemented, Sylow
65C3×Q8—181no—
66C3×Q8—181no—
67C2×C3×C4—61no—
68C2×C3×C4—61no—
69C2×C3×C4—92no—
70C2×C3×C4—121no—
71C2×C3×C4D3211yescharacteristic
72C2×Dic3—32no—
73C2×Dic3—182no—
74Dic6—61no—
75Dic6—61no—
76Dic6—181no—
77Dic6—181no—
78C3×C8—181no—
79C3×C8—181no—
80C3⋊C8—61no—
81C3⋊C8—61no—
82C3⋊C8—62no—
83C3⋊C8—61no—
84C3⋊C8—181no—
85C2×C32—21no—
86C2×C32—21no—
87C2×C32—21no—
88C2×C32—41no—
89C2×C32(C2×D3)⋊C411yescharacteristic
90C2×C32#21711yescharacteristic
91C2×Q8—91no—
92M16—91no—
93M16—91no—
94C22×C3—61no—
95C22×C3—61no—
96C22×C3—121no—
97C22×C3D3×Dic311yescharacteristic
98C3×C4—61no—
99C3×C4—61no—
100C3×C4—61no—
101C3×C4—61no—
102C3×C4—121no—
103C3×C4—121no—
104C3×C4—182no—
105C3×C4#62511yescharacteristic
106C3×C4#62611yescharacteristic
107Dic3—62no—
108Dic3—182no—
109C32—21no—
110C32#82411yescharacteristic, complemented
111C32#83611yescharacteristic, complemented
112Q8—181no—
113Q8—181no—
114C2×C4—92no—
115C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
116C8—181no—
117C8—181no—
118C2×C3—21no—
119C2×C3—61no—
120C2×C3—61no—
121C2×C3—61no—
122C2×C3—121no—
123C2×C3—121no—
124C2×C3—121no—
125C2×C3(C2×D3)⋊Dic311yescharacteristic
126C22((C3⋊D3)⋊C3)⋊C411yescharacteristic
127C4—182no—
128C4#619811yescharacteristic
129C4#620011yescharacteristic
130C3—61no—
131C3—61no—
132C3—121no—
133C3#6356011yescharacteristic
134C2—21no—
135C2#8845011yescharacteristic
subgroupG/H
centerC2#88450
commutator, prime radicalC22×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC2×C3(C2×D3)⋊Dic3
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#101
LCSG ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
64. H(3,𝔽3)#101semidirect
110. C32#824semidirect
111. C32#836semidirect

Maximal subgroups

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

subgroupindex
C2×#634⋊C32
#885382
#3748⋊C32
#635613
#635603