Finite Group Databasebeta

← #11720313

Subgroups — #11720313

Order 864.

169 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 50 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
2C2×#6180C211yesmaximal, characteristic
3C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
4#63567—31nomaximal
5#63566—31nomaximal
6C2×C4×H(3,𝔽3)C2211yescharacteristic
7C2×(C3⋊Dic3)⋊C3—21no—
8C2×(C3⋊Dic3)⋊C3—21no—
9#634⋊C3—21no—
10#634⋊C3C412yescomplemented
11#6180—21no—
12C2×#634D311yescharacteristic
13C2×C3×Dic6—31no—
14C4⋊(C3⋊Dic3)D311yescharacteristic
15C3×C4⋊Dic3—31no—
16C2×C3×C3⋊C8—31no—
17C2×C3×C3⋊C8—31no—
18C2×C3×C3⋊C8—61no—
19C22×H(3,𝔽3)D411yescharacteristic
20C4×H(3,𝔽3)C2×C411yescharacteristic
21C4×H(3,𝔽3)D411yescharacteristic
22(C3⋊Dic3)⋊C3—22no—
23(C3⋊Dic3)⋊C3—41no—
24#835—91no—
25#816—91no—
26#810—91no—
27C2×C32×C4—21no—
28C2×C32×C4C2×D311yescharacteristic
29C2×C32×C4C2×D311yescharacteristic
30C2×C3⋊Dic3—21no—
31C2×C3⋊Dic3—21no—
32#634—21no—
33#634Dic312yescomplemented
34C2×C3×Dic3—61no—
35C2×C3×Dic3—61no—
36C3×Dic6—32no—
37C3×Dic6—61no—
38C3×C3⋊C8—62no—
39C3×C3⋊C8—61no—
40C3×C3⋊C8—61no—
41C2×H(3,𝔽3)C22⋊C411yescharacteristic
42C2×H(3,𝔽3)SD1611yescharacteristic
43C2×H(3,𝔽3)Q1611yescharacteristic
44C2×C3×Q8—91no—
45C2×Dic6—31no—
46C2×Dic6—91no—
47C2×C3×C8—91no—
48C3×C4⋊C4—91no—
49C4⋊Dic3—31no—
50C4⋊Dic3—181no—
51C2×C3⋊C8—31no—
52C2×C3⋊C8—31no—
53C2×C3⋊C8—61no—
54C22×C32—21no—
55C22×C32#6711yescharacteristic
56C22×C32#6711yescharacteristic
57C32×C4—21no—
58C32×C4—21no—
59C32×C4C4×D311yescharacteristic
60C32×C4D1211yescharacteristic
61C32×C4C2×Dic311yescharacteristic
62C32×C4#6711yescharacteristic
63C3⋊Dic3—22no—
64C3⋊Dic3—41no—
65C3×Dic3—62no—
66C3×Dic3—121no—
67#103—271noSylow
68H(3,𝔽3)#10311yescharacteristic, complemented, Sylow
69C3×Q8—92no—
70C3×Q8—181no—
71C2×C3×C4—31no—
72C2×C3×C4—61no—
73C2×C3×C4—121no—
74C2×C3×C4—181no—
75C2×C3×C4—181no—
76C2×C3×C4D3211yescharacteristic
77C2×Dic3—61no—
78C2×Dic3—61no—
79C2×Dic3—181no—
80C2×Dic3—181no—
81Dic6—32no—
82Dic6—61no—
83Dic6—92no—
84Dic6—181no—
85C3×C8—181no—
86C3⋊C8—61no—
87C3⋊C8—61no—
88C3⋊C8—62no—
89C2×C32—21no—
90C2×C32—21no—
91C2×C32—21no—
92C2×C32(C2×D3)⋊C411yescharacteristic
93C2×C32#21411yescharacteristic
94C2×C32#21511yescharacteristic
95C2×C32#21611yescharacteristic
96C2×C32#21611yescharacteristic
97C2×C32#21711yescharacteristic
98C2×Q8—91no—
99C2×C8—91no—
100C4⋊C4—91no—
101C22×C3—31no—
102C22×C3—61no—
103C22×C3—121no—
104C22×C3#62511yescharacteristic
105C3×C4—31no—
106C3×C4—31no—
107C3×C4—61no—
108C3×C4—61no—
109C3×C4—121no—
110C3×C4—121no—
111C3×C4—182no—
112C3×C4—361no—
113C3×C4D3×Dic311yescharacteristic
114C3×C4#62611yescharacteristic
115Dic3—62no—
116Dic3—121no—
117Dic3—182no—
118Dic3—361no—
119C32—21no—
120C32#81011yescharacteristic, complemented
121C32#83511yescharacteristic, complemented
122Q8—92no—
123Q8—181no—
124C2×C4—181no—
125C2×C4—181no—
126C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
127C8—181no—
128C2×C3—31no—
129C2×C3—31no—
130C2×C3—31no—
131C2×C3—61no—
132C2×C3—61no—
133C2×C3—61no—
134C2×C3—121no—
135C2×C3—121no—
136C2×C3—121no—
137C2×C3#371511yescharacteristic
138C2×C3#371811yescharacteristic
139C2×C3(C2×D3)⋊Dic311yescharacteristic
140C22#619811yescharacteristic
141C4—182no—
142C4—361no—
143C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
144C4#620011yescharacteristic
145C3—31no—
146C3—61no—
147C3—121no—
148C3#6356511yescharacteristic
149C2#8843111yescharacteristic
150C2#8843311yescharacteristic
151C2#8845011yescharacteristic
subgroupG/H
centerC22#6198
commutatorC4×H(3,𝔽3)C2×C4
FrattiniC2×C3×C4D32
Fitting, cosocleC2×C4×H(3,𝔽3)C22
socleC22×C3#625
prime radicalC22×H(3,𝔽3)D4
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#103
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
10. #634⋊C3C4semidirect
33. #634Dic3semidirect
68. H(3,𝔽3)#103semidirect
120. C32#810semidirect
121. C32#835semidirect

Maximal subgroups

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

subgroupindex
C2×#634⋊C32
C2×#61802
C4⋊((C3⋊Dic3)⋊C3)2
#635673
#635663