Finite Group Databasebeta

← #13135907

Subgroups — #13135907

Order 1728.

374 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 73 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
1H(3,𝔽3)×#123C211yesmaximal, characteristic
2#11720580C211yesmaximal, characteristic
3#11720579C211yesmaximal, characteristic
4C3×#4539—34nomaximal
5C2×D4×H(3,𝔽3)C2211yescharacteristic
6C22×C4×H(3,𝔽3)C411yescharacteristic, complemented
7C4⋊C4×H(3,𝔽3)—21no—
8C22⋊C4×H(3,𝔽3)—21no—
9C22⋊C4×H(3,𝔽3)C411yescharacteristic, complemented
10#88552—21no—
11#88538—21no—
12C32×#123D314yes—
13C3×#834—34no—
14C3×#833—34no—
15C23×H(3,𝔽3)C2×C411yescharacteristic
16C23×H(3,𝔽3)D411yescharacteristic
17D4×H(3,𝔽3)—41no—
18C2×C4×H(3,𝔽3)—21no—
19C2×C4×H(3,𝔽3)—21no—
20C2×C4×H(3,𝔽3)—41no—
21C2×C4×H(3,𝔽3)D411yescharacteristic
22C2×#1084—41no—
23#6180—41no—
24C3×#352—91no—
25#4539—34no—
26C2×C32×D4C2×D314yes—
27C22×C32×C4Dic314yescomplemented
28C32×C4⋊C4—24no—
29C32×C22⋊C4—24no—
30C32×C22⋊C4Dic314yescomplemented
31C3×#217—64no—
32C3×#208—64no—
33C22×H(3,𝔽3)—21no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)—41no—
36C22×H(3,𝔽3)—41no—
37C22×H(3,𝔽3)C22⋊C411yescharacteristic
38C4×H(3,𝔽3)—21no—
39C4×H(3,𝔽3)—41no—
40C4×H(3,𝔽3)—41no—
41#1084—81no—
42C3×#123—34no—
43C3×#123C3⋊D311yescharacteristic
44C3×#100—91no—
45C3×#99—91no—
46#834—34no—
47#833—34no—
48C23×C32C2×Dic314yes—
49C23×C32#6714yes—
50C32×D4—44no—
51C2×C32×C4—24no—
52C2×C32×C4—24no—
53C2×C32×C4—44no—
54C2×C32×C4#6714yes—
55C2×C3×Dic3—124no—
56C3×C3⋊C8—124no—
57#352—91noSylow
58C2×H(3,𝔽3)—21no—
59C2×H(3,𝔽3)—41no—
60C2×H(3,𝔽3)—41no—
61C2×H(3,𝔽3)#9911yescharacteristic
62C2×C3×D4—34no—
63C2×C3×D4C2×C3⋊D311yescharacteristic
64C22×C3×C4—34no—
65C22×C3×C4C3⋊Dic311yescharacteristic
66C3×M16—181no—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—64no—
69C3×C22⋊C4—21no—
70C3×C22⋊C4—34no—
71C3×C22⋊C4—64no—
72C3×C22⋊C4—181no—
73C3×C22⋊C4C3⋊Dic311yescharacteristic
74#217—64no—
75#208—64no—
76C22×C32—24no—
77C22×C32—24no—
78C22×C32—44no—
79C22×C32—44no—
80C22×C32#21714yes—
81C32×C4—24no—
82C32×C4—44no—
83C32×C4—44no—
84C3×Dic3—244no—
85#123#27211yescharacteristic
86#100—91no—
87#99—91no—
88H(3,𝔽3)#35211yescharacteristic, complemented, Sylow
89C23×C3—34no—
90C23×C3—34no—
91C23×C3C2×C3⋊Dic311yescharacteristic
92C23×C3#63811yescharacteristic
93C3×D4—41no—
94C3×D4—124no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—34no—
98C2×C3×C4—41no—
99C2×C3×C4—64no—
100C2×C3×C4—64no—
101C2×C3×C4—124no—
102C2×C3×C4—361no—
103C2×C3×C4#63811yescharacteristic
104C2×Dic3—124no—
105C3×C8—361no—
106C3⋊C8—124no—
107C2×C32—24no—
108C2×C32—44no—
109C2×C32—44no—
110C2×C32#83414yes—
111C2×D4C2×#27211yescharacteristic
112C22×C4#108411yescharacteristic, complemented
113M16—181no—
114C4⋊C4—21no—
115C22⋊C4—21no—
116C22⋊C4—181no—
117C22⋊C4#108411yescharacteristic, complemented
118C22×C3—21no—
119C22×C3—21no—
120C22×C3—34no—
121C22×C3—41no—
122C22×C3—41no—
123C22×C3—64no—
124C22×C3—64no—
125C22×C3—124no—
126C22×C3—124no—
127C22×C3#375711yescharacteristic
128C3×C4—21no—
129C3×C4—41no—
130C3×C4—41no—
131C3×C4—64no—
132C3×C4—124no—
133C3×C4—124no—
134C3×C4—721no—
135Dic3—244no—
136C32#453914yescomplemented
137C23C2×#108411yescharacteristic
138C23#623511yescharacteristic
139D4—41no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—361no—
144C2×C4#623511yescharacteristic
145C8—361no—
146C2×C3—21no—
147C2×C3—34no—
148C2×C3—41no—
149C2×C3—41no—
150C2×C3—64no—
151C2×C3—124no—
152C2×C3—124no—
153C2×C3#6366111yescharacteristic
154C22—21no—
155C22—21no—
156C22—41no—
157C22—41no—
158C22#8855211yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—721no—
163C3—34no—
164C3#1058940511yescharacteristic
165C2—21no—
166C2—41no—
167C2—41no—
168C2#1172058011yescharacteristic
subgroupG/H
center, socleC2×C3#63661
commutatorC23×H(3,𝔽3)C2×C4
FrattiniC2×C3×D4C2×C3⋊D3
FittingH(3,𝔽3)×#123C2
prime radical, cosocleC2×D4×H(3,𝔽3)C22
hypercenterC3×#123C3⋊D3
hypocenterH(3,𝔽3)#352
LCSG ▷ C23×H(3,𝔽3) ▷ C22×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#123

Complements

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

NH ≅ G/Ntype
6. C22×C4×H(3,𝔽3)C4semidirect
9. C22⋊C4×H(3,𝔽3)C4semidirect
27. C22×C32×C4Dic3semidirect
30. C32×C22⋊C4Dic3semidirect
88. H(3,𝔽3)#352semidirect
112. C22×C4#1084semidirect
117. C22⋊C4#1084semidirect
136. C32#4539semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1232
#117205802
#117205792
C3×#45393