Finite Group Databasebeta

← #13144521

Subgroups — #13144521

Order 1728.

605 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 109 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)×#125C211yesmaximal, characteristic, complemented
2#11721720C211yesmaximal, characteristic
3#11720531C214yesmaximal, complemented
4#11720520C211yesmaximal, characteristic, complemented
5C3×#5067—34nomaximal
6C2×C4⋊#272—22no—
7C2×C4⋊#272C2212yes—
8C4⋊C4×H(3,𝔽3)—22no—
9C4⋊C4×H(3,𝔽3)C2212yes—
10C42×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2212yes—
12C32×#125D314yescomplemented
13C3×#874—34no—
14C3×#809—316no—
15C3×#803—34no—
16C22×#272—22no—
17C2×C4×H(3,𝔽3)—22no—
18C2×C4×H(3,𝔽3)D412yes—
19C2×C4×H(3,𝔽3)C2311yescharacteristic
20C4⋊#272—24no—
21C4⋊#272—44no—
22#6180—22no—
23C3×#490—91no—
24#5067—34no—
25C2×C3×D12—38no—
26C2×C3×D12—68no—
27C32×C4⋊C4—28no—
28C32×C4⋊C4C2×D318yes—
29C32×C42C2×D314yes—
30C2×C3×C3⋊C8—38no—
31C22×H(3,𝔽3)C2×D411yescharacteristic
32C2×#272—42no—
33C2×#272—44no—
34C4×H(3,𝔽3)—22no—
35C4×H(3,𝔽3)—42no—
36C4×H(3,𝔽3)#4112yes—
37C3×#127—91no—
38C3×#125—34no—
39C3×#125C3⋊D311yescharacteristic
40#874—34no—
41C3×#102—94no—
42C3×#97—91no—
43#809—316no—
44#803—34no—
45C22×C3×D3—68no—
46C2×C32×C4—28no—
47C2×C32×C4#6718yes—
48C2×C32×C4C22×D314yes—
49C3×D12—616no—
50C3×D12—1216no—
51C3×C3⋊C8—68no—
52#490—91noSylow
53C2×H(3,𝔽3)#12411yescharacteristic
54C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
55#272—82no—
56C2×C3×D4—92no—
57C2×C3×D4—182no—
58C2×D12—38no—
59C2×D12—68no—
60C2×C3×C8—92no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—38no—
63C3×C4⋊C4—68no—
64C3×C4⋊C4C2×C3⋊D312yes—
65C3×C42—34no—
66C3×C42C2×C3⋊D311yescharacteristic
67C2×C3⋊C8—38no—
68C22×C32C2×#6714yes—
69C2×C3×D3—128no—
70C2×C3×D3—1216no—
71C32×C4—28no—
72C32×C4—48no—
73C32×C4#23918yes—
74#127—91no—
75#125#27211yescharacteristic, complemented
76#102—94no—
77#97—91no—
78H(3,𝔽3)#49011yescharacteristic, complemented, Sylow
79C23×C3—182no—
80C22×D3—68no—
81C3×D4—184no—
82C3×D4—364no—
83C2×C3×C4—22no—
84C2×C3×C4—38no—
85C2×C3×C4—34no—
86C2×C3×C4—68no—
87C2×C3×C4#63812yes—
88C2×C3×C4C22×C3⋊D311yescharacteristic
89D12—616no—
90D12—1216no—
91C3×C8—182no—
92C3⋊C8—68no—
93C2×C32#94714yes—
94C2×C32#94918yes—
95C3×D3—248no—
96C2×D4—92no—
97C2×D4—182no—
98C2×C8—92no—
99C4⋊C4—22no—
100C4⋊C4C2×#27212yes—
101C42C2×#27211yescharacteristic
102C22×C3—34no—
103C22×C3—362no—
104C22×C3—364no—
105C22×C3C2×#63811yescharacteristic
106C2×D3—128no—
107C2×D3—1216no—
108C3×C4—22no—
109C3×C4—38no—
110C3×C4—42no—
111C3×C4—68no—
112C3×C4—128no—
113C3×C4#383212yes—
114C32#506714yescomplemented
115C23—182no—
116D4—184no—
117D4—364no—
118C2×C4—22no—
119C2×C4#623512yes—
120C2×C4C22×#27211yescharacteristic
121C8—182no—
122C2×C3—34no—
123C2×C3—38no—
124C2×C3—722no—
125C2×C3#6415611yescharacteristic
126C2×C3#6415812yes—
127D3—248no—
128C22—362no—
129C22—364no—
130C22C2×#623511yescharacteristic
131C4—22no—
132C4—42no—
133C4#8874812yes—
134C3—34no—
135C3#1059276611yescharacteristic
136C2—722no—
137C2#1172186411yescharacteristic
138C2#1172187012yes—
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#125C2
prime radical#11721720C2
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#490
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
3. #11720531C2semidirect
4. #11720520C2semidirect
12. C32×#125D3semidirect
75. #125#272semidirect
78. H(3,𝔽3)#490semidirect
114. C32#5067semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217202
#117205312
#117205202
C3×#50673