Finite Group Databasebeta

← #13144522

Subgroups — #13144522

Order 1728.

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

normal subgroups only

This group has 528 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#11721721C211yesmaximal, characteristic
3#11720532C212yesmaximal, complemented
4#11720531C212yesmaximal
5C4×#6180C211yesmaximal, characteristic, complemented
6C3×#5068—34nomaximal
7C2×C4⋊#272C2211yescharacteristic
8C2×#6229C2211yescharacteristic
9C4⋊C4×H(3,𝔽3)—22no—
10C4⋊C4×H(3,𝔽3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12#88542—22no—
13C2×#6180C2212yes—
14C32×#125D314yescomplemented
15C3×#875—34no—
16C3×#810—38no—
17C3×#809—38no—
18C3×C4×C3⋊C8—34no—
19C22×#272—21no—
20C2×C4×H(3,𝔽3)—22no—
21C2×C4×H(3,𝔽3)D412yes—
22C2×C4×H(3,𝔽3)C2311yescharacteristic
23C2×#1084—21no—
24C4⋊#272—22no—
25#6229—22no—
26#6180—22no—
27C3×#488—91no—
28#5068—34no—
29C2×C3×D12—34no—
30C2×C3×Dic6—34no—
31C32×C4⋊C4—28no—
32C32×C4⋊C4C2×D318yes—
33C32×C42C2×D314yes—
34C3×(C2×D3)⋊C4—68no—
35C2×C3×C3⋊C8—38no—
36C22×H(3,𝔽3)C2×D411yescharacteristic
37C2×#272—41no—
38C2×#272—42no—
39C4×H(3,𝔽3)—22no—
40C4×H(3,𝔽3)—42no—
41C4×H(3,𝔽3)#4112yes—
42#1084—41no—
43C3×#125—34no—
44C3×#125C3⋊D311yescharacteristic
45C3×#124—91no—
46#875—34no—
47C3×#103—92no—
48C3×#102—92no—
49C3×C4×C8—91no—
50#810—38no—
51#809—38no—
52C4×C3⋊C8—34no—
53C22×C3×D3—64no—
54C2×C32×C4—28no—
55C2×C32×C4#6718yes—
56C2×C32×C4C22×D314yes—
57C2×C3×Dic3—64no—
58C3×D12—68no—
59C3×Dic6—68no—
60C3×C3⋊C8—68no—
61#488—91noSylow
62C2×H(3,𝔽3)#12411yescharacteristic
63C2×H(3,𝔽3)#13512yes—
64#272—81no—
65C2×C3×Q8—91no—
66C2×C3×D4—91no—
67C2×D12—34no—
68C2×Dic6—34no—
69C2×C3×C8—92no—
70C3×C4⋊C4—22no—
71C3×C4⋊C4—38no—
72C3×C4⋊C4—68no—
73C3×C4⋊C4C2×C3⋊D312yes—
74C3×C22⋊C4—182no—
75C3×C42—34no—
76C3×C42C2×C3⋊D311yescharacteristic
77(C2×D3)⋊C4—68no—
78C2×C3⋊C8—38no—
79C22×C32C2×#6714yes—
80C2×C3×D3—124no—
81C2×C3×D3—128no—
82C32×C4—28no—
83C32×C4—48no—
84C32×C4#23918yes—
85C3×Dic3—124no—
86#125#27211yescharacteristic, complemented
87#124—91no—
88#103—92no—
89#102—92no—
90C4×C8—91no—
91H(3,𝔽3)#48811yescharacteristic, complemented, Sylow
92C23×C3—181no—
93C22×D3—64no—
94C3×Q8—182no—
95C3×D4—182no—
96C2×C3×C4—22no—
97C2×C3×C4—38no—
98C2×C3×C4—34no—
99C2×C3×C4—68no—
100C2×C3×C4—181no—
101C2×C3×C4#63812yes—
102C2×C3×C4C22×C3⋊D311yescharacteristic
103C2×Dic3—64no—
104D12—68no—
105Dic6—68no—
106C3×C8—182no—
107C3⋊C8—68no—
108C2×C32#94714yes—
109C2×C32#95018yes—
110C3×D3—244no—
111C2×Q8—91no—
112C2×D4—91no—
113C2×C8—92no—
114C4⋊C4—22no—
115C4⋊C4C2×#27212yes—
116C22⋊C4—182no—
117C42C2×#27211yescharacteristic
118C22×C3—34no—
119C22×C3—361no—
120C22×C3—362no—
121C22×C3C2×#63811yescharacteristic
122C2×D3—124no—
123C2×D3—128no—
124C3×C4—22no—
125C3×C4—38no—
126C3×C4—42no—
127C3×C4—68no—
128C3×C4—128no—
129C3×C4—361no—
130C3×C4#383212yes—
131Dic3—124no—
132C32#506814yescomplemented
133C23—181no—
134Q8—182no—
135D4—182no—
136C2×C4—22no—
137C2×C4—181no—
138C2×C4#623512yes—
139C2×C4C22×#27211yescharacteristic
140C8—182no—
141C2×C3—34no—
142C2×C3—38no—
143C2×C3—721no—
144C2×C3#6415611yescharacteristic
145C2×C3#6415912yes—
146D3—244no—
147C22—361no—
148C22—362no—
149C22C2×#623511yescharacteristic
150C4—22no—
151C4—42no—
152C4—361no—
153C4#8874812yes—
154C3—34no—
155C3#1059276711yescharacteristic
156C2—721no—
157C2#1172186411yescharacteristic
158C2#1172187112yes—
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊#272C22
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#488
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. #11720532C2semidirect
5. C4×#6180C2semidirect
14. C32×#125D3semidirect
86. #125#272semidirect
91. H(3,𝔽3)#488semidirect
132. C32#5068semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217212
#117205322
#117205312
C4×#61802
C3×#50683