Finite Group Databasebeta

← #13144523

Subgroups — #13144523

Order 1728.

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

normal subgroups only

This group has 495 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
2#11721715C211yesmaximal, characteristic
3#11720532C214yesmaximal
4#11720520C211yesmaximal, characteristic
5C3×#5069—34nomaximal
6C2×#6229C2212yes—
7C4⋊C4×H(3,𝔽3)—22no—
8C4⋊C4×H(3,𝔽3)C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084—22no—
11C2×#6180C2212yes—
12C32×#125D314yes—
13C3×#869—34no—
14C3×#810—316no—
15C3×#803—34no—
16C2×C4×H(3,𝔽3)—22no—
17C2×C4×H(3,𝔽3)D412yes—
18C2×C4×H(3,𝔽3)C2311yescharacteristic
19C2×#1084—22no—
20#6229—24no—
21#6180—22no—
22C3×#491—91no—
23#5069—34no—
24C2×C3×Dic6—38no—
25C32×C4⋊C4—28no—
26C32×C4⋊C4C2×D318yes—
27C32×C42C2×D314yes—
28C3×C4⋊Dic3—68no—
29C2×C3×C3⋊C8—38no—
30C22×H(3,𝔽3)C2×D411yescharacteristic
31C4×H(3,𝔽3)—22no—
32C4×H(3,𝔽3)—42no—
33C4×H(3,𝔽3)#4112yes—
34#1084—42no—
35C3×C4⋊Q8—91no—
36C3×#125—34no—
37C3×#125C3⋊D311yescharacteristic
38#869—34no—
39C3×#103—94no—
40C3×#97—91no—
41#810—316no—
42#803—34no—
43C2×C32×C4—28no—
44C2×C32×C4#6718yes—
45C2×C32×C4C22×D314yes—
46C2×C3×Dic3—68no—
47C3×Dic6—616no—
48C3×C3⋊C8—68no—
49#491—91noSylow
50C2×H(3,𝔽3)#12411yescharacteristic
51C2×H(3,𝔽3)#13712yes—
52C2×C3×Q8—92no—
53C2×Dic6—38no—
54C2×C3×C8—92no—
55C3×C4⋊C4—22no—
56C3×C4⋊C4—38no—
57C3×C4⋊C4—68no—
58C3×C4⋊C4—182no—
59C3×C4⋊C4C2×C3⋊D312yes—
60C3×C42—34no—
61C3×C42C2×C3⋊D311yescharacteristic
62C4⋊Dic3—68no—
63C2×C3⋊C8—38no—
64C22×C32C2×#6714yes—
65C32×C4—28no—
66C32×C4—48no—
67C32×C4#23918yes—
68C3×Dic3—128no—
69C4⋊Q8—91no—
70#125#27211yescharacteristic
71#103—94no—
72#97—91no—
73H(3,𝔽3)#49111yescharacteristic, complemented, Sylow
74C3×Q8—184no—
75C2×C3×C4—22no—
76C2×C3×C4—38no—
77C2×C3×C4—34no—
78C2×C3×C4—68no—
79C2×C3×C4—182no—
80C2×C3×C4#63812yes—
81C2×C3×C4C22×C3⋊D311yescharacteristic
82C2×Dic3—68no—
83Dic6—616no—
84C3×C8—182no—
85C3⋊C8—68no—
86C2×C32#94714yes—
87C2×C32#95118yes—
88C2×Q8—92no—
89C2×C8—92no—
90C4⋊C4—22no—
91C4⋊C4—182no—
92C4⋊C4C2×#27212yes—
93C42C2×#27211yescharacteristic
94C22×C3—34no—
95C22×C3C2×#63811yescharacteristic
96C3×C4—22no—
97C3×C4—38no—
98C3×C4—42no—
99C3×C4—68no—
100C3×C4—128no—
101C3×C4—362no—
102C3×C4#383212yes—
103Dic3—128no—
104C32#506914yescomplemented
105Q8—184no—
106C2×C4—22no—
107C2×C4—182no—
108C2×C4#623512yes—
109C2×C4C22×#27211yescharacteristic
110C8—182no—
111C2×C3—34no—
112C2×C3—38no—
113C2×C3#6415611yescharacteristic
114C2×C3#6416012yes—
115C22C2×#623511yescharacteristic
116C4—22no—
117C4—42no—
118C4—362no—
119C4#8874812yes—
120C3—34no—
121C3#1059276811yescharacteristic
122C2#1172186411yescharacteristic
123C2#1172187212yes—
subgroupG/H
center, socleC22×C3C2×#638
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C22×C3⋊D3
FittingH(3,𝔽3)×#125C2
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC3×#125C3⋊D3
hypocenterH(3,𝔽3)#491
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
73. H(3,𝔽3)#491semidirect
104. C32#5069semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217152
#117205322
#117205202
C3×#50693