Finite Group Databasebeta

← #13144519

Subgroups — #13144519

Order 1728.

473 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 473 conjugacy classes of subgroups — too many to draw. Only its 113 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#11721716C211yesmaximal, characteristic
3#11720530C212yesmaximal
4#11720529C212yesmaximal
5C4×#6180C211yesmaximal, characteristic
6C3×#5065—34nomaximal
7C4⋊C4×H(3,𝔽3)—22no—
8C4⋊C4×H(3,𝔽3)C2212yes—
9C42×H(3,𝔽3)C2211yescharacteristic
10C4⋊#1084C2212yes—
11#88540—22no—
12C2×#6180C2211yescharacteristic
13C2×#6180C2211yescharacteristic
14C32×#125D314yes—
15C3×#870—34no—
16C3×#808—38no—
17C3×#807—38no—
18C3×C4×C3⋊C8—34no—
19C2×C4×H(3,𝔽3)—22no—
20C2×C4×H(3,𝔽3)D412yes—
21C2×C4×H(3,𝔽3)C2311yescharacteristic
22C2×#1084—22no—
23#6180Q812yes—
24#6180Q812yes—
25C3×#499—91no—
26#5065—34no—
27C32×C4⋊C4—28no—
28C32×C4⋊C4C2×D318yes—
29C32×C42C2×D314yes—
30C3×C4⋊Dic3—38no—
31C3×Dic3⋊C4—68no—
32C2×C3×C3⋊C8—34no—
33C2×C3×C3⋊C8—34no—
34C22×H(3,𝔽3)C2×D411yescharacteristic
35C4×H(3,𝔽3)—22no—
36C4×H(3,𝔽3)—42no—
37C4×H(3,𝔽3)C2×Q811yescharacteristic
38C4×H(3,𝔽3)C2×Q811yescharacteristic
39#1084—42no—
40C3×#125—34no—
41C3×#125—91no—
42C3×#125C3⋊D311yescharacteristic
43#870—34no—
44C3×#107—92no—
45C3×#106—92no—
46C3×C4×C8—91no—
47#808—38no—
48#807—38no—
49C4×C3⋊C8—34no—
50C2×C32×C4—28no—
51C2×C32×C4#6718yes—
52C2×C32×C4C22×D314yes—
53C2×C3×Dic3—68no—
54C3×C3⋊C8—38no—
55C3×C3⋊C8—38no—
56#499—91noSylow
57C2×H(3,𝔽3)C4⋊Q811yescharacteristic
58C2×H(3,𝔽3)#13512yes—
59C2×C3×C8—91no—
60C2×C3×C8—91no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—38no—
63C3×C4⋊C4—68no—
64C3×C4⋊C4—92no—
65C3×C4⋊C4—182no—
66C3×C4⋊C4C2×C3⋊D312yes—
67C3×C42—34no—
68C3×C42C2×C3⋊D311yescharacteristic
69C4⋊Dic3—38no—
70Dic3⋊C4—68no—
71C2×C3⋊C8—34no—
72C2×C3⋊C8—34no—
73C22×C32C2×#6714yes—
74C32×C4—28no—
75C32×C4—48no—
76C32×C4D3×Q814yes—
77C32×C4D3×Q814yes—
78C3×Dic3—128no—
79#125—91no—
80#125#27211yescharacteristic
81#107—92no—
82#106—92no—
83C4×C8—91no—
84H(3,𝔽3)#49911yescharacteristic, complemented, Sylow
85C2×C3×C4—22no—
86C2×C3×C4—38no—
87C2×C3×C4—34no—
88C2×C3×C4—68no—
89C2×C3×C4—182no—
90C2×C3×C4#63812yes—
91C2×C3×C4C22×C3⋊D311yescharacteristic
92C2×Dic3—68no—
93C3×C8—92no—
94C3×C8—92no—
95C3⋊C8—38no—
96C3⋊C8—38no—
97C2×C32#94414yes—
98C2×C32#95018yes—
99C2×C8—91no—
100C2×C8—91no—
101C4⋊C4—22no—
102C4⋊C4—92no—
103C4⋊C4—182no—
104C4⋊C4C2×#27212yes—
105C42C2×#27211yescharacteristic
106C22×C3—34no—
107C22×C3C2×#63811yescharacteristic
108C3×C4—22no—
109C3×C4—34no—
110C3×C4—34no—
111C3×C4—42no—
112C3×C4—68no—
113C3×C4—128no—
114C3×C4—362no—
115C3×C4Q8×C3⋊D311yescharacteristic
116C3×C4Q8×C3⋊D311yescharacteristic
117Dic3—128no—
118C32#506514yescomplemented
119C2×C4—22no—
120C2×C4—182no—
121C2×C4#623512yes—
122C2×C4C22×#27211yescharacteristic
123C8—92no—
124C8—92no—
125C2×C3—34no—
126C2×C3—38no—
127C2×C3#6415311yescharacteristic
128C2×C3#6415912yes—
129C22C2×#623511yescharacteristic
130C4—22no—
131C4—42no—
132C4—362no—
133C4Q8×#27211yescharacteristic
134C4Q8×#27211yescharacteristic
135C3—34no—
136C3#1059276411yescharacteristic
137C2#1172186111yescharacteristic
138C2#1172187112yes—
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)#499
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
84. H(3,𝔽3)#499semidirect
118. C32#5065semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#117217162
#117205302
#117205292
C4×#61802
C3×#50653