Finite Group Databasebeta

← #13144209

Subgroups — #13144209

Order 1728.

462 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 71 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
1#11721794C212yesmaximal, complemented
2H(3,𝔽3)×D16C211yesmaximal, characteristic, complemented
3#11720566C212yesmaximal, complemented
4#11720514C211yesmaximal, characteristic, complemented
5C16×#272C211yesmaximal, characteristic, complemented
6C3×#4912—34nomaximal
7#88744—22no—
8D8×H(3,𝔽3)C2212yes—
9#88546—22no—
10#88530C2212yes—
11C8×#272C2211yescharacteristic
12C16×H(3,𝔽3)C2211yescharacteristic
13#88386C2211yescharacteristic
14C3×#912—38no—
15C32×D16D314yescomplemented
16C3×#827—38no—
17C3×Dic24—34no—
18C3×D3×C16—34no—
19D4×H(3,𝔽3)—22no—
20#6235—42no—
21C2×#1084—42no—
22C4×#272D411yescharacteristic
23#6229—22no—
24C8×H(3,𝔽3)C2311yescharacteristic
25#6180D411yescharacteristic
26C3×#508—91no—
27#4912—34no—
28C3×#237—68no—
29C32×D8C2×D318yes—
30C3×#214—68no—
31C3×Dic12—38no—
32C3×D3×C8—34no—
33C32×C16C2×D314yes—
34C3×C3⋊C16—34no—
35C22×H(3,𝔽3)—42no—
36C2×#272D811yescharacteristic
37C4×H(3,𝔽3)C2×D411yescharacteristic
38#1084—42no—
39#1084D811yescharacteristic
40C3×#135—92no—
41#912—38no—
42C3×Q32—91no—
43C3×SD32—92no—
44C3×D16—34no—
45C3×D16C3⋊D311yescharacteristic
46C2×C3×C16—91no—
47#827—38no—
48Dic24—34no—
49D3×C16—34no—
50C32×D4—28no—
51C3×#67—128no—
52C2×C3×Dic3—128no—
53C3×C4×D3—34no—
54C3×Dic6—68no—
55C32×C8C22×D314yes—
56C3×C3⋊C8—34no—
57#508—91noSylow
58C2×H(3,𝔽3)—82no—
59C2×H(3,𝔽3)C2×D811yescharacteristic
60#272—21no—
61C3×#41—182no—
62#237—68no—
63C3×Q16—92no—
64C3×SD16—182no—
65C3×D8—38no—
66C3×D8C2×C3⋊D312yes—
67C2×C3×C8—91no—
68#214—68no—
69Dic12—38no—
70D3×C8—34no—
71C3×C16—34no—
72C3×C16—91no—
73C3×C16C2×C3⋊D311yescharacteristic
74C3⋊C16—34no—
75C22×C32—48no—
76C2×C3×D3—34no—
77C32×C4D3×D414yes—
78C3×Dic3—34no—
79C3×Dic3—128no—
80#135—92no—
81Q32—91no—
82SD32—92no—
83D16#27211yescharacteristic, complemented
84C2×C16—91no—
85H(3,𝔽3)#50811yescharacteristic, complemented, Sylow
86C3×Q8—182no—
87C3×D4—22no—
88C3×D4—68no—
89C3×D4—362no—
90C2×C3×C4—91no—
91C2×C3×C4—362no—
92#67—128no—
93C2×Dic3—128no—
94C4×D3—34no—
95Dic6—68no—
96C3×C8—34no—
97C3×C8—91no—
98C3×C8C22×C3⋊D311yescharacteristic
99C3⋊C8—34no—
100C2×C32—88no—
101C2×C32D3×D814yes—
102C3×D3—64no—
103#41—182no—
104Q16—92no—
105SD16—182no—
106D8C2×#27212yes—
107C2×C8—91no—
108C16—91no—
109C16C2×#27211yescharacteristic
110C22×C3—42no—
111C22×C3—91no—
112C22×C3—128no—
113C2×D3—34no—
114C3×C4—34no—
115C3×C4—91no—
116C3×C4—362no—
117C3×C4D4×C3⋊D311yescharacteristic
118Dic3—34no—
119Dic3—128no—
120C32#491214yescomplemented
121Q8—182no—
122D4—22no—
123D4—362no—
124C2×C4—91no—
125C2×C4—362no—
126C8—91no—
127C8C22×#27211yescharacteristic
128C2×C3—34no—
129C2×C3—82no—
130C2×C3—181no—
131C2×C3—248no—
132C2×C3D8×C3⋊D311yescharacteristic
133D3—64no—
134C22—42no—
135C22—91no—
136C4—91no—
137C4—362no—
138C4D4×#27211yescharacteristic
139C3—34no—
140C3#1059261111yescharacteristic
141C2—82no—
142C2—181no—
143C2D8×#27211yescharacteristic
subgroupG/H
center, socleC2×C3D8×C3⋊D3
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C22×C3⋊D3
FittingH(3,𝔽3)×D16C2
hypercenterC3×D16C3⋊D3
hypocenterH(3,𝔽3)#508
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×C8 ◁ C3×D16

Complements

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

NH ≅ G/Ntype
1. #11721794C2semidirect
2. H(3,𝔽3)×D16C2semidirect
3. #11720566C2semidirect
4. #11720514C2semidirect
5. C16×#272C2semidirect
15. C32×D16D3semidirect
83. D16#272semidirect
85. H(3,𝔽3)#508semidirect
120. C32#4912semidirect

Maximal subgroups

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

subgroupindex
#117217942
H(3,𝔽3)×D162
#117205662
#117205142
C16×#2722
C3×#49123