Finite Group Databasebeta

← #13136730

Subgroups — #13136730

Order 1728.

231 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×C4×#1088C211yesmaximal, characteristic
2C2×#88626C212yesmaximal
3C22×#1088C412yes—
4C2×C4×#272C2211yescharacteristic
5C2×#6188—22no—
6C4×#1088—22no—
7#88626—22no—
8#88626C414yescomplemented
9C22×#272C2×C411yescharacteristic
10C2×#1088—22no—
11C2×#1088—22no—
12C2×#1088C814yescomplemented
13C2×C4×H(3,𝔽3)Q811yescharacteristic
14C2×#1084D411yescharacteristic
15C4×#272C2×C412yes—
16C4×#272D412yes—
17#6188—24no—
18C3×#336—91nomaximal
19C2×C3×C4×D3—121no—
20C22×H(3,𝔽3)C4⋊C411yescharacteristic
21C2×#272C4211yescharacteristic
22C2×#272C22⋊C411yescharacteristic
23C2×#272C2×C812yes—
24C2×#272M1612yes—
25#1088—24no—
26C4×H(3,𝔽3)C4⋊C412yes—
27#1084C22⋊C412yes—
28C22×C3×C8—92no—
29C2×C3×C42—91no—
30C22×C3×D3—121no—
31C2×C32×C4—41no—
32C2×C3×Dic3—121no—
33C3×C4×D3—122no—
34C3×C4×D3—122no—
35#336—91noSylow
36C2×H(3,𝔽3)#9511yescharacteristic
37C2×H(3,𝔽3)C4⋊C812yes—
38#272C4×C811yescharacteristic
39#272#9711yescharacteristic
40#272C22⋊C812yes—
41C22×C3×C4—91no—
42C22×C3×C4—92no—
43C2×C4×D3—121no—
44C2×C3×C8—94no—
45C2×C3×C8—182no—
46C2×C3×C8—182no—
47C3×C42—182no—
48C22×C32—41no—
49C2×C3×D3—121no—
50C2×C3×D3—121no—
51C2×C3×D3—122no—
52C2×C3×D3—122no—
53C32×C4—42no—
54C3×Dic3—122no—
55C22×C8—92no—
56C2×C42—91no—
57H(3,𝔽3)#33611yescharacteristic, complemented, Sylow
58C23×C3—91no—
59C22×D3—121no—
60C2×C3×C4—91no—
61C2×C3×C4—92no—
62C2×C3×C4—92no—
63C2×C3×C4—94no—
64C2×C3×C4—121no—
65C2×C3×C4—182no—
66C2×C3×C4—182no—
67C2×C3×C4PSU(3,𝔽2)11yescharacteristic
68C2×Dic3—121no—
69C4×D3—122no—
70C4×D3—122no—
71C3×C8—184no—
72C2×C32—42no—
73C2×C32—41no—
74C3×D3—121no—
75C3×D3—121no—
76C3×D3—122no—
77C22×C4—91no—
78C22×C4—92no—
79C2×C8—94no—
80C2×C8—182no—
81C2×C8—182no—
82C42—182no—
83C22×C3—92no—
84C22×C3—92no—
85C22×C3—91no—
86C22×C3—91no—
87C22×C3—121no—
88C22×C3#377711yescharacteristic
89C2×D3—121no—
90C2×D3—121no—
91C2×D3—122no—
92C2×D3—122no—
93C3×C4—92no—
94C3×C4—122no—
95C3×C4—184no—
96C3×C4#377712yes—
97Dic3—122no—
98C32—41no—
99C23—91no—
100C2×C4—91no—
101C2×C4—92no—
102C2×C4—92no—
103C2×C4—94no—
104C2×C4—182no—
105C2×C4—182no—
106C2×C4SU(3,𝔽2)11yescharacteristic
107C8—184no—
108C2×C3—92no—
109C2×C3—91no—
110C2×C3—91no—
111C2×C3—122no—
112C2×C3—121no—
113C2×C3#6374412yes—
114C2×C3#6374711yescharacteristic
115D3—122no—
116D3—121no—
117D3—121no—
118C22—92no—
119C22—92no—
120C22—91no—
121C22—91no—
122C22#8859211yescharacteristic
123C4—92no—
124C4—184no—
125C4#8859212yes—
126C3—121no—
127C3#1059009011yescharacteristic
128C2—92no—
129C2—91no—
130C2—91no—
131C2#1172070812yes—
132C2#1172071111yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C3×C4PSU(3,𝔽2)
commutator#272C4×C8
FittingC2×C4×H(3,𝔽3)Q8
socleC22×C3#3777
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypocenterH(3,𝔽3)#336
LCSG ▷ #272 ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
8. #88626C4semidirect
12. C2×#1088C8semidirect
57. H(3,𝔽3)#336semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#10882
C2×#886262
C3×#3369