Finite Group Databasebeta

← #13136734

Subgroups — #13136734

Order 1728.

201 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 41 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
2#11722445C212yesmaximal
3C22×#1088C412yes—
4C2×C4×#272C2211yescharacteristic
5#88631—22no—
6C4⋊#1088—22no—
7#88627C414yescomplemented
8#88626—22no—
9C22×#272C2×C411yescharacteristic
10C2×#1088—22no—
11C2×#1088—42no—
12C2×C4×H(3,𝔽3)Q811yescharacteristic
13C2×#1084D411yescharacteristic
14C4×#272C2×C412yes—
15C4×#272D412yes—
16#6188—24no—
17C3×#343—91nomaximal
18C2×C3×C4×D3—121no—
19C22×H(3,𝔽3)C4⋊C411yescharacteristic
20C2×#272—21no—
21C2×#272—21no—
22C2×#272C4211yescharacteristic
23C2×#272C22⋊C411yescharacteristic
24#1088—42no—
25C4×H(3,𝔽3)C4⋊C412yes—
26#1084C22⋊C412yes—
27C2×C3×M16—92no—
28C2×C3×C4⋊C4—91no—
29C22×C3×D3—121no—
30C2×C32×C4—41no—
31C2×C3×Dic3—121no—
32C3×C4×D3—122no—
33C3×C4×D3—122no—
34#343—91noSylow
35C2×H(3,𝔽3)—21no—
36C2×H(3,𝔽3)#9511yescharacteristic
37#272—21no—
38#272#10011yescharacteristic
39#272#10111yescharacteristic
40C22×C3×C4—91no—
41C22×C3×C4—92no—
42C2×C4×D3—121no—
43C3×M16—94no—
44C3×M16—182no—
45C2×C3×C8—182no—
46C3×C4⋊C4—182no—
47C22×C32—41no—
48C2×C3×D3—121no—
49C2×C3×D3—121no—
50C2×C3×D3—122no—
51C2×C3×D3—122no—
52C32×C4—42no—
53C3×Dic3—122no—
54C2×M16—92no—
55C2×C4⋊C4—91no—
56H(3,𝔽3)#34311yescharacteristic, complemented, Sylow
57C23×C3—91no—
58C22×D3—121no—
59C2×C3×C4—91no—
60C2×C3×C4—92no—
61C2×C3×C4—92no—
62C2×C3×C4—121no—
63C2×C3×C4—182no—
64C2×C3×C4—362no—
65C2×C3×C4PSU(3,𝔽2)11yescharacteristic
66C2×Dic3—121no—
67C4×D3—122no—
68C4×D3—122no—
69C3×C8—184no—
70C2×C32—42no—
71C2×C32—41no—
72C3×D3—121no—
73C3×D3—121no—
74C3×D3—122no—
75C22×C4—91no—
76C22×C4—92no—
77M16—94no—
78M16—182no—
79C2×C8—182no—
80C4⋊C4—182no—
81C22×C3—91no—
82C22×C3—91no—
83C22×C3—121no—
84C22×C3—181no—
85C22×C3—181no—
86C22×C3#377711yescharacteristic
87C2×D3—121no—
88C2×D3—121no—
89C2×D3—122no—
90C2×D3—122no—
91C3×C4—92no—
92C3×C4—122no—
93C3×C4—362no—
94C3×C4#377712yes—
95Dic3—122no—
96C32—41no—
97C23—91no—
98C2×C4—91no—
99C2×C4—92no—
100C2×C4—92no—
101C2×C4—182no—
102C2×C4—362no—
103C2×C4SU(3,𝔽2)11yescharacteristic
104C8—184no—
105C2×C3—21no—
106C2×C3—91no—
107C2×C3—91no—
108C2×C3—122no—
109C2×C3—121no—
110C2×C3—181no—
111C2×C3#6374711yescharacteristic
112D3—121no—
113D3—121no—
114D3—122no—
115C22—91no—
116C22—91no—
117C22—181no—
118C22—181no—
119C22#8859211yescharacteristic
120C4—92no—
121C4—362no—
122C4#8859212yes—
123C3—121no—
124C3#1059009411yescharacteristic
125C2—21no—
126C2—91no—
127C2—91no—
128C2—181no—
129C2#1172071111yescharacteristic
subgroupG/H
center, socleC2×C3#63747
commutatorC2×#272C42
Frattini, hypercenterC2×C3×C4PSU(3,𝔽2)
FittingC2×C4×H(3,𝔽3)Q8
prime radicalC22×#272C2×C4
cosocleC2×C4×#272C22
hypocenterH(3,𝔽3)#343
LCSG ▷ C2×#272 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
7. #88627C4semidirect
56. H(3,𝔽3)#343semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10882
#117224452
C3×#3439