Finite Group Databasebeta

← #13136732

Subgroups — #13136732

Order 1728.

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

normal subgroups only

This group has 219 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
1#11722448C211yesmaximal, characteristic
2C2×C4×#1088C211yesmaximal, characteristic
3#11722445C211yesmaximal, characteristic
4C22×#1088—21no—
5C2×C4×#272C2211yescharacteristic
6C22⋊#1088—21no—
7#88631C412yescomplemented
8C4⋊#1088C412yescomplemented
9C4×#1088—21no—
10C4×#1088—21no—
11C4×#1088C412yescomplemented
12#88627—21no—
13#88626—21no—
14C22×#272D411yescharacteristic
15C2×#1088—24no—
16C2×#1088—22no—
17C2×#1088—22no—
18C2×C4×H(3,𝔽3)Q811yescharacteristic
19C2×#1084C2×C411yescharacteristic
20C4×#272C2×C411yescharacteristic
21C4×#272C2×C411yescharacteristic
22C4×#272D411yescharacteristic
23C4×#272D411yescharacteristic
24#6188—22no—
25C3×#339—91nomaximal
26C2×C3×C4×D3—121no—
27C22×H(3,𝔽3)C4⋊C411yescharacteristic
28C2×#272—22no—
29C2×#272C22⋊C411yescharacteristic
30C2×#272C22⋊C411yescharacteristic
31#1088—24no—
32#1088—42no—
33C4×H(3,𝔽3)C4⋊C411yescharacteristic
34C4×H(3,𝔽3)C4⋊C411yescharacteristic
35#1084C4211yescharacteristic
36#1084C22⋊C411yescharacteristic
37C2×C3×M16—91no—
38C3×#117—91no—
39C2×C3×C42—91no—
40C22×C3×D3—121no—
41C2×C32×C4—41no—
42C2×C3×Dic3—121no—
43C3×C4×D3—121no—
44C3×C4×D3—121no—
45C3×C4×D3—121no—
46C3×C4×D3—121no—
47#339—91noSylow
48C2×H(3,𝔽3)—21no—
49C2×H(3,𝔽3)#9511yescharacteristic
50#272—21no—
51#272#10412yes—
52C22×C3×C4—91no—
53C22×C3×C4—181no—
54C2×C4×D3—121no—
55C3×M16—92no—
56C3×M16—181no—
57C2×C3×C8—181no—
58C3×C4⋊C4—92no—
59C3×C22⋊C4—181no—
60C3×C42—92no—
61C3×C42—181no—
62C3×C42—181no—
63C22×C32—41no—
64C2×C3×D3—121no—
65C2×C3×D3—121no—
66C2×C3×D3—124no—
67C32×C4—41no—
68C32×C4—41no—
69C3×Dic3—121no—
70C3×Dic3—121no—
71C2×M16—91no—
72#117—91no—
73C2×C42—91no—
74H(3,𝔽3)#33911yescharacteristic, complemented, Sylow
75C23×C3—91no—
76C22×D3—121no—
77C2×C3×C4—91no—
78C2×C3×C4—91no—
79C2×C3×C4—91no—
80C2×C3×C4—91no—
81C2×C3×C4—91no—
82C2×C3×C4—121no—
83C2×C3×C4—182no—
84C2×C3×C4—184no—
85C2×C3×C4—182no—
86C2×C3×C4PSU(3,𝔽2)11yescharacteristic
87C2×Dic3—121no—
88C4×D3—121no—
89C4×D3—121no—
90C4×D3—121no—
91C4×D3—121no—
92C3×C8—182no—
93C2×C32—42no—
94C2×C32—41no—
95C3×D3—122no—
96C3×D3—122no—
97C22×C4—91no—
98C22×C4—181no—
99M16—92no—
100M16—181no—
101C2×C8—181no—
102C4⋊C4—92no—
103C22⋊C4—181no—
104C42—92no—
105C42—181no—
106C42—181no—
107C22×C3—91no—
108C22×C3—91no—
109C22×C3—121no—
110C22×C3—182no—
111C22×C3#377711yescharacteristic
112C2×D3—121no—
113C2×D3—121no—
114C2×D3—124no—
115C3×C4—91no—
116C3×C4—91no—
117C3×C4—121no—
118C3×C4—121no—
119C3×C4—184no—
120C3×C4—362no—
121C3×C4#377711yescharacteristic
122C3×C4#377711yescharacteristic
123Dic3—121no—
124Dic3—121no—
125C32—41no—
126C23—91no—
127C2×C4—91no—
128C2×C4—91no—
129C2×C4—91no—
130C2×C4—91no—
131C2×C4—91no—
132C2×C4—182no—
133C2×C4—184no—
134C2×C4—182no—
135C2×C4SU(3,𝔽2)11yescharacteristic
136C8—182no—
137C2×C3—21no—
138C2×C3—92no—
139C2×C3—122no—
140C2×C3—121no—
141C2×C3—181no—
142C2×C3#6374711yescharacteristic
143D3—122no—
144D3—122no—
145C22—91no—
146C22—91no—
147C22—182no—
148C22#8859211yescharacteristic
149C4—91no—
150C4—91no—
151C4—184no—
152C4—362no—
153C4#8859211yescharacteristic
154C4#8859211yescharacteristic
155C3—121no—
156C3#1059009211yescharacteristic
157C2—21no—
158C2—92no—
159C2—181no—
160C2#1172071111yescharacteristic
subgroupG/H
centerC3×C4#3777
commutator#1084C42
Frattini, hypercenterC2×C3×C4PSU(3,𝔽2)
FittingC2×C4×H(3,𝔽3)Q8
socleC2×C3#63747
prime radicalC22×#272D4
cosocleC2×C4×#272C22
hypocenterH(3,𝔽3)#339
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
7. #88631C4semidirect
8. C4⋊#1088C4semidirect
11. C4×#1088C4semidirect
74. H(3,𝔽3)#339semidirect

Maximal subgroups

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

subgroupindex
#117224482
C2×C4×#10882
#117224452
C3×#3399