Finite Group Databasebeta

← #13136735

Subgroups — #13136735

Order 1728.

189 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 39 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
2#11722445C211yesmaximal, characteristic
3C2×#88627C211yesmaximal, characteristic
4C2×C4×#272C2211yescharacteristic
5C22⋊#1088C412yes—
6#88631—21no—
7C2×#6188—21no—
8C4⋊#1088—21no—
9C4×#1088—21no—
10#88627—21no—
11#88627—21no—
12#88627C412yescomplemented
13#88626C412yescomplemented
14C22×#272C2×C411yescharacteristic
15C2×#1088—22no—
16C2×C4×H(3,𝔽3)Q811yescharacteristic
17C2×#1084D411yescharacteristic
18C4×#272C2×C411yescharacteristic
19C4×#272C2×C411yescharacteristic
20C4×#272D411yescharacteristic
21C4×#272D411yescharacteristic
22#6188—22no—
23#6188—22no—
24C3×#344—91nomaximal
25C2×C3×C4×D3—121no—
26C22×H(3,𝔽3)C4⋊C411yescharacteristic
27C2×#272—41no—
28C2×#272C4211yescharacteristic
29C2×#272C22⋊C411yescharacteristic
30#1088—42no—
31C4×H(3,𝔽3)C4⋊C411yescharacteristic
32C4×H(3,𝔽3)C4⋊C411yescharacteristic
33#1084C22⋊C411yescharacteristic
34#1084C22⋊C411yescharacteristic
35C2×C3×M16—91no—
36C2×C3×M16—91no—
37C3×#117—91no—
38C22×C3×D3—121no—
39C2×C32×C4—41no—
40C2×C3×Dic3—121no—
41C3×C4×D3—121no—
42C3×C4×D3—121no—
43C3×C4×D3—121no—
44C3×C4×D3—121no—
45#344—91noSylow
46C2×H(3,𝔽3)—21no—
47C2×H(3,𝔽3)#9511yescharacteristic
48#272—21no—
49#272—21no—
50C22×C3×C4—91no—
51C2×C4×D3—121no—
52C3×M16—92no—
53C3×M16—181no—
54C3×M16—181no—
55C3×M16—181no—
56C2×C3×C8—92no—
57C2×C3×C8—181no—
58C3×C4⋊C4—181no—
59C3×C22⋊C4—92no—
60C3×C42—181no—
61C22×C32—41no—
62C2×C3×D3—121no—
63C2×C3×D3—121no—
64C2×C3×D3—124no—
65C32×C4—41no—
66C32×C4—41no—
67C3×Dic3—121no—
68C3×Dic3—121no—
69C2×M16—91no—
70C2×M16—91no—
71#117—91no—
72H(3,𝔽3)#34411yescharacteristic, complemented, Sylow
73C23×C3—91no—
74C22×D3—121no—
75C2×C3×C4—91no—
76C2×C3×C4—91no—
77C2×C3×C4—91no—
78C2×C3×C4—91no—
79C2×C3×C4—91no—
80C2×C3×C4—121no—
81C2×C3×C4—182no—
82C2×C3×C4PSU(3,𝔽2)11yescharacteristic
83C2×Dic3—121no—
84C4×D3—121no—
85C4×D3—121no—
86C4×D3—121no—
87C4×D3—121no—
88C3×C8—182no—
89C3×C8—182no—
90C2×C32—42no—
91C2×C32—41no—
92C3×D3—122no—
93C3×D3—122no—
94C22×C4—91no—
95M16—92no—
96M16—181no—
97M16—181no—
98M16—181no—
99C2×C8—92no—
100C2×C8—181no—
101C4⋊C4—181no—
102C22⋊C4—92no—
103C42—181no—
104C22×C3—91no—
105C22×C3—91no—
106C22×C3—121no—
107C22×C3—361no—
108C22×C3#377711yescharacteristic
109C2×D3—121no—
110C2×D3—121no—
111C2×D3—124no—
112C3×C4—91no—
113C3×C4—91no—
114C3×C4—121no—
115C3×C4—121no—
116C3×C4—362no—
117C3×C4#377711yescharacteristic
118C3×C4#377711yescharacteristic
119Dic3—121no—
120Dic3—121no—
121C32—41no—
122C23—91no—
123C2×C4—91no—
124C2×C4—91no—
125C2×C4—91no—
126C2×C4—91no—
127C2×C4—91no—
128C2×C4—182no—
129C2×C4SU(3,𝔽2)11yescharacteristic
130C8—182no—
131C8—182no—
132C2×C3—21no—
133C2×C3—122no—
134C2×C3—121no—
135C2×C3—181no—
136C2×C3—181no—
137C2×C3#6374711yescharacteristic
138D3—122no—
139D3—122no—
140C22—91no—
141C22—91no—
142C22—361no—
143C22#8859211yescharacteristic
144C4—91no—
145C4—91no—
146C4—362no—
147C4#8859211yescharacteristic
148C4#8859211yescharacteristic
149C3—121no—
150C3#1059009511yescharacteristic
151C2—21no—
152C2—181no—
153C2—181no—
154C2#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)#344
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
12. #88627C4semidirect
13. #88626C4semidirect
72. H(3,𝔽3)#344semidirect

Maximal subgroups

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

subgroupindex
#117224482
#117224452
C2×#886272
C3×#3449