Finite Group Databasebeta

← #13155169

Subgroups — #13155169

Order 1728.

266 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 63 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#11724511C212yesmaximal
2C2×C3×C4⋊#157C211yesmaximal, characteristic
3#10590094—31nomaximal
4C22×C3×#157C412yes—
5C2×C3×C4×C3⋊D3C2211yescharacteristic
6#88993—22no—
7#88989C414yescomplemented
8#88988—22no—
9C3×C4⋊#157—22no—
10C2×C4⋊#157D311yescharacteristic
11#64283—32no—
12C22×C3×C3⋊D3C2×C411yescharacteristic
13C2×C3×#157—22no—
14C2×C3×#157—42no—
15C2×C33×C4Q811yescharacteristic
16C2×C3×C3⋊Dic3D411yescharacteristic
17C3×C4×C3⋊D3C2×C412yes—
18C3×C4×C3⋊D3D412yes—
19C3⋊#622—24no—
20#4530—91nomaximal
21C22×#157Dic312yes—
22C2×C4×C3⋊D3C2×D311yescharacteristic
23C2×C3×C4×D3—121no—
24#3792—62no—
25C4⋊#157—22no—
26#3788—34no—
27#3787—62no—
28C22×C33C4⋊C411yescharacteristic
29C2×C3×C3⋊D3—21no—
30C2×C3×C3⋊D3—21no—
31C2×C3×C3⋊D3C4211yescharacteristic
32C2×C3×C3⋊D3C22⋊C411yescharacteristic
33C3×#157—42no—
34C33×C4C4⋊C412yes—
35C3×C3⋊Dic3C22⋊C412yes—
36C2×C3×C4⋊C4—91no—
37C2×#208—92no—
38C22×C3⋊D3C2×Dic311yescharacteristic
39C22×C3×D3—121no—
40C2×#157—22no—
41C2×#157—42no—
42C2×C32×C4—41no—
43C2×C32×C4—81no—
44C2×C32×C4Dic611yescharacteristic
45C2×C3⋊Dic3D1211yescharacteristic
46C4×C3⋊D3C4×D312yes—
47C4×C3⋊D3#6712yes—
48C2×C3×Dic3—121no—
49C3×C4×D3—122no—
50C3×C4×D3—122no—
51#622—64no—
52#343—271noSylow
53C2×C33—21no—
54C2×C33#9511yescharacteristic
55C3×C3⋊D3—21no—
56C3×C3⋊D3#10011yescharacteristic
57C3×C3⋊D3#10111yescharacteristic
58C22×C3×C4—91no—
59C22×C3×C4—92no—
60C2×C4×D3—121no—
61C3×C4⋊C4—182no—
62#208—94no—
63#208—182no—
64C2×C3⋊C8—182no—
65C22×C32—41no—
66C22×C32—81no—
67C22×C32C4⋊Dic311yescharacteristic
68C2×C3⋊D3—21no—
69C2×C3⋊D3—21no—
70C2×C3⋊D3C4×Dic311yescharacteristic
71C2×C3⋊D3#21711yescharacteristic
72C2×C3×D3—121no—
73C2×C3×D3—121no—
74C2×C3×D3—122no—
75C2×C3×D3—122no—
76#157—42no—
77C32×C4—42no—
78C32×C4—82no—
79C32×C4Dic3⋊C412yes—
80C3⋊Dic3(C2×D3)⋊C412yes—
81C3×Dic3—122no—
82C2×M16—272no—
83C2×C4⋊C4—91no—
84C33#34311yescharacteristic, complemented, Sylow
85C23×C3—91no—
86C22×D3—121no—
87C2×C3×C4—41no—
88C2×C3×C4—81no—
89C2×C3×C4—91no—
90C2×C3×C4—92no—
91C2×C3×C4—92no—
92C2×C3×C4—182no—
93C2×C3×C4—362no—
94C2×C3×C4PSU(3,𝔽2)11yescharacteristic
95C2×Dic3—121no—
96C4×D3—122no—
97C4×D3—122no—
98C3⋊C8—184no—
99C2×C32—21no—
100C2×C32—42no—
101C2×C32—41no—
102C2×C32—82no—
103C2×C32—81no—
104C2×C32C3⋊#9511yescharacteristic
105C3⋊D3—21no—
106C3⋊D3#83311yescharacteristic
107C3⋊D3#83611yescharacteristic
108C3×D3—121no—
109C3×D3—121no—
110C3×D3—122no—
111C22×C4—91no—
112C22×C4—92no—
113M16—274no—
114M16—542no—
115C2×C8—542no—
116C4⋊C4—182no—
117C22×C3—41no—
118C22×C3—81no—
119C22×C3—91no—
120C22×C3—91no—
121C22×C3—181no—
122C22×C3—181no—
123C22×C3#377711yescharacteristic
124C2×D3—121no—
125C2×D3—121no—
126C2×D3—122no—
127C2×D3—122no—
128C3×C4—42no—
129C3×C4—82no—
130C3×C4—92no—
131C3×C4—362no—
132C3×C4#377712yes—
133Dic3—122no—
134C32—41no—
135C32—81no—
136C32#453011yescharacteristic, complemented
137C23—91no—
138C2×C4—91no—
139C2×C4—92no—
140C2×C4—92no—
141C2×C4—182no—
142C2×C4—362no—
143C2×C4C3⋊PSU(3,𝔽2)11yescharacteristic
144C8—544no—
145C2×C3—21no—
146C2×C3—42no—
147C2×C3—41no—
148C2×C3—82no—
149C2×C3—81no—
150C2×C3—91no—
151C2×C3—91no—
152C2×C3—181no—
153C2×C3#6374711yescharacteristic
154D3—121no—
155D3—121no—
156D3—122no—
157C22—91no—
158C22—91no—
159C22—181no—
160C22—181no—
161C22#8894611yescharacteristic
162C4—92no—
163C4—362no—
164C4#8894512yes—
165C3—41no—
166C3—81no—
167C3#1059009411yescharacteristic, complemented
168C2—21no—
169C2—91no—
170C2—91no—
171C2—181no—
172C2C3⋊#6374711yescharacteristic
subgroupG/H
centerC2C3⋊#63747
commutatorC2×C3×C3⋊D3C42
Frattini, hypercenterC2×C4C3⋊PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#343
LCSG ▷ C2×C3×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
7. #88989C4semidirect
84. C33#343semidirect
136. C32#4530semidirect
167. C3#10590094semidirect

Maximal subgroups

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

subgroupindex
#117245112
C2×C3×C4⋊#1572
#105900943
#45309