Finite Group Databasebeta

← #13155166

Subgroups — #13155166

Order 1728.

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

normal subgroups only

This group has 290 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#11724514C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic
3C2×C3×C4×#157C211yesmaximal, characteristic
4#10590092—31nomaximal
5C22×C3×#157—21no—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7#88994—21no—
8#88993C412yescomplemented
9C4⋊(C3⋊#157)C412yescomplemented
10C4×C3⋊#157—21no—
11#88989—21no—
12#88988—21no—
13C3×C4×#157—21no—
14C3×C4×#157C412yescomplemented
15#64286—31no—
16C2×C4×#157D311yescharacteristic
17#64283—31no—
18C22×C3×C3⋊D3D411yescharacteristic
19C2×C3⋊#157—22no—
20C2×C3×#157—24no—
21C2×C3×#157—22no—
22C2×C33×C4Q811yescharacteristic
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C3×C4×C3⋊D3C2×C411yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3⋊#622—22no—
29#4523—91nomaximal
30C22×#157—21no—
31C2×C4×C3⋊D3C2×D311yescharacteristic
32C2×C3×C4×D3—121no—
33C22⋊#157—61no—
34#3792—32no—
35C4⋊#157—32no—
36C4×#157—21no—
37C4×#157—61no—
38C4×#157Dic312yescomplemented
39#3788—61no—
40#3787—61no—
41C22×C33C4⋊C411yescharacteristic
42C2×C3×C3⋊D3—22no—
43C2×C3×C3⋊D3C22⋊C411yescharacteristic
44C2×C3×C3⋊D3C22⋊C411yescharacteristic
45C3⋊#157—42no—
46C3×#157—24no—
47C33×C4C4⋊C411yescharacteristic
48C33×C4C4⋊C411yescharacteristic
49C3×C3⋊Dic3C4211yescharacteristic
50C3×C3⋊Dic3C22⋊C411yescharacteristic
51C2×C3×C42—91no—
52#926—91no—
53C2×#208—91no—
54C22×C3⋊D3#6711yescharacteristic
55C22×C3×D3—121no—
56C2×#157—24no—
57C2×#157—22no—
58C2×#157—62no—
59C2×C32×C4—41no—
60C2×C32×C4—81no—
61C2×C32×C4Dic611yescharacteristic
62C2×C3⋊Dic3C4×D311yescharacteristic
63C4×C3⋊D3C4×D311yescharacteristic
64C4×C3⋊D3D1211yescharacteristic
65C4×C3⋊D3C2×Dic311yescharacteristic
66C4×C3⋊D3#6711yescharacteristic
67C2×C3×Dic3—121no—
68C3×C4×D3—121no—
69C3×C4×D3—121no—
70C3×C4×D3—121no—
71C3×C4×D3—121no—
72#622—62no—
73#339—271noSylow
74C2×C33—21no—
75C2×C33#9511yescharacteristic
76C3×C3⋊D3—21no—
77C3×C3⋊D3#10412yes—
78C22×C3×C4—91no—
79C22×C3×C4—181no—
80C2×C4×D3—121no—
81C3×C42—92no—
82C3×C42—181no—
83#217—181no—
84C4⋊Dic3—92no—
85C4×Dic3—181no—
86#208—92no—
87#208—181no—
88C2×C3⋊C8—181no—
89C22×C32—41no—
90C22×C32—81no—
91C22×C32Dic3⋊C411yescharacteristic
92C2×C3⋊D3—22no—
93C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
94C2×C3⋊D3#21711yescharacteristic
95C2×C3×D3—121no—
96C2×C3×D3—121no—
97C2×C3×D3—124no—
98#157—24no—
99#157—122no—
100C32×C4—41no—
101C32×C4—41no—
102C32×C4—81no—
103C32×C4—81no—
104C32×C4Dic3⋊C411yescharacteristic
105C32×C4C4⋊Dic311yescharacteristic
106C3⋊Dic3C4×Dic311yescharacteristic
107C3⋊Dic3(C2×D3)⋊C411yescharacteristic
108C3×Dic3—121no—
109C3×Dic3—121no—
110C2×M16—271no—
111#117—271no—
112C2×C42—91no—
113C33#33911yescharacteristic, complemented, Sylow
114C23×C3—91no—
115C22×D3—121no—
116C2×C3×C4—41no—
117C2×C3×C4—81no—
118C2×C3×C4—91no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—182no—
124C2×C3×C4—184no—
125C2×C3×C4PSU(3,𝔽2)11yescharacteristic
126C2×Dic3—121no—
127C2×Dic3—182no—
128C4×D3—121no—
129C4×D3—121no—
130C4×D3—121no—
131C4×D3—121no—
132C3⋊C8—182no—
133C2×C32—21no—
134C2×C32—42no—
135C2×C32—41no—
136C2×C32—82no—
137C2×C32—81no—
138C2×C32C3⋊#9511yescharacteristic
139C3⋊D3—21no—
140C3⋊D3#80512yes—
141C3×D3—122no—
142C3×D3—122no—
143C22×C4—91no—
144C22×C4—181no—
145M16—272no—
146M16—541no—
147C2×C8—541no—
148C4⋊C4—272no—
149C22⋊C4—541no—
150C42—92no—
151C42—181no—
152C42—541no—
153C22×C3—41no—
154C22×C3—81no—
155C22×C3—91no—
156C22×C3—91no—
157C22×C3—182no—
158C22×C3#377711yescharacteristic
159C2×D3—121no—
160C2×D3—121no—
161C2×D3—124no—
162C3×C4—41no—
163C3×C4—41no—
164C3×C4—81no—
165C3×C4—81no—
166C3×C4—91no—
167C3×C4—91no—
168C3×C4—184no—
169C3×C4#377711yescharacteristic
170C3×C4#377711yescharacteristic
171Dic3—121no—
172Dic3—121no—
173Dic3—362no—
174C32—41no—
175C32—81no—
176C32#452311yescharacteristic, complemented
177C23—91no—
178C2×C4—91no—
179C2×C4—91no—
180C2×C4—91no—
181C2×C4—91no—
182C2×C4—91no—
183C2×C4—182no—
184C2×C4—184no—
185C2×C4—542no—
186C2×C4C3⋊PSU(3,𝔽2)11yescharacteristic
187C8—542no—
188C2×C3—21no—
189C2×C3—42no—
190C2×C3—41no—
191C2×C3—82no—
192C2×C3—81no—
193C2×C3—92no—
194C2×C3—181no—
195C2×C3#6374711yescharacteristic
196D3—122no—
197D3—122no—
198C22—91no—
199C22—91no—
200C22—182no—
201C22#8894511yescharacteristic
202C4—91no—
203C4—91no—
204C4—184no—
205C4—1082no—
206C4#8894511yescharacteristic
207C4#8894611yescharacteristic
208C3—41no—
209C3—81no—
210C3#1059009211yescharacteristic, complemented
211C2—21no—
212C2—92no—
213C2—181no—
214C2C3⋊#6374711yescharacteristic
subgroupG/H
centerC4#88946
commutatorC3×C3⋊Dic3C42
Frattini, hypercenterC2×C4C3⋊PSU(3,𝔽2)
FittingC2×C33×C4Q8
socleC2×C33#95
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3×C4×C3⋊D3C22
hypocenterC33#339
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
8. #88993C4semidirect
9. C4⋊(C3⋊#157)C4semidirect
14. C3×C4×#157C4semidirect
38. C4×#157Dic3semidirect
113. C33#339semidirect
176. C32#4523semidirect
210. C3#10590092semidirect

Maximal subgroups

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

subgroupindex
#117245142
#117245112
C2×C3×C4×#1572
#105900923
#45239