Finite Group Databasebeta

← #13157205

Subgroups — #13157205

Order 1728.

372 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 65 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⋊(C3⋊#157)C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic
3C3×M16×C3⋊D3C211yesmaximal, characteristic
4#10590188—31nomaximal
5C22×C3⋊#157C412yes—
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7#88993C412yescomplemented
8C4⋊(C3⋊#157)—21no—
9C4⋊(C3⋊#157)—21no—
10#88989—21no—
11#88988—21no—
12C33×M16C411yescharacteristic, complemented
13C3×#3748C411yescharacteristic, complemented
14C3×#3743—21no—
15C3×C8×C3⋊D3—21no—
16C2×C4⋊#157—31no—
17#64283—31no—
18M16×C3⋊D3D311yescharacteristic
19C3×D3×M16—62no—
20C22×C3×C3⋊D3C2×C411yescharacteristic
21C2×C3⋊#157—22no—
22C2×C3⋊#157—42no—
23C2×C33×C4C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C3×C4×C3⋊D3D411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3D411yescharacteristic
28C3×C4×C3⋊D3Q811yescharacteristic
29C3⋊#622—22no—
30C33×C8—21no—
31C3×C3⋊(C3⋊C8)—21no—
32#4554—91nomaximal
33C22×#157—32no—
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36#3792—32no—
37C4⋊#157—61no—
38C4⋊#157—61no—
39#3788—61no—
40#3787—61no—
41C32×M16—22no—
42C32×M16—42no—
43C32×M16Dic311yescharacteristic, complemented
44#3748Dic311yescharacteristic, complemented
45#3743—21no—
46C8×C3⋊D3—21no—
47C3×#208—62no—
48C3×ΓL(1,𝔽25)—64no—
49C3×D3×C8—64no—
50C22×C33C4211yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—21no—
53C2×C3×C3⋊D3C22⋊C411yescharacteristic
54C2×C3×C3⋊D3C4⋊C411yescharacteristic
55C3⋊#157—42no—
56C33×C4C22⋊C411yescharacteristic
57C33×C4C4⋊C411yescharacteristic
58C3×C3⋊Dic3C22⋊C411yescharacteristic
59C3×C3⋊Dic3C4⋊C411yescharacteristic
60C2×C3×M16—91no—
61C2×C4⋊Dic3—91no—
62C2×#208—91no—
63D3×M16—62no—
64C22×C3⋊D3C4×D311yescharacteristic
65C22×C3×D3—62no—
66C2×#157—62no—
67C2×#157—122no—
68C2×C32×C4—22no—
69C2×C32×C4—42no—
70C2×C32×C4C2×Dic311yescharacteristic
71C2×C3⋊Dic3C4×D311yescharacteristic
72C4×C3⋊D3Dic611yescharacteristic
73C4×C3⋊D3D1211yescharacteristic
74C4×C3⋊D3#6711yescharacteristic
75C4×C3⋊D3#6711yescharacteristic
76C2×C3×Dic3—62no—
77C3×C4×D3—62no—
78C3×C4×D3—62no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81#622—62no—
82C32×C8—21no—
83C32×C8—24no—
84C32×C8—44no—
85C3⋊(C3⋊C8)—21no—
86C3×C3⋊C8—64no—
87#343—271noSylow
88C2×C33—21no—
89C2×C33#9511yescharacteristic
90C3×C3⋊D3—21no—
91C3×C3⋊D3#10011yescharacteristic
92C3×C3⋊D3#10111yescharacteristic
93C22×C3×C4—91no—
94C22×Dic3—92no—
95C2×C4×D3—62no—
96C3×M16—22no—
97C3×M16—42no—
98C3×M16—91no—
99C3×M16—181no—
100C3×M16#15711yescharacteristic, complemented
101C2×C3×C8—181no—
102C4⋊Dic3—181no—
103C4⋊Dic3—181no—
104#208—62no—
105#208—92no—
106#208—181no—
107C2×C3⋊C8—181no—
108ΓL(1,𝔽25)—64no—
109D3×C8—64no—
110C22×C32—22no—
111C22×C32—42no—
112C22×C32C4×Dic311yescharacteristic
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3Dic3⋊C411yescharacteristic
116C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
117C2×C3×D3—62no—
118C2×C3×D3—62no—
119C2×C3×D3—122no—
120C2×C3×D3—122no—
121#157—122no—
122C32×C4—22no—
123C32×C4—22no—
124C32×C4—42no—
125C32×C4—42no—
126C32×C4C4⋊Dic311yescharacteristic
127C32×C4#21711yescharacteristic
128C3⋊Dic3Dic3⋊C411yescharacteristic
129C3⋊Dic3(C2×D3)⋊C411yescharacteristic
130C3×Dic3—62no—
131C3×Dic3—62no—
132C2×M16—91no—
133C2×M16—271no—
134C2×C4⋊C4—271no—
135C33#34311yescharacteristic, complemented, Sylow
136C23×C3—91no—
137C22×D3—62no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4—91no—
141C2×C3×C4—91no—
142C2×C3×C4—91no—
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4C2×#15711yescharacteristic
146C2×Dic3—62no—
147C2×Dic3—182no—
148C2×Dic3—362no—
149C4×D3—62no—
150C4×D3—62no—
151C4×D3—62no—
152C4×D3—62no—
153C3×C8—24no—
154C3×C8—21no—
155C3×C8—44no—
156C3×C8—181no—
157C3⋊C8—64no—
158C3⋊C8—182no—
159C2×C32—22no—
160C2×C32—21no—
161C2×C32—42no—
162C2×C32—42no—
163C2×C32—82no—
164C2×C32C3⋊#9511yescharacteristic
165C3⋊D3—21no—
166C3⋊D3#82311yescharacteristic
167C3⋊D3#82411yescharacteristic
168C3×D3—62no—
169C3×D3—62no—
170C3×D3—122no—
171C22×C4—91no—
172C22×C4—272no—
173M16—91no—
174M16—181no—
175M16—272no—
176M16—541no—
177M16C3⋊#15711yescharacteristic, complemented
178C2×C8—181no—
179C2×C8—541no—
180C4⋊C4—541no—
181C4⋊C4—541no—
182C22×C3—22no—
183C22×C3—42no—
184C22×C3—91no—
185C22×C3—91no—
186C22×C3—181no—
187C22×C3—181no—
188C22×C3C4×#15711yescharacteristic
189C2×D3—62no—
190C2×D3—62no—
191C2×D3—122no—
192C2×D3—122no—
193C3×C4—22no—
194C3×C4—22no—
195C3×C4—42no—
196C3×C4—42no—
197C3×C4—91no—
198C3×C4—91no—
199C3×C4C4⋊#15711yescharacteristic
200C3×C4C22⋊#15711yescharacteristic
201Dic3—62no—
202Dic3—62no—
203Dic3—362no—
204C32—22no—
205C32—42no—
206C32#455411yescharacteristic, complemented
207C23—91no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—542no—
214C2×C4—1082no—
215C2×C4C2×C3⋊#15711yescharacteristic
216C8—21no—
217C8—181no—
218C8—542no—
219C2×C3—21no—
220C2×C3—22no—
221C2×C3—42no—
222C2×C3—42no—
223C2×C3—82no—
224C2×C3—91no—
225C2×C3—91no—
226C2×C3—181no—
227C2×C3#6378111yescharacteristic
228D3—62no—
229D3—62no—
230D3—122no—
231C22—91no—
232C22—91no—
233C22—181no—
234C22—181no—
235C22C4×C3⋊#15711yescharacteristic
236C4—91no—
237C4—91no—
238C4—1082no—
239C4C4⋊(C3⋊#157)11yescharacteristic
240C4#8899411yescharacteristic
241C3—22no—
242C3—42no—
243C3#1059018811yescharacteristic, complemented
244C2—21no—
245C2—91no—
246C2—91no—
247C2—181no—
248C2#1172310611yescharacteristic
subgroupG/H
centerC2#11723106
commutatorC22×C33C42
FrattiniC2×C4C2×C3⋊#157
FittingC33×M16C4
socleC2×C33#95
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C3×C4×C3⋊D3C22
hypercenterM16C3⋊#157
hypocenterC33#343
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
7. #88993C4semidirect
12. C33×M16C4semidirect
13. C3×#3748C4semidirect
43. C32×M16Dic3semidirect
44. #3748Dic3semidirect
100. C3×M16#157semidirect
135. C33#343semidirect
177. M16C3⋊#157semidirect
206. C32#4554semidirect
243. C3#10590188semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊#157)2
#117245112
C3×M16×C3⋊D32
#105901883
#45549