Finite Group Databasebeta

← #13154851

Subgroups — #13154851

Order 1728.

488 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 68 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×#88993C211yesmaximal, characteristic
2C2×C3×#3792C211yesmaximal, characteristic
3C2×#88807C211yesmaximal, characteristic
4#10590208—31nomaximal
5C22×C3⋊(C3⋊Dic3)C411yescharacteristic
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C22×C32×Dic3C411yescharacteristic
8#88993—21no—
9#88993C412yescomplemented
10C2×C3⋊#622—21no—
11C3×#3792—21no—
12C3×#3792C412yescomplemented
13C2×C3×#622—21no—
14#88807—21no—
15#88807—21no—
16C2×#3792—31no—
17C2×#3792D311yescharacteristic
18C2×#3750—31no—
19C2×Dic3⋊Dic3—62no—
20C23×C33C2×C411yescharacteristic
21C2×C3⋊(C3⋊Dic3)—21no—
22C2×C3⋊(C3⋊Dic3)—41no—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3D411yescharacteristic
28C2×C3×C3⋊Dic3Q811yescharacteristic
29C2×C32×Dic3—21no—
30C2×C32×Dic3—41no—
31C3⋊#622—22no—
32C3×#622—22no—
33#4554—91nomaximal
34C22×C32×C4—31no—
35C22×C3⋊Dic3—31no—
36C22×C3⋊Dic3—62no—
37C22×C3⋊Dic3—122no—
38C22×C3⋊Dic3C2×D311yescharacteristic
39C22×C3×Dic3—22no—
40C22×C3×Dic3—62no—
41#3792—21no—
42#3792—32no—
43#3792—61no—
44#3792Dic312yescomplemented
45C2×#622—21no—
46C2×#622—61no—
47#3750—61no—
48#3750—61no—
49Dic3⋊Dic3—64no—
50Dic3⋊Dic3—64no—
51C22×C33—21no—
52C22×C33—21no—
53C22×C33C4211yescharacteristic
54C22×C33C22⋊C411yescharacteristic
55C22×C33C4⋊C411yescharacteristic
56C3⋊(C3⋊Dic3)—41no—
57C3×C3⋊Dic3C22⋊C411yescharacteristic
58C3×C3⋊Dic3C22⋊C411yescharacteristic
59C3×C3⋊Dic3C4⋊C411yescharacteristic
60C3×C3⋊Dic3C4⋊C411yescharacteristic
61C32×Dic3—41no—
62C2×C3×M16—91no—
63C2×C4⋊Dic3—91no—
64C2×Dic3⋊C4—182no—
65C2×#208—91no—
66C23×C32—22no—
67C23×C32—42no—
68C23×C32C4×D311yescharacteristic
69C2×C32×C4—61no—
70C2×C32×C4—121no—
71C2×C3⋊Dic3—61no—
72C2×C3⋊Dic3—64no—
73C2×C3⋊Dic3—124no—
74C2×C3⋊Dic3—124no—
75C2×C3⋊Dic3—128no—
76C2×C3⋊Dic3—121no—
77C2×C3⋊Dic3Dic611yescharacteristic
78C2×C3⋊Dic3C4×D311yescharacteristic
79C2×C3⋊Dic3D1211yescharacteristic
80C2×C3⋊Dic3C2×Dic311yescharacteristic
81C2×C3⋊Dic3#6711yescharacteristic
82C2×C3⋊Dic3#6711yescharacteristic
83C2×C3×Dic3—24no—
84C2×C3×Dic3—44no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—62no—
90C2×C3×Dic3—62no—
91#622—22no—
92#622—62no—
93#343—271noSylow
94C2×C33—21no—
95C2×C33—21no—
96C2×C33#9511yescharacteristic
97C2×C33#10011yescharacteristic
98C2×C33#10111yescharacteristic
99C22×C3×C4—62no—
100C22×C3×C4—91no—
101C22×Dic3—62no—
102C22×Dic3—91no—
103C22×Dic3—182no—
104C22×Dic3—362no—
105C22×Dic3#15711yescharacteristic
106C3×M16—92no—
107C3×M16—181no—
108C2×C3×C8—181no—
109C4⋊Dic3—181no—
110C4⋊Dic3—181no—
111Dic3⋊C4—184no—
112Dic3⋊C4—184no—
113#208—92no—
114#208—181no—
115C2×C3⋊C8—181no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32—24no—
119C22×C32—24no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—22no—
123C22×C32—44no—
124C22×C32—44no—
125C22×C32—42no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32C4×Dic311yescharacteristic
129C22×C32Dic3⋊C411yescharacteristic
130C22×C32(C2×D3)⋊C411yescharacteristic
131C32×C4—121no—
132C3⋊Dic3—124no—
133C3⋊Dic3—128no—
134C3⋊Dic3—121no—
135C3⋊Dic3Dic3⋊C411yescharacteristic
136C3⋊Dic3C4⋊Dic311yescharacteristic
137C3⋊Dic3(C2×D3)⋊C411yescharacteristic
138C3⋊Dic3#21711yescharacteristic
139C3×Dic3—44no—
140C3×Dic3—62no—
141C3×Dic3—62no—
142C3×Dic3—62no—
143C3×Dic3—62no—
144C2×M16—91no—
145C2×M16—271no—
146C2×C4⋊C4—271no—
147C33#34311yescharacteristic, complemented, Sylow
148C23×C3—22no—
149C23×C3—42no—
150C23×C3C2×#15711yescharacteristic
151C2×C3×C4—64no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—91no—
155C2×C3×C4—91no—
156C2×C3×C4—91no—
157C2×C3×C4—91no—
158C2×C3×C4—124no—
159C2×Dic3—21no—
160C2×Dic3—41no—
161C2×Dic3—62no—
162C2×Dic3—62no—
163C2×Dic3—62no—
164C2×Dic3—62no—
165C2×Dic3—62no—
166C2×Dic3—62no—
167C2×Dic3—184no—
168C2×Dic3—181no—
169C2×Dic3—364no—
170C2×Dic3—364no—
171C2×Dic3—368no—
172C2×Dic3—361no—
173C3×C8—182no—
174C3⋊C8—182no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32—24no—
178C2×C32—24no—
179C2×C32—22no—
180C2×C32—22no—
181C2×C32—22no—
182C2×C32—44no—
183C2×C32—44no—
184C2×C32—42no—
185C2×C32—42no—
186C2×C32—42no—
187C2×C32#82311yescharacteristic
188C2×C32#82411yescharacteristic
189C2×C32C3⋊#9511yescharacteristic
190C22×C4—31no—
191C22×C4—91no—
192C22×C4—271no—
193M16—92no—
194M16—181no—
195M16—272no—
196M16—541no—
197C2×C8—181no—
198C2×C8—541no—
199C4⋊C4—541no—
200C4⋊C4—541no—
201C22×C3—24no—
202C22×C3—24no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—44no—
209C22×C3—44no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3C4×#15711yescharacteristic
214C22×C3C4⋊#15711yescharacteristic
215C22×C3C22⋊#15711yescharacteristic
216C3×C4—91no—
217C3×C4—91no—
218C3×C4—91no—
219C3×C4—91no—
220C3×C4—124no—
221Dic3—41no—
222Dic3—62no—
223Dic3—62no—
224Dic3—62no—
225Dic3—62no—
226Dic3—364no—
227Dic3—368no—
228Dic3—361no—
229C32—22no—
230C32—42no—
231C32#455411yescharacteristic, complemented
232C23D3×#15711yescharacteristic
233C2×C4—61no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—121no—
241C2×C4—541no—
242C2×C4—1081no—
243C8—182no—
244C8—542no—
245C2×C3—24no—
246C2×C3—24no—
247C2×C3—21no—
248C2×C3—21no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—22no—
252C2×C3—44no—
253C2×C3—44no—
254C2×C3—42no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3#6378011yescharacteristic
258C2×C3#6378111yescharacteristic
259C2×C3#6378811yescharacteristic
260C22—21no—
261C22—21no—
262C22Dic3×#15711yescharacteristic
263C22(C2×D3)⋊#15711yescharacteristic
264C22Dic3⋊#15711yescharacteristic
265C4—91no—
266C4—91no—
267C4—91no—
268C4—91no—
269C4—121no—
270C4—1081no—
271C3—22no—
272C3—42no—
273C3#1059020811yescharacteristic, complemented
274C2—21no—
275C2—21no—
276C2#1172272211yescharacteristic
277C2#1172272911yescharacteristic
278C2#1172273011yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC22×C33C42
Frattini, hypercenterC23D3×#157
Fitting, prime radicalC23×C33C2×C4
socleC22×C33C22⋊C4
cosocleC22×C3×C3⋊Dic3C22
hypocenterC33#343
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
9. #88993C4semidirect
12. C3×#3792C4semidirect
44. #3792Dic3semidirect
147. C33#343semidirect
231. C32#4554semidirect
273. C3#10590208semidirect

Maximal subgroups

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

subgroupindex
C2×#889932
C2×C3×#37922
C2×#888072
#105902083
#45549