Finite Group Databasebeta

← #13160252

Subgroups — #13160252

Order 1728.

1132 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2#11722201C211yesmaximal, characteristic
3C32×C3⋊#95C211yesmaximal, characteristic
4#11722061C212yesmaximal
5#11722061C212yesmaximal
6#10592350—31nomaximal
7#10591223—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C3×C4×C3⋊Dic3C414yescomplemented
14C2×C4×C3⋊Dic3D311yescharacteristic
15C2×C3×C4×Dic3—34no—
16C32×#95—31no—
17#63658—31no—
18#63658—32no—
19#63658—32no—
20#63658—34no—
21#63658—64no—
22C3×C3⋊#95D314yes—
23#63579—38no—
24#63579—38no—
25C23×C33C2311yescharacteristic
26C2×C33×C4—22no—
27C2×C33×C4C2×C412yes—
28C2×C3⋊(C3⋊Dic3)—22no—
29C2×C3⋊(C3⋊Dic3)—22no—
30C2×C3⋊(C3⋊Dic3)—22no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C412yes—
36C2×C32×Dic3—22no—
37C2×C32×Dic3—22no—
38C2×C32×Dic3—22no—
39#4937—91no—
40#4651—94no—
41C22×C32×C4—24no—
42C22×C32×C4—32no—
43C22×C32×C4C2×D314yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—32no—
46C22×C3⋊Dic3—38no—
47C22×C3⋊Dic3—68no—
48C22×C3⋊Dic3C2×D311yescharacteristic
49C22×C3⋊Dic3C2×D311yescharacteristic
50C22×C3×Dic3—34no—
51C22×C3×Dic3—34no—
52C22×C3×Dic3C2×D318yes—
53C4×C3⋊Dic3Dic314yescomplemented
54C3×C4×Dic3—316no—
55C22×C33C22×C411yescharacteristic
56C22×C33#4112yes—
57C22×C33#4111yescharacteristic
58C22×C33#4112yes—
59C22×C33#4111yescharacteristic
60C33×C4—22no—
61C3⋊(C3⋊Dic3)—42no—
62C3×C3⋊Dic3—22no—
63C3×C3⋊Dic3—22no—
64C32×Dic3—42no—
65C2×C3×C42—91no—
66C2×C4×Dic3—34no—
67C3×#95—34no—
68C3⋊#95—98no—
69C3⋊#95—98no—
70C3⋊#95—94no—
71C3⋊#95—91no—
72C3⋊#95—92no—
73C3⋊#95—92no—
74C3⋊#95—184no—
75C3⋊#95C3⋊D311yescharacteristic
76C23×C32—24no—
77C23×C32C22×D314yes—
78C23×C32C22×D311yescharacteristic
79C2×C32×C4—22no—
80C2×C32×C4—28no—
81C2×C32×C4—28no—
82C2×C32×C4—48no—
83C2×C32×C4—62no—
84C2×C32×C4—62no—
85C2×C32×C4—62no—
86C2×C32×C4C4×D318yes—
87C2×C32×C4C2×Dic312yes—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—616no—
92C2×C3⋊Dic3—68no—
93C2×C3⋊Dic3—62no—
94C2×C3⋊Dic3—62no—
95C2×C3⋊Dic3—62no—
96C2×C3⋊Dic3—68no—
97C2×C3⋊Dic3—68no—
98C2×C3⋊Dic3—128no—
99C2×C3⋊Dic3—128no—
100C2×C3⋊Dic3C2×Dic312yes—
101C2×C3⋊Dic3C2×Dic312yes—
102C2×C3×Dic3—28no—
103C2×C3×Dic3—28no—
104C2×C3×Dic3—28no—
105C2×C3×Dic3—38no—
106C2×C3×Dic3—38no—
107C2×C3×Dic3—68no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—64no—
110#383—271noSylow
111C2×C33#11711yescharacteristic
112C2×C33#11711yescharacteristic
113C2×C33#11711yescharacteristic
114C2×C33#12612yes—
115C2×C33#12612yes—
116C22×C3×C4—24no—
117C22×C3×C4—38no—
118C22×C3×C4—91no—
119C22×C3×C4—91no—
120C22×C3×C4D3214yes—
121C22×C3×C4C2×C3⋊D311yescharacteristic
122C22×Dic3—34no—
123C22×Dic3—34no—
124C22×Dic3—98no—
125C22×Dic3—92no—
126C22×Dic3—188no—
127C22×Dic3C2×C3⋊D312yes—
128C3×C42—94no—
129C4×Dic3—316no—
130C22×C32—24no—
131C22×C32—28no—
132C22×C32—24no—
133C22×C32—24no—
134C22×C32—28no—
135C22×C32C2×C4×D314yes—
136C22×C32#23518yes—
137C22×C32#23511yescharacteristic
138C22×C32#23512yes—
139C22×C32#23511yescharacteristic
140C22×C32#23512yes—
141C22×C32#23714yes—
142C22×C32#23718yes—
143C22×C32#23914yes—
144C22×C32C22×Dic311yescharacteristic
145C32×C4—22no—
146C32×C4—28no—
147C32×C4—48no—
148C32×C4—122no—
149C3⋊Dic3—22no—
150C3⋊Dic3—22no—
151C3⋊Dic3—1216no—
152C3⋊Dic3—122no—
153C3⋊Dic3—128no—
154C3×Dic3—48no—
155C3×Dic3—68no—
156C3×Dic3—68no—
157C2×C42—91no—
158#95—31no—
159#95—272no—
160#95—272no—
161#95—271no—
162C33#38311yescharacteristic, complemented, Sylow
163C23×C3—24no—
164C23×C3C2×D3214yes—
165C23×C3C22×C3⋊D311yescharacteristic
166C2×C3×C4—28no—
167C2×C3×C4—22no—
168C2×C3×C4—28no—
169C2×C3×C4—48no—
170C2×C3×C4—68no—
171C2×C3×C4—68no—
172C2×C3×C4—68no—
173C2×C3×C4—92no—
174C2×C3×C4—92no—
175C2×C3×C4—182no—
176C2×C3×C4—181no—
177C2×C3×C4—181no—
178C2×C3×C4D3×Dic318yes—
179C2×C3×C4C4×C3⋊D312yes—
180C2×Dic3—22no—
181C2×Dic3—22no—
182C2×Dic3—22no—
183C2×Dic3—38no—
184C2×Dic3—38no—
185C2×Dic3—64no—
186C2×Dic3—68no—
187C2×Dic3—64no—
188C2×Dic3—1816no—
189C2×Dic3—188no—
190C2×Dic3—188no—
191C2×Dic3—188no—
192C2×Dic3—182no—
193C2×Dic3—182no—
194C2×Dic3—182no—
195C2×Dic3—368no—
196C2×Dic3—368no—
197C2×C32—24no—
198C2×C32—24no—
199C2×C32—28no—
200C2×C32—28no—
201C2×C32—24no—
202C2×C32#87214yes—
203C2×C32#87612yes—
204C2×C32#87612yes—
205C2×C32#87714yes—
206C2×C32#87918yes—
207C2×C32#89214yes—
208C2×C32#89818yes—
209C2×C32#92611yescharacteristic
210C2×C32#92611yescharacteristic
211C2×C32#92611yescharacteristic
212C22×C4—32no—
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—272no—
216C22×C4D3×C3⋊D311yescharacteristic
217C42—94no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—28no—
221C22×C3—28no—
222C22×C3—24no—
223C22×C3#379514yes—
224C22×C3#379818yes—
225C22×C3#379914yes—
226C22×C3C2×D3×Dic314yes—
227C22×C3#380418yes—
228C22×C3C2×C4×C3⋊D311yescharacteristic
229C22×C3#382812yes—
230C22×C3#383011yescharacteristic
231C22×C3#383012yes—
232C22×C3#383211yescharacteristic
233C3×C4—22no—
234C3×C4—28no—
235C3×C4—48no—
236C3×C4—128no—
237C3×C4—182no—
238C3×C4—182no—
239Dic3—42no—
240Dic3—68no—
241Dic3—68no—
242Dic3—3616no—
243Dic3—362no—
244Dic3—368no—
245C32—24no—
246C32#465114yescomplemented
247C32#493711yescharacteristic, complemented
248C23C2×D3×C3⋊D311yescharacteristic
249C2×C4—22no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—92no—
254C2×C4—92no—
255C2×C4—181no—
256C2×C4—181no—
257C2×C4—182no—
258C2×C4—542no—
259C2×C4—542no—
260C2×C4—542no—
261C2×C4Dic3×C3⋊D312yes—
262C2×C3—24no—
263C2×C3—24no—
264C2×C3—28no—
265C2×C3—24no—
266C2×C3—28no—
267C2×C3#6384114yes—
268C2×C3#6385514yes—
269C2×C3#6385918yes—
270C2×C3#6386818yes—
271C2×C3#6395514yes—
272C2×C3#6408111yescharacteristic
273C2×C3#6408611yescharacteristic
274C2×C3#6408812yes—
275C2×C3#6410111yescharacteristic
276C2×C3#6410712yes—
277C22#8901411yescharacteristic
278C22#8901811yescharacteristic
279C22#8902012yes—
280C22#8902812yes—
281C22C2×Dic3×C3⋊D311yescharacteristic
282C4—22no—
283C4—122no—
284C4—182no—
285C4—182no—
286C4—1082no—
287C3—24no—
288C3#1059122314yescomplemented
289C3#1059235011yescharacteristic, complemented
290C2#1172340411yescharacteristic
291C2#1172342812yes—
292C2#1172343711yescharacteristic
293C2#1172344612yes—
294C2#1172359511yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, prime radical, cosocleC23×C33C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#383
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
13. C3×C4×C3⋊Dic3C4semidirect
53. C4×C3⋊Dic3Dic3semidirect
162. C33#383semidirect
246. C32#4651semidirect
247. C32#4937semidirect
288. C3#10591223semidirect
289. C3#10592350semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊Dic32
#117222012
C32×C3⋊#952
#117220612
#105923503
#105912233