Finite Group Databasebeta

← C3×#10591229 #13158078

Subgroups — C3×#10591229 #13158078

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C32×C3⋊#95C212yesmaximal
3C3×#63579C212yesmaximal
4C3×#63579C212yesmaximal
5C3×#4651—32nomaximal
6#10591229C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3C2214yes—
11C3×C4×C3⋊Dic3C414yescomplemented
12C2×C4×C3⋊Dic3C2×C311yescharacteristic
13C2×C3×C4×Dic3—32no—
14C2×C3×C4×Dic3—61no—
15C32×#95—32no—
16C3×C3⋊#95—22no—
17C3×C3⋊#95—34no—
18C3×C3⋊#95—34no—
19C3×C3⋊#95D312yes—
20C3×C3⋊#95C2×C312yes—
21#63579C2×C312yes—
22#63579C2×C312yes—
23C23×C33C2311yescharacteristic
24C2×C33×C4—22no—
25C2×C33×C4C2×C412yes—
26C2×C3×C3⋊Dic3—22no—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3C2×C412yes—
29C2×C3×C3⋊Dic3C2×C412yes—
30C2×C32×Dic3—24no—
31C2×C32×Dic3—24no—
32C2×C32×Dic3—24no—
33C3×#383—91no—
34#4651—32no—
35C22×C32×C4—21no—
36C22×C32×C4—22no—
37C22×C32×C4—34no—
38C22×C32×C4—41no—
39C22×C32×C4C2×D312yes—
40C22×C32×C4C22×C311yescharacteristic
41C22×C3⋊Dic3C22×C311yescharacteristic
42C22×C3⋊Dic3C22×C311yescharacteristic
43C22×C3×Dic3—24no—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—61no—
48C22×C3×Dic3C2×D314yes—
49C22×C3×Dic3C22×C314yes—
50C4×C3⋊Dic3C3×C414yescomplemented
51C3×C4×Dic3—38no—
52C3×C4×Dic3—64no—
53C22×C33C22×C411yescharacteristic
54C22×C33#4112yes—
55C22×C33#4112yes—
56C22×C33#4112yes—
57C33×C4—22no—
58C3×C3⋊Dic3—22no—
59C3×C3⋊Dic3—22no—
60C32×Dic3—44no—
61C2×C3×C42—91no—
62C2×C4×Dic3—32no—
63C2×C4×Dic3—61no—
64C3×#95—32no—
65C3×#95—32no—
66C3×#95—62no—
67C3×#95—92no—
68C3×#95—92no—
69C3⋊#95—34no—
70C3⋊#95—34no—
71C3⋊#95C3×D312yes—
72C23×C32—21no—
73C23×C32—22no—
74C23×C32—41no—
75C23×C32C22×D312yes—
76C23×C32C23×C311yescharacteristic
77C2×C32×C4—22no—
78C2×C32×C4—24no—
79C2×C32×C4—24no—
80C2×C32×C4—22no—
81C2×C32×C4—24no—
82C2×C32×C4—42no—
83C2×C32×C4—44no—
84C2×C32×C4—44no—
85C2×C32×C4—64no—
86C2×C32×C4—64no—
87C2×C32×C4—64no—
88C2×C32×C4C4×D314yes—
89C2×C32×C4C2×C3×C412yes—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3C2×C3×C412yes—
93C2×C3⋊Dic3C2×C3×C412yes—
94C2×C3×Dic3—28no—
95C2×C3×Dic3—24no—
96C2×C3×Dic3—24no—
97C2×C3×Dic3—24no—
98C2×C3×Dic3—24no—
99C2×C3×Dic3—24no—
100C2×C3×Dic3—24no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—44no—
104C2×C3×Dic3—44no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—64no—
108C2×C3×Dic3—64no—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—62no—
112#383—91noSylow
113C2×C33#11711yescharacteristic
114C2×C33#11711yescharacteristic
115C2×C33#11711yescharacteristic
116C2×C33#12614yes—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—34no—
120C22×C3×C4—34no—
121C22×C3×C4—41no—
122C22×C3×C4—64no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4D3211yescharacteristic
126C22×C3×C4C2×C3×D312yes—
127C22×Dic3—32no—
128C22×Dic3—32no—
129C22×Dic3—61no—
130C22×Dic3—61no—
131C22×Dic3C2×C3×D314yes—
132C3×C42—94no—
133C4×Dic3—38no—
134C4×Dic3—64no—
135C22×C32—22no—
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—21no—
139C22×C32—22no—
140C22×C32—24no—
141C22×C32—22no—
142C22×C32—24no—
143C22×C32—22no—
144C22×C32—41no—
145C22×C32—42no—
146C22×C32—42no—
147C22×C32—42no—
148C22×C32C2×C4×D312yes—
149C22×C32#23514yes—
150C22×C32#23712yes—
151C22×C32#23714yes—
152C22×C32#23912yes—
153C22×C32C22×C3×C411yescharacteristic
154C22×C32C3×#4112yes—
155C22×C32C3×#4112yes—
156C22×C32C3×#4112yes—
157C32×C4—22no—
158C32×C4—24no—
159C32×C4—24no—
160C32×C4—44no—
161C32×C4—44no—
162C32×C4—124no—
163C3⋊Dic3—22no—
164C3⋊Dic3—22no—
165C3×Dic3—48no—
166C3×Dic3—44no—
167C3×Dic3—44no—
168C3×Dic3—64no—
169C3×Dic3—64no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C2×C42—91no—
173#95—32no—
174#95—92no—
175#95—92no—
176C33#38311yescharacteristic, complemented, Sylow
177C23×C3—22no—
178C23×C3—21no—
179C23×C3—41no—
180C23×C3C2×D3211yescharacteristic
181C23×C3C22×C3×D312yes—
182C2×C3×C4—24no—
183C2×C3×C4—22no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—24no—
187C2×C3×C4—44no—
188C2×C3×C4—44no—
189C2×C3×C4—42no—
190C2×C3×C4—68no—
191C2×C3×C4—64no—
192C2×C3×C4—64no—
193C2×C3×C4—64no—
194C2×C3×C4—64no—
195C2×C3×C4—64no—
196C2×C3×C4—64no—
197C2×C3×C4—92no—
198C2×C3×C4—92no—
199C2×C3×C4—124no—
200C2×C3×C4—124no—
201C2×C3×C4—182no—
202C2×C3×C4—182no—
203C2×C3×C4#62412yes—
204C2×C3×C4C3×C4×D314yes—
205C2×Dic3—24no—
206C2×Dic3—24no—
207C2×Dic3—24no—
208C2×Dic3—34no—
209C2×Dic3—34no—
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—64no—
213C2×Dic3—64no—
214C2×Dic3—64no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×C32—22no—
218C2×C32—24no—
219C2×C32—22no—
220C2×C32—24no—
221C2×C32—21no—
222C2×C32—24no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—22no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—44no—
230C2×C32#87212yes—
231C2×C32#87712yes—
232C2×C32#87914yes—
233C2×C32#89212yes—
234C2×C32#89814yes—
235C2×C32C3×#11711yescharacteristic
236C2×C32C3×#11711yescharacteristic
237C2×C32C3×#11711yescharacteristic
238C2×C32C3×#12614yes—
239C22×C4—34no—
240C22×C4—91no—
241C22×C4—91no—
242C22×C4C3×D3211yescharacteristic
243C42—94no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—22no—
250C22×C3—24no—
251C22×C3—21no—
252C22×C3—22no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—41no—
257C22×C3#379612yes—
258C22×C3#379812yes—
259C22×C3#380512yes—
260C22×C3C2×#62411yescharacteristic
261C22×C3C2×C3×C4×D312yes—
262C22×C3C3×#23514yes—
263C22×C3C3×#23712yes—
264C22×C3C3×#23714yes—
265C22×C3C3×#23912yes—
266C3×C4—24no—
267C3×C4—24no—
268C3×C4—22no—
269C3×C4—44no—
270C3×C4—44no—
271C3×C4—128no—
272C3×C4—124no—
273C3×C4—124no—
274C3×C4—182no—
275C3×C4—182no—
276Dic3—44no—
277Dic3—64no—
278Dic3—64no—
279Dic3—64no—
280Dic3—64no—
281C32—22no—
282C32—21no—
283C32—41no—
284C32#465112yescomplemented
285C32C3×#38311yescharacteristic, complemented
286C23C2×C3×D3211yescharacteristic
287C2×C4—22no—
288C2×C4—64no—
289C2×C4—64no—
290C2×C4—64no—
291C2×C4—92no—
292C2×C4—92no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4C3×#62412yes—
296C2×C3—21no—
297C2×C3—22no—
298C2×C3—21no—
299C2×C3—22no—
300C2×C3—24no—
301C2×C3—24no—
302C2×C3—24no—
303C2×C3—21no—
304C2×C3—22no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—44no—
308C2×C3—41no—
309C2×C3#6384911yescharacteristic
310C2×C3#6386214yes—
311C2×C3#6387411yescharacteristic
312C2×C3#6395711yescharacteristic
313C2×C3C3×#87212yes—
314C2×C3C3×#87712yes—
315C2×C3C3×#87914yes—
316C2×C3C3×#89212yes—
317C2×C3C3×#89814yes—
318C22C3×#379612yes—
319C22C3×#379812yes—
320C22C3×#380512yes—
321C22C2×C3×#62411yescharacteristic
322C4—22no—
323C4—124no—
324C4—182no—
325C4—182no—
326C3—22no—
327C3—21no—
328C3—41no—
329C3#1059122911yescharacteristic, direct product factor
330C3C3×#465112yescomplemented
331C2C3×#6384911yescharacteristic
332C2C3×#6386214yes—
333C2C3×#6387411yescharacteristic
334C2C3×#6395711yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#383
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. #10591229C3direct (N × H)
11. C3×C4×C3⋊Dic3C4semidirect
50. C4×C3⋊Dic3C3×C4semidirect
176. C33#383semidirect
284. C32#4651semidirect
285. C32C3×#383semidirect
329. C3#10591229direct (N × H)
330. C3C3×#4651semidirect

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
C32×C3⋊#952
C3×#635792
C3×#46513
#105912293