Finite Group Databasebeta

← C3×C4×#3724 #13158263

Subgroups — C3×C4×#3724 #13158263

Order 1728.

1154 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
2C2×C32×C4×Dic3C212yesmaximal
3C2×C3×#3724C212yesmaximal
4C3×#63579C212yesmaximal
5C3×C4×Dic3⋊C4—32nomaximal
6C4×#3724C311yesmaximal, characteristic, direct product factor
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C32×Dic3C2214yes—
11C3×C4×C3⋊Dic3C414yescomplemented
12C32×C4×Dic3—24no—
13C3×#3724C418yesdirect product factor
14C2×C32×C42—32no—
15C2×C4×C3⋊Dic3C2×C311yescharacteristic
16C2×C3×Dic3⋊C4—34no—
17C2×C3×C4×Dic3—22no—
18C2×C3×C4×Dic3—32no—
19C2×C3×C4×Dic3—61no—
20C2×C3×C4×Dic3D312yes—
21C2×C3×C4×Dic3C2×C312yes—
22C2×#3724C2×C312yes—
23C3×C3⋊#95—34no—
24#63579C2×C312yes—
25C23×C33C2311yescharacteristic
26C2×C33×C4C2×C412yes—
27C2×C33×C4D412yes—
28C2×C33×C4Q812yes—
29C2×C3×C3⋊Dic3—22no—
30C2×C3×C3⋊Dic3C2×C412yes—
31C2×C3×C3⋊Dic3C2×C412yes—
32C2×C3×C3⋊Dic3C2×C414yes—
33C2×C32×Dic3—24no—
34C2×C32×Dic3—24no—
35C2×C32×Dic3C2×C418yes—
36C3×C4×C4⋊C4—91no—
37C4×Dic3⋊C4—32no—
38C22×C32×C4—21no—
39C22×C32×C4—22no—
40C22×C32×C4—34no—
41C22×C32×C4—41no—
42C22×C32×C4C2×D312yes—
43C22×C32×C4C22×C311yescharacteristic
44C22×C3⋊Dic3C22×C311yescharacteristic
45C22×C3⋊Dic3C22×C311yescharacteristic
46C22×C3×Dic3—24no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—61no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3C2×D314yes—
52C22×C3×Dic3C22×C314yes—
53C32×C42—64no—
54C4×C3⋊Dic3C3×C414yescomplemented
55C3×Dic3⋊C4—316no—
56C3×C4×Dic3—28no—
57C3×C4×Dic3—24no—
58C3×C4×Dic3—24no—
59C3×C4×Dic3—38no—
60C3×C4×Dic3—64no—
61#3724C3×C418yesdirect product factor
62C22×C33C22×C411yescharacteristic
63C22×C33C22×C412yes—
64C22×C33C2×D411yescharacteristic
65C22×C33C2×Q811yescharacteristic
66C22×C33#4112yes—
67C33×C4C4⋊C414yescomplemented
68C3×C3⋊Dic3—22no—
69C3×C3⋊Dic3C4214yescomplemented
70C32×Dic3—28no—
71C2×C3×C4⋊C4—92no—
72C2×C3×C42—32no—
73C2×C3×C42—32no—
74C2×C3×C42—62no—
75C2×C3×C42—91no—
76C2×Dic3⋊C4—34no—
77C2×C4×Dic3—32no—
78C2×C4×Dic3—61no—
79C2×C4×Dic3C3×D312yes—
80C3×#95—92no—
81C3⋊#95—34no—
82C23×C32—21no—
83C23×C32—22no—
84C23×C32—41no—
85C23×C32C22×D312yes—
86C23×C32C23×C311yescharacteristic
87C2×C32×C4—22no—
88C2×C32×C4—22no—
89C2×C32×C4—22no—
90C2×C32×C4—24no—
91C2×C32×C4—24no—
92C2×C32×C4—24no—
93C2×C32×C4—38no—
94C2×C32×C4—42no—
95C2×C32×C4—42no—
96C2×C32×C4—42no—
97C2×C32×C4—64no—
98C2×C32×C4—64no—
99C2×C32×C4Dic614yes—
100C2×C32×C4C4×D314yes—
101C2×C32×C4#6714yes—
102C2×C32×C4C2×C3×C412yes—
103C2×C32×C4C3×D412yes—
104C2×C32×C4C3×Q812yes—
105C2×C3⋊Dic3—22no—
106C2×C3⋊Dic3C2×C3×C412yes—
107C2×C3⋊Dic3C2×C3×C414yes—
108C2×C3⋊Dic3C2×C3×C412yes—
109C2×C3×Dic3—28no—
110C2×C3×Dic3—28no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—24no—
115C2×C3×Dic3—28no—
116C2×C3×Dic3—34no—
117C2×C3×Dic3—38no—
118C2×C3×Dic3—34no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C2×C3×Dic3C4×D318yes—
125C2×C3×Dic3C2×C3×C418yes—
126C4×C4⋊C4—91noSylow
127C2×C33C2×C4211yescharacteristic
128C2×C33C2×C4⋊C411yescharacteristic
129C2×C33#11711yescharacteristic
130C2×C33C4×D412yes—
131C2×C33C4×Q812yes—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—34no—
135C22×C3×C4—34no—
136C22×C3×C4—41no—
137C22×C3×C4—64no—
138C22×C3×C4—91no—
139C22×C3×C4—91no—
140C22×C3×C4D3211yescharacteristic
141C22×C3×C4C2×C3×D312yes—
142C22×Dic3—32no—
143C22×Dic3—32no—
144C22×Dic3—61no—
145C22×Dic3—61no—
146C22×Dic3C2×C3×D314yes—
147C3×C4⋊C4—98no—
148C3×C42—68no—
149C3×C42—64no—
150C3×C42—64no—
151C3×C42—94no—
152Dic3⋊C4—316no—
153C4×Dic3—24no—
154C4×Dic3—38no—
155C4×Dic3—64no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—22no—
162C22×C32—24no—
163C22×C32—22no—
164C22×C32—24no—
165C22×C32—22no—
166C22×C32—41no—
167C22×C32—42no—
168C22×C32—41no—
169C22×C32—41no—
170C22×C32—42no—
171C22×C32C2×Dic612yes—
172C22×C32C2×C4×D314yes—
173C22×C32C2×C4×D312yes—
174C22×C32#23514yes—
175C22×C32C2×#6712yes—
176C22×C32C22×C3×C411yescharacteristic
177C22×C32C22×C3×C412yes—
178C22×C32C2×C3×D411yescharacteristic
179C22×C32C2×C3×Q811yescharacteristic
180C22×C32C3×#4112yes—
181C32×C4—24no—
182C32×C4—28no—
183C32×C4—44no—
184C32×C4—68no—
185C32×C4Dic3⋊C418yescomplemented
186C32×C4C3×C4⋊C414yescomplemented
187C3⋊Dic3—22no—
188C3⋊Dic3C3×C4214yescomplemented
189C3×Dic3—216no—
190C3×Dic3—28no—
191C3×Dic3—28no—
192C3×Dic3—38no—
193C3×Dic3—64no—
194C3×Dic3—64no—
195C3×Dic3—64no—
196C2×C4⋊C4—92no—
197C2×C42—32no—
198C2×C42—91no—
199#95—92no—
200C33C4×C4⋊C411yescharacteristic, complemented, Sylow
201C23×C3—22no—
202C23×C3—21no—
203C23×C3—41no—
204C23×C3C2×D3211yescharacteristic
205C23×C3C22×C3×D312yes—
206C2×C3×C4—24no—
207C2×C3×C4—22no—
208C2×C3×C4—24no—
209C2×C3×C4—22no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—38no—
213C2×C3×C4—38no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—68no—
218C2×C3×C4—68no—
219C2×C3×C4—68no—
220C2×C3×C4—64no—
221C2×C3×C4—64no—
222C2×C3×C4—64no—
223C2×C3×C4—64no—
224C2×C3×C4—94no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4—182no—
228C2×C3×C4#62412yes—
229C2×C3×C4#62512yes—
230C2×C3×C4#62712yes—
231C2×C3×C4C3×Dic614yes—
232C2×C3×C4C3×C4×D314yes—
233C2×C3×C4C3×#6714yes—
234C2×Dic3—24no—
235C2×Dic3—24no—
236C2×Dic3—34no—
237C2×Dic3—38no—
238C2×Dic3—34no—
239C2×Dic3—62no—
240C2×Dic3—62no—
241C2×Dic3—64no—
242C2×Dic3—64no—
243C2×Dic3—64no—
244C2×Dic3C3×C4×D318yes—
245C2×C32—22no—
246C2×C32—24no—
247C2×C32—21no—
248C2×C32—21no—
249C2×C32—22no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32—24no—
253C2×C32—22no—
254C2×C32—22no—
255C2×C32—41no—
256C2×C32—41no—
257C2×C32—41no—
258C2×C32—42no—
259C2×C32—42no—
260C2×C32C4×Dic614yes—
261C2×C32C42×D312yes—
262C2×C32#87212yes—
263C2×C32C2×Dic3⋊C412yes—
264C2×C32C4×#6714yes—
265C2×C32C2×C3×C4211yescharacteristic
266C2×C32C2×C3×C4⋊C411yescharacteristic
267C2×C32C3×#11711yescharacteristic
268C2×C32C3×C4×D412yes—
269C2×C32C3×C4×Q812yes—
270C22×C4—34no—
271C22×C4—91no—
272C22×C4—91no—
273C22×C4C3×D3211yescharacteristic
274C4⋊C4—98no—
275C42—64no—
276C42—94no—
277C22×C3—22no—
278C22×C3—24no—
279C22×C3—22no—
280C22×C3—24no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—42no—
288C22×C3—42no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3#379812yes—
293C22×C3C4×D3212yes—
294C22×C3C2×#62411yescharacteristic
295C22×C3C2×#62511yescharacteristic
296C22×C3C2×#62711yescharacteristic
297C22×C3C2×C3×Dic612yes—
298C22×C3C2×C3×C4×D314yes—
299C22×C3C2×C3×C4×D312yes—
300C22×C3C3×#23514yes—
301C22×C3C2×C3×#6712yes—
302C3×C4—28no—
303C3×C4—24no—
304C3×C4—44no—
305C3×C4—616no—
306C3×C4—68no—
307C3×C4—68no—
308C3×C4—94no—
309C3×C4—182no—
310C3×C4#372414yesdirect product factor
311C3×C4C3×Dic3⋊C418yescomplemented
312Dic3—28no—
313Dic3—38no—
314Dic3—64no—
315Dic3—64no—
316Dic3—64no—
317C32—22no—
318C32—21no—
319C32—41no—
320C32C4×Dic3⋊C412yescomplemented
321C32C3×C4×C4⋊C411yescharacteristic, complemented
322C23C2×C3×D3211yescharacteristic
323C2×C4—38no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—94no—
327C2×C4—92no—
328C2×C4—92no—
329C2×C4—182no—
330C2×C4C3×#62412yes—
331C2×C4C3×#62512yes—
332C2×C4C3×#62712yes—
333C2×C3—22no—
334C2×C3—21no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—21no—
338C2×C3—22no—
339C2×C3—24no—
340C2×C3—22no—
341C2×C3—24no—
342C2×C3—22no—
343C2×C3—41no—
344C2×C3—41no—
345C2×C3—41no—
346C2×C3—42no—
347C2×C3—42no—
348C2×C3#6387411yescharacteristic
349C2×C3C4×#62411yescharacteristic
350C2×C3C4×#62512yes—
351C2×C3C4×#62712yes—
352C2×C3C2×#372411yescharacteristic
353C2×C3C3×C4×Dic614yes—
354C2×C3C3×C42×D312yes—
355C2×C3C3×#87212yes—
356C2×C3C2×C3×Dic3⋊C412yes—
357C2×C3C3×C4×#6714yes—
358C22C3×#379812yes—
359C22C3×C4×D3212yes—
360C22C2×C3×#62411yescharacteristic
361C22C2×C3×#62511yescharacteristic
362C22C2×C3×#62711yescharacteristic
363C4—68no—
364C4—94no—
365C4—182no—
366C4C3×#372414yesdirect product factor
367C3—22no—
368C3—21no—
369C3—41no—
370C3C4×#372411yescharacteristic, direct product factor
371C3C3×C4×Dic3⋊C412yescomplemented
372C2C3×#6387411yescharacteristic
373C2C3×C4×#62411yescharacteristic
374C2C3×C4×#62512yes—
375C2C3×C4×#62712yes—
376C2C2×C3×#372411yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
6. C4×#3724C3direct (N × H)
11. C3×C4×C3⋊Dic3C4semidirect
13. C3×#3724C4direct (N × H)
54. C4×C3⋊Dic3C3×C4semidirect
61. #3724C3×C4direct (N × H)
67. C33×C4C4⋊C4semidirect
69. C3×C3⋊Dic3C42semidirect
185. C32×C4Dic3⋊C4semidirect
186. C32×C4C3×C4⋊C4semidirect
188. C3⋊Dic3C3×C42semidirect
200. C33C4×C4⋊C4semidirect
310. C3×C4#3724direct (N × H)
311. C3×C4C3×Dic3⋊C4semidirect
320. C32C4×Dic3⋊C4semidirect
321. C32C3×C4×C4⋊C4semidirect
366. C4C3×#3724direct (N × H)
370. C3C4×#3724direct (N × H)
371. C3C3×C4×Dic3⋊C4semidirect

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
C2×C32×C4×Dic32
C2×C3×#37242
C3×#635792
C3×C4×Dic3⋊C43
C4×#37243