Finite Group Databasebeta

← C3×C4×#3722 #13158206

Subgroups — C3×C4×#3722 #13158206

Order 1728.

1538 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C32×C4×Dic3C212yesmaximal, complemented
3C2×C3×#3722C212yesmaximal
4C3×#63579C212yesmaximal, complemented
5C3×C4×(C2×D3)⋊C4—32nomaximal
6C4×#3722C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×C4×C3⋊D3—24no—
11C2×C3×C4×C3⋊D3C414yescomplemented
12C22×C32×Dic3C2214yes—
13C32×C4×Dic3—24no—
14C3×#3722C418yesdirect product factor
15C22×C4×C3⋊D3C2×C311yescharacteristic
16C22×C3×C4×D3—32no—
17C22×C3×C4×D3—61no—
18C2×C32×C42—32no—
19C2×C3×(C2×D3)⋊C4—34no—
20C2×C3×C4×Dic3—22no—
21C2×C3×C4×Dic3D312yescomplemented
22C2×C3×C4×Dic3C2×C312yescomplemented
23C2×#3722C2×C312yes—
24C3×C3⋊#95—34no—
25#63579C2×C312yescomplemented
26C23×C33C2311yescharacteristic
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3C2×C414yes—
30C22×C3×C3⋊D3C2×C412yes—
31C2×C33×C4C2×C412yes—
32C2×C33×C4D414yes—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3C2×C412yes—
35C3×C4×C3⋊D3—28no—
36C2×C32×Dic3—24no—
37C2×C32×Dic3—24no—
38C2×C32×Dic3C2×C418yescomplemented
39C3×C4×C22⋊C4—91no—
40C4×(C2×D3)⋊C4—32no—
41C23×C3⋊D3C22×C311yescharacteristic
42C23×C3×D3—32no—
43C23×C3×D3—61no—
44C22×C32×C4—21no—
45C22×C32×C4—22no—
46C22×C32×C4—34no—
47C22×C32×C4—41no—
48C22×C32×C4C2×D312yes—
49C22×C32×C4C22×C311yescharacteristic
50C22×C3⋊Dic3C22×C311yescharacteristic
51C2×C4×C3⋊D3—24no—
52C2×C4×C3⋊D3C3×C414yescomplemented
53C22×C3×Dic3—24no—
54C22×C3×Dic3—32no—
55C22×C3×Dic3—61no—
56C22×C3×Dic3C2×D314yes—
57C22×C3×Dic3C22×C314yes—
58C2×C3×C4×D3—38no—
59C2×C3×C4×D3—64no—
60C2×C3×C4×D3—68no—
61C2×C3×C4×D3—64no—
62C2×C3×C4×D3—64no—
63C32×C42—64no—
64C3×(C2×D3)⋊C4—316no—
65C3×C4×Dic3—28no—
66C3×C4×Dic3—24no—
67C3×C4×Dic3—24no—
68#3722C3×C418yesdirect product factor
69C22×C33C22×C411yescharacteristic
70C22×C33C22×C412yes—
71C22×C33C2×D412yes—
72C22×C33#4112yes—
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—28no—
76C2×C3×C3⋊D3C4214yescomplemented
77C33×C4C22⋊C414yescomplemented
78C3×C3⋊Dic3—22no—
79C32×Dic3—28no—
80C23×C3×C4—91no—
81C22×C4×D3—32no—
82C22×C4×D3—61no—
83C2×C3×C22⋊C4—92no—
84C2×C3×C42—32no—
85C2×C3×C42—32no—
86C2×C3×C42—62no—
87C2×(C2×D3)⋊C4—34no—
88C2×C4×Dic3C3×D312yescomplemented
89C3×#95—92no—
90C3⋊#95—34no—
91C23×C32—21no—
92C23×C32—22no—
93C23×C32—41no—
94C23×C32C22×D312yes—
95C23×C32C23×C311yescharacteristic
96C22×C3⋊D3—22no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3C2×C3×C414yes—
99C22×C3⋊D3C2×C3×C412yes—
100C22×C3×D3—38no—
101C22×C3×D3—34no—
102C22×C3×D3—64no—
103C22×C3×D3—62no—
104C22×C3×D3—64no—
105C22×C3×D3—64no—
106C22×C3×D3—62no—
107C22×C3×D3—64no—
108C22×C3×D3—62no—
109C2×C32×C4—22no—
110C2×C32×C4—24no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—24no—
114C2×C32×C4—38no—
115C2×C32×C4—42no—
116C2×C32×C4—44no—
117C2×C32×C4—64no—
118C2×C32×C4—64no—
119C2×C32×C4C4×D314yes—
120C2×C32×C4D1214yes—
121C2×C32×C4#6714yes—
122C2×C32×C4C2×C3×C412yes—
123C2×C32×C4C3×D414yes—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3C2×C3×C412yes—
126C4×C3⋊D3—28no—
127C2×C3×Dic3—28no—
128C2×C3×Dic3—28no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—28no—
134C2×C3×Dic3—34no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—62no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3C4×D318yescomplemented
140C2×C3×Dic3C2×C3×C418yescomplemented
141C3×C4×D3—616no—
142C3×C4×D3—616no—
143C4×C22⋊C4—91noSylow
144C2×C33C2×C4211yescharacteristic
145C2×C33C2×C22⋊C411yescharacteristic
146C2×C33#11711yescharacteristic
147C2×C33C4×D414yes—
148C3×C3⋊D3—24no—
149C24×C3—91no—
150C23×D3—32no—
151C23×D3—61no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—34no—
155C22×C3×C4—34no—
156C22×C3×C4—41no—
157C22×C3×C4—64no—
158C22×C3×C4—91no—
159C22×C3×C4—94no—
160C22×C3×C4—184no—
161C22×C3×C4D3211yescharacteristic
162C22×C3×C4C2×C3×D312yes—
163C22×Dic3—32no—
164C22×Dic3—61no—
165C22×Dic3C2×C3×D314yes—
166C2×C4×D3—38no—
167C2×C4×D3—64no—
168C2×C4×D3—68no—
169C2×C4×D3—64no—
170C2×C4×D3—64no—
171C3×C22⋊C4—98no—
172C3×C42—68no—
173C3×C42—64no—
174C3×C42—64no—
175(C2×D3)⋊C4—316no—
176C4×Dic3—24no—
177C22×C32—22no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32—22no—
181C22×C32—24no—
182C22×C32—22no—
183C22×C32—22no—
184C22×C32—24no—
185C22×C32—22no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—42no—
190C22×C32C2×C4×D314yes—
191C22×C32C2×C4×D312yes—
192C22×C32C2×D1212yes—
193C22×C32#23514yes—
194C22×C32C2×#6712yes—
195C22×C32C22×C3×C411yescharacteristic
196C22×C32C22×C3×C412yes—
197C22×C32C2×C3×D412yes—
198C22×C32C3×#4112yes—
199C2×C3⋊D3—22no—
200C2×C3⋊D3—22no—
201C2×C3⋊D3—28no—
202C2×C3⋊D3C3×C4214yescomplemented
203C2×C3×D3—38no—
204C2×C3×D3—64no—
205C2×C3×D3—64no—
206C2×C3×D3—64no—
207C2×C3×D3—616no—
208C2×C3×D3—64no—
209C2×C3×D3—68no—
210C2×C3×D3—64no—
211C2×C3×D3—68no—
212C32×C4—24no—
213C32×C4—28no—
214C32×C4—44no—
215C32×C4—68no—
216C32×C4(C2×D3)⋊C418yescomplemented
217C32×C4C3×C22⋊C414yescomplemented
218C3⋊Dic3—22no—
219C3×Dic3—216no—
220C3×Dic3—28no—
221C3×Dic3—28no—
222C3×Dic3—64no—
223C3×Dic3—64no—
224C23×C4—91no—
225C2×C22⋊C4—92no—
226C2×C42—32no—
227#95—92no—
228C33C4×C22⋊C411yescharacteristic, complemented, Sylow
229C23×C3—22no—
230C23×C3—21no—
231C23×C3—41no—
232C23×C3—94no—
233C23×C3—92no—
234C23×C3—182no—
235C23×C3—182no—
236C23×C3C2×D3211yescharacteristic
237C23×C3C22×C3×D312yes—
238C22×D3—38no—
239C22×D3—34no—
240C22×D3—62no—
241C22×D3—64no—
242C22×D3—64no—
243C22×D3—64no—
244C22×D3—62no—
245C22×D3—64no—
246C22×D3—62no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—38no—
253C2×C3×C4—38no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—68no—
257C2×C3×C4—68no—
258C2×C3×C4—68no—
259C2×C3×C4—64no—
260C2×C3×C4—64no—
261C2×C3×C4—64no—
262C2×C3×C4—64no—
263C2×C3×C4—92no—
264C2×C3×C4—182no—
265C2×C3×C4—188no—
266C2×C3×C4#62412yes—
267C2×C3×C4#62614yes—
268C2×C3×C4C3×C4×D314yes—
269C2×C3×C4C3×D1214yes—
270C2×C3×C4C3×#6714yes—
271C2×Dic3—24no—
272C2×Dic3—24no—
273C2×Dic3—34no—
274C2×Dic3—62no—
275C2×Dic3—64no—
276C2×Dic3—62no—
277C2×Dic3—62no—
278C2×Dic3C3×C4×D318yescomplemented
279C4×D3—616no—
280C4×D3—616no—
281C2×C32—22no—
282C2×C32—24no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—24no—
286C2×C32—21no—
287C2×C32—22no—
288C2×C32—22no—
289C2×C32—24no—
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—44no—
294C2×C32C42×D312yes—
295C2×C32#87212yes—
296C2×C32C4×D1214yes—
297C2×C32C2×(C2×D3)⋊C412yes—
298C2×C32C4×#6714yes—
299C2×C32C2×C3×C4211yescharacteristic
300C2×C32C2×C3×C22⋊C411yescharacteristic
301C2×C32C3×#11711yescharacteristic
302C2×C32C3×C4×D414yes—
303C3⋊D3—24no—
304C3×D3—68no—
305C3×D3—68no—
306C24—91no—
307C22×C4—34no—
308C22×C4—91no—
309C22×C4—94no—
310C22×C4—184no—
311C22×C4C3×D3211yescharacteristic
312C22⋊C4—98no—
313C42—64no—
314C22×C3—24no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—94no—
328C22×C3—182no—
329C22×C3—188no—
330C22×C3—182no—
331C22×C3#379812yes—
332C22×C3C4×D3212yes—
333C22×C3C2×#62411yescharacteristic
334C22×C3C2×#62612yes—
335C22×C3C2×C3×C4×D314yes—
336C22×C3C2×C3×C4×D312yes—
337C22×C3C2×C3×D1212yes—
338C22×C3C3×#23514yes—
339C22×C3C2×C3×#6712yes—
340C2×D3—38no—
341C2×D3—64no—
342C2×D3—64no—
343C2×D3—64no—
344C2×D3—616no—
345C2×D3—64no—
346C2×D3—68no—
347C2×D3—64no—
348C2×D3—68no—
349C3×C4—24no—
350C3×C4—28no—
351C3×C4—44no—
352C3×C4—616no—
353C3×C4—68no—
354C3×C4—68no—
355C3×C4—182no—
356C3×C4#372214yesdirect product factor
357C3×C4C3×(C2×D3)⋊C418yescomplemented
358Dic3—28no—
359Dic3—64no—
360Dic3—64no—
361C32—22no—
362C32—21no—
363C32—41no—
364C32C4×(C2×D3)⋊C412yescomplemented
365C32C3×C4×C22⋊C411yescharacteristic, complemented
366C23—94no—
367C23—92no—
368C23—182no—
369C23—182no—
370C23C2×C3×D3211yescharacteristic
371C2×C4—38no—
372C2×C4—64no—
373C2×C4—64no—
374C2×C4—92no—
375C2×C4—182no—
376C2×C4—188no—
377C2×C4C3×#62412yes—
378C2×C4C3×#62614yes—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—21no—
382C2×C3—22no—
383C2×C3—21no—
384C2×C3—22no—
385C2×C3—24no—
386C2×C3—24no—
387C2×C3—24no—
388C2×C3—41no—
389C2×C3—41no—
390C2×C3—41no—
391C2×C3—44no—
392C2×C3—184no—
393C2×C3#6387411yescharacteristic
394C2×C3C4×#62411yescharacteristic
395C2×C3C4×#62614yes—
396C2×C3C2×#372211yescharacteristic
397C2×C3C3×C42×D312yes—
398C2×C3C3×#87212yes—
399C2×C3C3×C4×D1214yes—
400C2×C3C2×C3×(C2×D3)⋊C412yes—
401C2×C3C3×C4×#6714yes—
402D3—68no—
403D3—68no—
404C22—94no—
405C22—182no—
406C22—182no—
407C22—188no—
408C22C3×#379812yes—
409C22C3×C4×D3212yes—
410C22C2×C3×#62411yescharacteristic
411C22C2×C3×#62612yes—
412C4—68no—
413C4—182no—
414C4C3×#372214yesdirect product factor
415C3—21no—
416C3—22no—
417C3—41no—
418C3C4×#372211yescharacteristic, direct product factor
419C3C3×C4×(C2×D3)⋊C412yescomplemented
420C2—184no—
421C2C3×#6387411yescharacteristic
422C2C3×C4×#62411yescharacteristic
423C2C3×C4×#62614yes—
424C2C2×C3×#372211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C2×C3×C42
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypocenterC32C3×C4×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×Dic3C2semidirect
4. C3×#63579C2semidirect
6. C4×#3722C3direct (N × H)
11. C2×C3×C4×C3⋊D3C4semidirect
14. C3×#3722C4direct (N × H)
21. C2×C3×C4×Dic3D3semidirect
22. C2×C3×C4×Dic3C2×C3semidirect
25. #63579C2×C3semidirect
38. C2×C32×Dic3C2×C4semidirect
52. C2×C4×C3⋊D3C3×C4semidirect
68. #3722C3×C4direct (N × H)
76. C2×C3×C3⋊D3C42semidirect
77. C33×C4C22⋊C4semidirect
88. C2×C4×Dic3C3×D3semidirect
139. C2×C3×Dic3C4×D3semidirect
140. C2×C3×Dic3C2×C3×C4semidirect
202. C2×C3⋊D3C3×C42semidirect
216. C32×C4(C2×D3)⋊C4semidirect
217. C32×C4C3×C22⋊C4semidirect
228. C33C4×C22⋊C4semidirect
278. C2×Dic3C3×C4×D3semidirect
356. C3×C4#3722direct (N × H)
357. C3×C4C3×(C2×D3)⋊C4semidirect
364. C32C4×(C2×D3)⋊C4semidirect
365. C32C3×C4×C22⋊C4semidirect
414. C4C3×#3722direct (N × H)
418. C3C4×#3722direct (N × H)
419. C3C3×C4×(C2×D3)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C32×C4×Dic32
C2×C3×#37222
C3×#635792
C3×C4×(C2×D3)⋊C43
C4×#37223