Finite Group Databasebeta

← C2×C3×#63841 #13174383

Subgroups — C2×C3×#63841 #13174383

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D3)⋊Dic3C212yesmaximal
5C2×C3×Dic32C212yesmaximal, complemented
6C3×#63841C218yesmaximal, direct product factor
7C2×C3×#926—31nomaximal
8C2×C3×#892—31nomaximal
9C2×#63841C311yesmaximal, characteristic, direct product factor
10C23×C32×D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2212yes—
13C22×C32×Dic3C2212yes—
14C22×C32×Dic3C2211yescharacteristic
15C2×C32×C4×D3C418yescomplemented
16C2×C32×C4×D3C2214yes—
17C3×C4⋊(C3⋊Dic3)C2214yescomplemented
18C32×C4⋊Dic3C2214yescomplemented
19C3×(C2×D3)⋊Dic3C2218yes—
20C3×Dic32C2218yescomplemented
21C23×C32×C4—31no—
22C22×C3×C4×D3—21no—
23C22×C3×C4×D3D311yescharacteristic
24C22×C3×C4×D3C2×C311yescharacteristic
25C2×C32×C4⋊C4—31no—
26C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
27C2×C3×#217—32no—
28C2×C3×(C2×D3)⋊C4—32no—
29C3×#926—38no—
30C2×C3×C4⋊Dic3—21no—
31C2×C3×C4⋊Dic3—31no—
32C2×C3×C4⋊Dic3—31no—
33C2×C3×C4⋊Dic3—61no—
34C2×C3×C4⋊Dic3D311yescharacteristic, complemented
35C2×C3×C4⋊Dic3C2×C311yescharacteristic, complemented
36C2×C3×C4×Dic3—32no—
37C2×C3×C4×Dic3—32no—
38C3×#892—38no—
39C2×(C2×D3)⋊Dic3C2×C312yes—
40C2×Dic32C2×C312yescomplemented
41#63841C2×C318yesdirect product factor
42C23×C33C2311yescharacteristic
43C22×C32×D3—22no—
44C22×C32×D3—22no—
45C22×C32×D3C2×C414yes—
46C22×C32×D3C2312yes—
47C2×C33×C4C2×C414yes—
48C2×C33×C4C2312yes—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3C2314yes—
51C2×C32×Dic3—24no—
52C2×C32×Dic3C2×C414yes—
53C2×C32×Dic3C2314yes—
54C2×C32×Dic3C2312yes—
55C32×C4×D3C2×C4116yescomplemented
56C2×C3×#117—91no—
57C2×#926—31no—
58C2×#892—31no—
59C24×C32—31no—
60C23×C3×D3—21no—
61C23×C3×D3C2×D311yescharacteristic
62C23×C3×D3C22×C311yescharacteristic
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—31no—
67C22×C32×C4—38no—
68C22×C32×C4—34no—
69C22×C32×C4—32no—
70C22×C32×C4—41no—
71C22×C32×C4C2×D311yescharacteristic
72C22×C32×C4C2×D311yescharacteristic
73C22×C32×C4C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C312yes—
75C22×C3×Dic3—21no—
76C22×C3×Dic3—22no—
77C22×C3×Dic3—32no—
78C22×C3×Dic3—32no—
79C22×C3×Dic3—62no—
80C22×C3×Dic3C2×D312yes—
81C22×C3×Dic3C2×D311yescharacteristic
82C22×C3×Dic3C22×C312yes—
83C22×C3×Dic3C22×C311yescharacteristic
84C2×C3×C4×D3—28no—
85C2×C3×C4×D3—24no—
86C2×C3×C4×D3Dic318yescomplemented
87C2×C3×C4×D3C3×C418yescomplemented
88C2×C3×C4×D3C2×D314yes—
89C2×C3×C4×D3C22×C314yes—
90C32×C4⋊C4—34no—
91C4⋊(C3⋊Dic3)C22×C314yescomplemented
92C3×#217—38no—
93C3×(C2×D3)⋊C4—38no—
94C3×C4⋊Dic3—24no—
95C3×C4⋊Dic3—34no—
96C3×C4⋊Dic3—34no—
97C3×C4⋊Dic3—64no—
98C3×C4⋊Dic3C2×D314yescomplemented
99C3×C4⋊Dic3C22×C314yescomplemented
100C3×C4×Dic3—38no—
101C3×C4×Dic3—38no—
102(C2×D3)⋊Dic3C22×C318yes—
103Dic32C22×C318yescomplemented
104C22×C33C22×C412yes—
105C22×C33#4112yes—
106C22×C33#4112yes—
107C22×C33C2411yescharacteristic
108C2×C32×D3—28no—
109C2×C32×D3—22no—
110C2×C32×D3—22no—
111C2×C32×D3C22×C414yes—
112C33×C4C22×C414yes—
113C3×C3⋊Dic3—24no—
114C32×Dic3—24no—
115C32×Dic3C22×C414yes—
116C23×C3×C4—31no—
117C23×C3×C4—31no—
118C23×C3×C4—61no—
119C22×C4×D3C3×D311yescharacteristic
120C3×#117—98no—
121C2×C3×C4⋊C4—31no—
122C2×C3×C4⋊C4—31no—
123C2×C3×C4⋊C4—61no—
124C2×C3×C4⋊C4—91no—
125C2×C3×C22⋊C4—92no—
126C2×C3×C42—92no—
127C2×#217—32no—
128C2×(C2×D3)⋊C4—32no—
129#926—38no—
130C2×C4⋊Dic3—31no—
131C2×C4⋊Dic3—31no—
132C2×C4⋊Dic3—61no—
133C2×C4⋊Dic3C3×D311yescharacteristic, complemented
134C2×C4×Dic3—32no—
135C2×C4×Dic3—32no—
136#892—38no—
137C23×C32—21no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—34no—
141C23×C32—32no—
142C23×C32—41no—
143C23×C32—62no—
144C23×C32—62no—
145C23×C32C22×D311yescharacteristic
146C23×C32C22×D311yescharacteristic
147C23×C32C23×C311yescharacteristic
148C22×C3×D3—24no—
149C22×C3×D3—24no—
150C22×C3×D3—22no—
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—24no—
155C22×C3×D3—22no—
156C22×C3×D3C2×Dic314yes—
157C22×C3×D3C2×C3×C414yes—
158C22×C3×D3C22×D312yes—
159C22×C3×D3C23×C312yes—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—24no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4—34no—
167C2×C32×C4—32no—
168C2×C32×C4—316no—
169C2×C32×C4—34no—
170C2×C32×C4—44no—
171C2×C32×C4—42no—
172C2×C32×C4—64no—
173C2×C32×C4C4×D314yes—
174C2×C32×C4C2×Dic314yes—
175C2×C32×C4C2×C3×C414yes—
176C2×C32×C4C22×D312yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C23×C312yes—
179C2×C3⋊Dic3—24no—
180C2×C3⋊Dic3C23×C314yes—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—34no—
188C2×C3×Dic3—44no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—124no—
193C2×C3×Dic3C2×Dic314yes—
194C2×C3×Dic3C2×C3×C414yes—
195C2×C3×Dic3C22×D314yes—
196C2×C3×Dic3C22×D312yes—
197C2×C3×Dic3C23×C314yes—
198C2×C3×Dic3C23×C312yes—
199C3×C4×D3—216no—
200C3×C4×D3C2×Dic3116yescomplemented
201C3×C4×D3C2×C3×C4116yescomplemented
202C2×#117—91noSylow
203C2×C33#11714yescomplemented
204C2×C33C23×C411yescharacteristic
205C2×C33C2×#4111yescharacteristic
206C2×C33C2×#4111yescharacteristic
207C32×D3—24no—
208C24×C3—31no—
209C24×C3—31no—
210C24×C3—61no—
211C23×D3C2×C3×D311yescharacteristic
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—31no—
216C22×C3×C4—31no—
217C22×C3×C4—38no—
218C22×C3×C4—38no—
219C22×C3×C4—34no—
220C22×C3×C4—34no—
221C22×C3×C4—32no—
222C22×C3×C4—32no—
223C22×C3×C4—41no—
224C22×C3×C4—68no—
225C22×C3×C4—64no—
226C22×C3×C4—61no—
227C22×C3×C4—62no—
228C22×C3×C4—92no—
229C22×C3×C4D3211yescharacteristic
230C22×C3×C4C2×C3×D311yescharacteristic
231C22×C3×C4C2×C3×D311yescharacteristic
232C22×Dic3—32no—
233C22×Dic3—32no—
234C22×Dic3—62no—
235C22×Dic3C2×C3×D312yes—
236C22×Dic3C2×C3×D311yescharacteristic
237C2×C4×D3C3×Dic318yescomplemented
238C2×C4×D3C2×C3×D314yes—
239C3×C4⋊C4—34no—
240C3×C4⋊C4—34no—
241C3×C4⋊C4—64no—
242C3×C4⋊C4—94no—
243C3×C22⋊C4—98no—
244C3×C42—98no—
245#217—38no—
246(C2×D3)⋊C4—38no—
247C4⋊Dic3—34no—
248C4⋊Dic3—34no—
249C4⋊Dic3—64no—
250C4⋊Dic3C2×C3×D314yescomplemented
251C4×Dic3—38no—
252C4×Dic3—38no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—22no—
260C22×C32—22no—
261C22×C32—22no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—34no—
266C22×C32—42no—
267C22×C32—42no—
268C22×C32—42no—
269C22×C32—41no—
270C22×C32—68no—
271C22×C32—62no—
272C22×C32—62no—
273C22×C32C2×C4×D312yes—
274C22×C32#23512yes—
275C22×C32#23512yes—
276C22×C32#23712yes—
277C22×C32#23912yes—
278C22×C32C22×Dic312yes—
279C22×C32C22×C3×C412yes—
280C22×C32C3×#4112yes—
281C22×C32C3×#4112yes—
282C22×C32C23×D311yescharacteristic
283C22×C32C23×D311yescharacteristic
284C22×C32C24×C311yescharacteristic
285C2×C3×D3—216no—
286C2×C3×D3—24no—
287C2×C3×D3—24no—
288C2×C3×D3—28no—
289C2×C3×D3—28no—
290C2×C3×D3—22no—
291C2×C3×D3—22no—
292C2×C3×D3—22no—
293C2×C3×D3—22no—
294C2×C3×D3—24no—
295C2×C3×D3C22×Dic314yes—
296C2×C3×D3C22×C3×C414yes—
297C32×C4—24no—
298C32×C4—24no—
299C32×C4—24no—
300C32×C4—34no—
301C32×C4—44no—
302C32×C4—64no—
303C32×C4C2×C4×D314yes—
304C32×C4C22×Dic314yes—
305C32×C4C22×C3×C414yes—
306C3⋊Dic3—24no—
307C3×Dic3—24no—
308C3×Dic3—24no—
309C3×Dic3—24no—
310C3×Dic3—44no—
311C3×Dic3—64no—
312C3×Dic3—64no—
313C3×Dic3—124no—
314C3×Dic3C22×Dic314yes—
315C3×Dic3C22×C3×C414yes—
316C23×C4—31no—
317#117—98no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—91no—
320C2×C22⋊C4—92no—
321C2×C42—92no—
322C33C2×#11711yescharacteristic, complemented, Sylow
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—32no—
327C23×C3—32no—
328C23×C3—34no—
329C23×C3—34no—
330C23×C3—41no—
331C23×C3—62no—
332C23×C3—64no—
333C23×C3—64no—
334C23×C3—64no—
335C23×C3—62no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—62no—
339C23×C3C2×D3211yescharacteristic
340C23×C3C22×C3×D311yescharacteristic
341C23×C3C22×C3×D311yescharacteristic
342C22×D3—22no—
343C22×D3—22no—
344C22×D3C2×C3×Dic314yes—
345C22×D3C22×C3×D312yes—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—34no—
355C2×C3×C4—34no—
356C2×C3×C4—316no—
357C2×C3×C4—316no—
358C2×C3×C4—34no—
359C2×C3×C4—34no—
360C2×C3×C4—44no—
361C2×C3×C4—42no—
362C2×C3×C4—616no—
363C2×C3×C4—62no—
364C2×C3×C4—64no—
365C2×C3×C4—64no—
366C2×C3×C4—64no—
367C2×C3×C4—64no—
368C2×C3×C4—94no—
369C2×C3×C4—124no—
370C2×C3×C4—184no—
371C2×C3×C4D3×Dic314yes—
372C2×C3×C4C3×C4×D314yes—
373C2×C3×C4C2×C3×Dic314yes—
374C2×C3×C4C2×D3212yes—
375C2×C3×C4C22×C3×D312yes—
376C2×C3×C4C22×C3×D312yes—
377C2×Dic3—24no—
378C2×Dic3—34no—
379C2×Dic3—34no—
380C2×Dic3—64no—
381C2×Dic3—64no—
382C2×Dic3—64no—
383C2×Dic3—124no—
384C2×Dic3C2×C3×Dic314yes—
385C2×Dic3C22×C3×D314yes—
386C2×Dic3C22×C3×D312yes—
387C4×D3C2×C3×Dic3116yescomplemented
388C2×C32—24no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—24no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—24no—
399C2×C32—21no—
400C2×C32—44no—
401C2×C32—41no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—64no—
405C2×C32#89214yescomplemented
406C2×C32#92614yescomplemented
407C2×C32C3×#11714yescomplemented
408C2×C32C22×C4×D311yescharacteristic
409C2×C32C2×#23511yescharacteristic
410C2×C32C2×#23511yescharacteristic
411C2×C32C2×#23711yescharacteristic
412C2×C32C2×#23911yescharacteristic
413C2×C32C23×Dic311yescharacteristic
414C2×C32C23×C3×C411yescharacteristic
415C2×C32C2×C3×#4111yescharacteristic
416C2×C32C2×C3×#4111yescharacteristic
417C3×D3—28no—
418C3×D3—24no—
419C3×D3—24no—
420C24—31no—
421C22×C4—31no—
422C22×C4—38no—
423C22×C4—34no—
424C22×C4—32no—
425C22×C4—92no—
426C22×C4C3×D3211yescharacteristic
427C4⋊C4—34no—
428C4⋊C4—94no—
429C22⋊C4—98no—
430C42—98no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—34no—
444C22×C3—34no—
445C22×C3—41no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—64no—
450C22×C3—616no—
451C22×C3—64no—
452C22×C3—64no—
453C22×C3—68no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—62no—
458C22×C3—68no—
459C22×C3#379512yes—
460C22×C3#379912yes—
461C22×C3C2×D3×Dic312yes—
462C22×C3C2×C3×C4×D312yes—
463C22×C3C3×#23512yes—
464C22×C3C3×#23512yes—
465C22×C3C3×#23712yes—
466C22×C3C3×#23912yes—
467C22×C3C22×C3×Dic312yes—
468C22×C3C22×D3211yescharacteristic
469C22×C3C23×C3×D311yescharacteristic
470C22×C3C23×C3×D311yescharacteristic
471C2×D3—28no—
472C2×D3—22no—
473C2×D3—22no—
474C2×D3C22×C3×Dic314yes—
475C3×C4—24no—
476C3×C4—24no—
477C3×C4—24no—
478C3×C4—34no—
479C3×C4—34no—
480C3×C4—44no—
481C3×C4—64no—
482C3×C4—64no—
483C3×C4—64no—
484C3×C4—124no—
485C3×C4—184no—
486C3×C4C2×D3×Dic314yes—
487C3×C4C2×C3×C4×D314yes—
488C3×C4C22×C3×Dic314yes—
489Dic3—24no—
490Dic3—64no—
491Dic3—64no—
492Dic3—124no—
493Dic3C22×C3×Dic314yes—
494C32—21no—
495C32—21no—
496C32—21no—
497C32—41no—
498C32C2×#89211yescharacteristic, complemented
499C32C2×#92611yescharacteristic, complemented
500C32C2×C3×#11711yescharacteristic, complemented
501C23—32no—
502C23—34no—
503C23—62no—
504C23—62no—
505C23C2×C3×D3211yescharacteristic
506C2×C4—32no—
507C2×C4—34no—
508C2×C4—316no—
509C2×C4—34no—
510C2×C4—64no—
511C2×C4—94no—
512C2×C4—184no—
513C2×C4C3×D3×Dic314yes—
514C2×C4C2×C3×D3212yes—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—24no—
519C2×C3—24no—
520C2×C3—24no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—44no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—68no—
532C2×C3—64no—
533C2×C3—64no—
534C2×C3#6384114yesdirect product factor
535C2×C3C3×#89214yescomplemented
536C2×C3C3×#92614yescomplemented
537C2×C3C2×#379511yescharacteristic
538C2×C3C2×#379911yescharacteristic
539C2×C3C22×D3×Dic311yescharacteristic
540C2×C3C22×C3×C4×D311yescharacteristic
541C2×C3C2×C3×#23511yescharacteristic
542C2×C3C2×C3×#23511yescharacteristic
543C2×C3C2×C3×#23711yescharacteristic
544C2×C3C2×C3×#23911yescharacteristic
545C2×C3C23×C3×Dic311yescharacteristic
546D3—24no—
547C22—34no—
548C22—68no—
549C22—62no—
550C22—62no—
551C22C3×#379512yes—
552C22C3×#379912yes—
553C22C2×C3×D3×Dic312yes—
554C22C22×C3×D3211yescharacteristic
555C4—34no—
556C4—64no—
557C4—184no—
558C4C2×C3×D3×Dic314yes—
559C3—21no—
560C3—21no—
561C3—21no—
562C3—41no—
563C3C2×#6384111yescharacteristic, direct product factor
564C3C2×C3×#89211yescharacteristic, complemented
565C3C2×C3×#92611yescharacteristic, complemented
566C2—64no—
567C2C3×#6384114yesdirect product factor
568C2C2×C3×#379511yescharacteristic
569C2C2×C3×#379911yescharacteristic
570C2C22×C3×D3×Dic311yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC2×C32C23×C3×C4
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C32×D3C22
cosocleC22×C32C24×C3
hypercenterC22×C3×C4D32
hypocenterC32C2×C3×#117
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C2×C32×C4⋊Dic3C2semidirect
5. C2×C3×Dic32C2semidirect
6. C3×#63841C2direct (N × H)
9. C2×#63841C3direct (N × H)
15. C2×C32×C4×D3C4semidirect
17. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C32×C4⋊Dic3C22semidirect
20. C3×Dic32C22semidirect
26. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
34. C2×C3×C4⋊Dic3D3semidirect
35. C2×C3×C4⋊Dic3C2×C3semidirect
40. C2×Dic32C2×C3semidirect
41. #63841C2×C3direct (N × H)
55. C32×C4×D3C2×C4semidirect
86. C2×C3×C4×D3Dic3semidirect
87. C2×C3×C4×D3C3×C4semidirect
91. C4⋊(C3⋊Dic3)C22×C3semidirect
98. C3×C4⋊Dic3C2×D3semidirect
99. C3×C4⋊Dic3C22×C3semidirect
103. Dic32C22×C3semidirect
133. C2×C4⋊Dic3C3×D3semidirect
200. C3×C4×D3C2×Dic3semidirect
201. C3×C4×D3C2×C3×C4semidirect
203. C2×C33#117semidirect
237. C2×C4×D3C3×Dic3semidirect
250. C4⋊Dic3C2×C3×D3semidirect
322. C33C2×#117semidirect
387. C4×D3C2×C3×Dic3semidirect
405. C2×C32#892semidirect
406. C2×C32#926semidirect
407. C2×C32C3×#117semidirect
498. C32C2×#892semidirect
499. C32C2×#926semidirect
500. C32C2×C3×#117semidirect
534. C2×C3#63841direct (N × H)
535. C2×C3C3×#892semidirect
536. C2×C3C3×#926semidirect
563. C3C2×#63841direct (N × H)
564. C3C2×C3×#892semidirect
565. C3C2×C3×#926semidirect
567. C2C3×#63841direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C4×D32
C2×C3×C4⋊(C3⋊Dic3)2
C2×C32×C4⋊Dic32
C2×C3×(C2×D3)⋊Dic32
C2×C3×Dic322
C3×#638412
C2×C3×#9263
C2×C3×#8923
C2×#638413