Finite Group Databasebeta

← C3×A4×#237 #13181440

Subgroups — C3×A4×#237 #13181440

Order 1728.

1182 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D4×A4C211yesmaximal, characteristic, complemented
2C3×A4×#67C212yesmaximal, complemented
3C2×C3×Dic3×A4C212yesmaximal, complemented
4C3×C4×D3×A4C211yesmaximal, characteristic, complemented
5C3×A4×Dic6C211yesmaximal, characteristic, complemented
6C22×C3×#237C311yesmaximal, characteristic, complemented
7C3×A4×#41—31nomaximal
8A4×#237C313yesmaximal, direct product factor
9C22×C32×A4C2212yes—
10C2×C3×D3×A4C2211yescharacteristic
11C32×#237—41nomaximal
12C32×C4×A4C2211yescharacteristic
13C3×Dic3×A4C2212yes—
14C3×Dic3×A4C2211yescharacteristic
15C22×C32×D4C2×C311yescharacteristic, complemented
16C22×C3×#67C2×C312yescomplemented
17C23×C3×Dic3C2×C312yescomplemented
18C2×C3×#237—31no—
19C2×C3×#237—31no—
20C2×C3×#237—31no—
21C2×C3×#237—31no—
22C2×C3×#237—32no—
23C2×C3×#237—32no—
24C22×C3×C4×D3C2×C311yescharacteristic, complemented
25C22×C3×Dic6C2×C311yescharacteristic, complemented
26C3×Q8×A4—31no—
27C3×D4×A4—21no—
28C3×D4×A4—23no—
29C3×D4×A4—32no—
30C3×D4×A4D311yescharacteristic, complemented
31C3×D4×A4C2×C313yescomplemented
32C2×C3×C4×A4—31no—
33C2×C3×C4×A4—32no—
34A4×#67C2×C316yescomplemented
35C2×Dic3×A4C2×C316yescomplemented
36C4×D3×A4C2×C313yescomplemented
37A4×Dic6C2×C313yescomplemented
38C2×C32×A4—22no—
39C2×C32×A4C2311yescharacteristic
40C3×D3×A4—21no—
41C33×D4—41no—
42C32×#67—42no—
43C2×C32×Dic3—42no—
44C32×C4×D3—41no—
45C32×Dic6—41no—
46C22×C3×#41—31no—
47C22×#237C3211yescharacteristic, complemented
48A4×#41—33no—
49C24×C32C22×C312yes—
50C23×C3×D3C22×C311yescharacteristic
51C22×C3×A4—22no—
52C22×C3×A4—26no—
53C22×C3×A4—31no—
54C22×C3×A4C2×D312yes—
55C22×C3×A4C22×C316yes—
56C2×D3×A4C22×C313yes—
57C32×#41—121no—
58C2×C32×D4—31no—
59C2×C32×D4—31no—
60C2×C32×D4—32no—
61C22×C32×C4C22×C311yescharacteristic
62C2×C3×#67—32no—
63C2×C3×#67—32no—
64C2×C3×#67—32no—
65C2×C3×#67—32no—
66C22×C3×Dic3—32no—
67C22×C3×Dic3—32no—
68C22×C3×Dic3—32no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3C22×C312yes—
71C22×C3×Dic3C22×C311yescharacteristic
72C3×#237—34no—
73C3×#237—32no—
74C3×#237—32no—
75C3×#237—32no—
76C3×#237—34no—
77C3×#237—31no—
78C3×#237—32no—
79C3×#237—32no—
80C3×#237—31no—
81C3×#237—31no—
82C3×#237—43no—
83C3×#237A411yescharacteristic, direct product factor
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—32no—
91C3×C4×A4—21no—
92C3×C4×A4—23no—
93C3×C4×A4—31no—
94C3×C4×A4—32no—
95C3×C4×A4C2×D311yescharacteristic
96C3×C4×A4C22×C313yes—
97Dic3×A4C22×C316yes—
98Dic3×A4C22×C313yes—
99C22×C33—42no—
100C2×C32×D3—41no—
101C32×A4#4111yescharacteristic, complemented
102C33×C4—41no—
103C32×Dic3—42no—
104C32×Dic3—41no—
105C2×C3×#41—91no—
106C2×C3×#41—91no—
107C2×C3×#41—92no—
108C2×C3×#41—91no—
109C2×C3×#41—92no—
110C2×C3×#41—91no—
111C22×C3×Q8—31no—
112C22×C3×D4—21no—
113C22×C3×D4—32no—
114C22×C3×D4C3×D311yescharacteristic, complemented
115C22×C3×D4C2×C3211yescharacteristic, complemented
116C23×C3×C4—32no—
117C23×C3×C4—31no—
118C22×#67C2×C3212yescomplemented
119C23×Dic3C2×C3212yescomplemented
120C2×#237—31no—
121C2×#237—31no—
122C2×#237—31no—
123C2×#237—31no—
124C2×#237—32no—
125C2×#237—32no—
126C22×C4×D3C2×C3211yescharacteristic, complemented
127C22×Dic6C2×C3211yescharacteristic, complemented
128Q8×A4—33no—
129D4×A4—23no—
130D4×A4—36no—
131D4×A4C3×D313yescomplemented
132C2×C4×A4—36no—
133C2×C4×A4—33no—
134C23×C32—22no—
135C23×C32—32no—
136C23×C32—32no—
137C23×C32—62no—
138C23×C32C23×C311yescharacteristic
139C22×C3×D3—21no—
140C22×C3×D3—31no—
141C22×C3×D3—31no—
142C22×C3×D3—61no—
143C2×C3×A4—22no—
144C2×C3×A4—21no—
145C2×C3×A4—23no—
146C2×C3×A4—26no—
147C2×C3×A4—46no—
148C2×C3×A4—42no—
149C2×C3×A4—61no—
150C2×C3×A4C22×D311yescharacteristic
151C2×C3×A4C23×C313yes—
152D3×A4—23no—
153C32×Q8—121no—
154C32×D4—32no—
155C32×D4—32no—
156C32×D4—31no—
157C32×D4—41no—
158C32×D4—43no—
159C32×D4—83no—
160C32×D4—81no—
161C32×D4—122no—
162C32×D4C2×A411yescharacteristic, complemented
163C2×C32×C4—31no—
164C2×C32×C4—31no—
165C2×C32×C4—121no—
166C2×C32×C4—122no—
167C3×#67—34no—
168C3×#67—32no—
169C3×#67—32no—
170C3×#67—32no—
171C3×#67—46no—
172C3×#67C2×A412yescomplemented
173C2×C3×Dic3—34no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—46no—
182C2×C3×Dic3C2×A412yescomplemented
183C3×C4×D3—32no—
184C3×C4×D3—31no—
185C3×C4×D3—31no—
186C3×C4×D3—31no—
187C3×C4×D3—43no—
188C3×C4×D3C2×A411yescharacteristic, complemented
189C3×Dic6—32no—
190C3×Dic6—31no—
191C3×Dic6—32no—
192C3×Dic6—43no—
193C3×Dic6C2×A411yescharacteristic, complemented
194C22×#41—31noSylow
195C2×C33—41no—
196C2×C33—82no—
197C32×D3—81no—
198C24×C3—22no—
199C24×C3—31no—
200C24×C3C2×C3×D312yes—
201C24×C3C22×C3212yes—
202C23×D3C22×C3211yescharacteristic
203C22×A4—26no—
204C22×A4—33no—
205C22×A4C2×C3×D316yes—
206C3×#41—31no—
207C3×#41—92no—
208C3×#41—91no—
209C3×#41—92no—
210C3×#41—91no—
211C3×#41—92no—
212C3×#41—92no—
213C3×#41—94no—
214C3×#41—94no—
215C3×#41—92no—
216C3×#41—91no—
217C3×#41—123no—
218C2×C3×Q8—91no—
219C2×C3×Q8—91no—
220C2×C3×Q8—92no—
221C2×C3×D4—31no—
222C2×C3×D4—31no—
223C2×C3×D4—31no—
224C2×C3×D4—32no—
225C2×C3×D4—31no—
226C2×C3×D4—32no—
227C2×C3×D4—61no—
228C2×C3×D4—61no—
229C2×C3×D4—62no—
230C2×C3×D4—92no—
231C2×C3×D4—92no—
232C2×C3×D4—92no—
233C2×C3×D4—92no—
234C22×C3×C4—21no—
235C22×C3×C4—31no—
236C22×C3×C4—32no—
237C22×C3×C4—91no—
238C22×C3×C4—92no—
239C22×C3×C4—91no—
240C22×C3×C4—91no—
241C22×C3×C4—91no—
242C22×C3×C4—92no—
243C22×C3×C4—92no—
244C22×C3×C4—92no—
245C22×C3×C4C2×C3×D311yescharacteristic
246C22×C3×C4C22×C3211yescharacteristic
247C2×#67—32no—
248C2×#67—32no—
249C2×#67—32no—
250C2×#67—32no—
251C22×Dic3—32no—
252C22×Dic3—32no—
253C22×Dic3—32no—
254C22×Dic3—32no—
255C22×Dic3C22×C3212yes—
256C22×Dic3C22×C3211yescharacteristic
257#237—34no—
258#237—32no—
259#237—32no—
260#237—32no—
261#237—34no—
262#237—31no—
263#237—32no—
264#237—32no—
265#237—31no—
266#237—31no—
267#237C3×A411yescharacteristic, direct product factor
268C2×C4×D3—31no—
269C2×C4×D3—31no—
270C2×C4×D3—31no—
271C2×C4×D3—31no—
272C2×Dic6—31no—
273C2×Dic6—31no—
274C2×Dic6—32no—
275C4×A4—23no—
276C4×A4—33no—
277C4×A4—36no—
278C4×A4C2×C3×D313yes—
279C22×C32—31no—
280C22×C32—31no—
281C22×C32—32no—
282C22×C32—42no—
283C22×C32—46no—
284C22×C32—62no—
285C22×C32—62no—
286C22×C32—62no—
287C22×C32—62no—
288C22×C32—86no—
289C22×C32—82no—
290C22×C32—121no—
291C22×C32C3×#4111yescharacteristic, complemented
292C22×C32C22×A412yes—
293C2×C3×D3—31no—
294C2×C3×D3—43no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3C22×A411yescharacteristic
300C3×A4—21no—
301C3×A4—23no—
302C3×A4#23711yescharacteristic, direct product factor
303C3×A4C3×#4113yescomplemented
304C32×C4—31no—
305C32×C4—41no—
306C32×C4—43no—
307C32×C4—83no—
308C32×C4—81no—
309C32×C4—121no—
310C32×C4—122no—
311C32×C4C22×A411yescharacteristic
312C3×Dic3—32no—
313C3×Dic3—31no—
314C3×Dic3—46no—
315C3×Dic3—43no—
316C3×Dic3C22×A412yes—
317C3×Dic3C22×A411yescharacteristic
318C2×#41—91no—
319C2×#41—91no—
320C2×#41—92no—
321C2×#41—91no—
322C2×#41—92no—
323C2×#41—91no—
324C22×Q8—31no—
325C22×D4—32no—
326C22×D4C32×D311yescharacteristic, complemented
327C23×C4—32no—
328C23×C4—31no—
329C33—41noSylow
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—32no—
334C23×C3—32no—
335C23×C3—32no—
336C23×C3—32no—
337C23×C3—42no—
338C23×C3—62no—
339C23×C3—62no—
340C23×C3—62no—
341C23×C3—62no—
342C23×C3—61no—
343C23×C3—91no—
344C23×C3—91no—
345C23×C3—122no—
346C23×C3—181no—
347C23×C3C22×C3×D311yescharacteristic
348C23×C3C23×C3211yescharacteristic
349C22×D3—21no—
350C22×D3—31no—
351C22×D3—31no—
352C22×D3—61no—
353C2×A4—26no—
354C2×A4—23no—
355C2×A4—46no—
356C2×A4—63no—
357C2×A4C22×C3×D313yes—
358C3×Q8—31no—
359C3×Q8—92no—
360C3×Q8—92no—
361C3×Q8—91no—
362C3×Q8—123no—
363C3×D4—21no—
364C3×D4—32no—
365C3×D4—32no—
366C3×D4—32no—
367C3×D4—32no—
368C3×D4—32no—
369C3×D4—31no—
370C3×D4—31no—
371C3×D4—43no—
372C3×D4—61no—
373C3×D4—62no—
374C3×D4—62no—
375C3×D4—83no—
376C3×D4—92no—
377C3×D4—92no—
378C3×D4—94no—
379C3×D4—92no—
380C3×D4—126no—
381C3×D4D3×A411yescharacteristic, complemented
382C3×D4C2×C3×A411yescharacteristic, complemented
383C2×C3×C4—32no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—31no—
389C2×C3×C4—61no—
390C2×C3×C4—61no—
391C2×C3×C4—91no—
392C2×C3×C4—91no—
393C2×C3×C4—91no—
394C2×C3×C4—92no—
395C2×C3×C4—92no—
396C2×C3×C4—92no—
397C2×C3×C4—92no—
398C2×C3×C4—94no—
399C2×C3×C4—92no—
400C2×C3×C4—91no—
401C2×C3×C4—91no—
402C2×C3×C4—92no—
403C2×C3×C4—123no—
404C2×C3×C4—126no—
405#67—34no—
406#67—32no—
407#67—32no—
408#67—32no—
409#67C2×C3×A412yescomplemented
410C2×Dic3—34no—
411C2×Dic3—32no—
412C2×Dic3—32no—
413C2×Dic3—31no—
414C2×Dic3—32no—
415C2×Dic3—32no—
416C2×Dic3—32no—
417C2×Dic3—31no—
418C2×Dic3C2×C3×A412yescomplemented
419C4×D3—32no—
420C4×D3—31no—
421C4×D3—31no—
422C4×D3—31no—
423C4×D3C2×C3×A411yescharacteristic, complemented
424Dic6—32no—
425Dic6—31no—
426Dic6—32no—
427Dic6C2×C3×A411yescharacteristic, complemented
428C2×C32—22no—
429C2×C32—31no—
430C2×C32—31no—
431C2×C32—41no—
432C2×C32—43no—
433C2×C32—62no—
434C2×C32—83no—
435C2×C32—82no—
436C2×C32—86no—
437C2×C32—81no—
438C2×C32—162no—
439C2×C32—166no—
440C2×C32—241no—
441C2×C32C23×A411yescharacteristic
442C3×D3—21no—
443C3×D3—61no—
444C3×D3—83no—
445C24—31no—
446C24C2×C32×D312yes—
447#41—31no—
448#41—92no—
449#41—91no—
450#41—92no—
451#41—91no—
452#41—92no—
453#41—92no—
454#41—94no—
455#41—94no—
456#41—92no—
457#41—91no—
458C2×Q8—91no—
459C2×Q8—91no—
460C2×Q8—92no—
461C2×D4—31no—
462C2×D4—31no—
463C2×D4—32no—
464C2×D4—92no—
465C2×D4—92no—
466C2×D4—92no—
467C2×D4—92no—
468C22×C4—31no—
469C22×C4—32no—
470C22×C4—91no—
471C22×C4—92no—
472C22×C4—91no—
473C22×C4—91no—
474C22×C4—91no—
475C22×C4—92no—
476C22×C4—92no—
477C22×C4—92no—
478C22×C4C2×C32×D311yescharacteristic
479C22×C3—22no—
480C22×C3—21no—
481C22×C3—31no—
482C22×C3—31no—
483C22×C3—31no—
484C22×C3—31no—
485C22×C3—31no—
486C22×C3—32no—
487C22×C3—32no—
488C22×C3—46no—
489C22×C3—61no—
490C22×C3—62no—
491C22×C3—61no—
492C22×C3—62no—
493C22×C3—62no—
494C22×C3—62no—
495C22×C3—62no—
496C22×C3—62no—
497C22×C3—62no—
498C22×C3—62no—
499C22×C3—62no—
500C22×C3—86no—
501C22×C3—91no—
502C22×C3—123no—
503C22×C3—122no—
504C22×C3—122no—
505C22×C3—122no—
506C22×C3—122no—
507C22×C3—181no—
508C22×C3—181no—
509C22×C3—181no—
510C22×C3—181no—
511C22×C3C3×#23711yescharacteristic, complemented
512C22×C3C32×#4111yescharacteristic, complemented
513C22×C3C2×D3×A412yes—
514C22×C3C22×C3×A412yes—
515C2×D3—31no—
516C2×D3—61no—
517C2×D3—61no—
518C2×D3—61no—
519C2×D3—61no—
520C2×D3C22×C3×A411yescharacteristic
521A4—23no—
522A4C3×#23713yesdirect product factor
523C3×C4—21no—
524C3×C4—31no—
525C3×C4—31no—
526C3×C4—32no—
527C3×C4—31no—
528C3×C4—43no—
529C3×C4—61no—
530C3×C4—83no—
531C3×C4—91no—
532C3×C4—92no—
533C3×C4—123no—
534C3×C4—126no—
535C3×C4C2×D3×A411yescharacteristic
536C3×C4C22×C3×A411yescharacteristic
537Dic3—32no—
538Dic3—31no—
539Dic3C22×C3×A412yes—
540Dic3C22×C3×A411yescharacteristic
541C32—41no—
542C32—43no—
543C32—83no—
544C32—81no—
545C32A4×#4111yescharacteristic, complemented
546C23—22no—
547C23—32no—
548C23—32no—
549C23—62no—
550C23—61no—
551C23—91no—
552C23—91no—
553C23—181no—
554C23C22×C32×D311yescharacteristic
555Q8—31no—
556Q8—92no—
557Q8—92no—
558Q8—91no—
559D4—32no—
560D4—32no—
561D4—32no—
562D4—31no—
563D4—92no—
564D4—92no—
565D4—94no—
566D4—92no—
567D4C3×D3×A411yescharacteristic, complemented
568C2×C4—32no—
569C2×C4—31no—
570C2×C4—31no—
571C2×C4—31no—
572C2×C4—91no—
573C2×C4—91no—
574C2×C4—92no—
575C2×C4—91no—
576C2×C4—92no—
577C2×C4—92no—
578C2×C4—92no—
579C2×C4—94no—
580C2×C4—92no—
581C2×C4—91no—
582C2×C4—91no—
583C2×C4—92no—
584C2×C3—22no—
585C2×C3—22no—
586C2×C3—21no—
587C2×C3—31no—
588C2×C3—31no—
589C2×C3—31no—
590C2×C3—31no—
591C2×C3—42no—
592C2×C3—43no—
593C2×C3—61no—
594C2×C3—61no—
595C2×C3—62no—
596C2×C3—61no—
597C2×C3—62no—
598C2×C3—86no—
599C2×C3—83no—
600C2×C3—122no—
601C2×C3—166no—
602C2×C3—181no—
603C2×C3—243no—
604C2×C3C22×D3×A411yescharacteristic
605C2×C3C23×C3×A411yescharacteristic
606D3—21no—
607D3—61no—
608C22—31no—
609C22—31no—
610C22—31no—
611C22—32no—
612C22—62no—
613C22—62no—
614C22—62no—
615C22—62no—
616C22—91no—
617C22—181no—
618C22—181no—
619C22—181no—
620C22—181no—
621C22C32×#23711yescharacteristic, complemented
622C22C2×C3×D3×A412yes—
623C4—31no—
624C4—31no—
625C4—32no—
626C4—91no—
627C4—92no—
628C4C2×C3×D3×A411yescharacteristic
629C3—21no—
630C3—43no—
631C3—83no—
632C3A4×#23711yescharacteristic, direct product factor
633C3C3×A4×#4111yescharacteristic, complemented
634C2—22no—
635C2—31no—
636C2—31no—
637C2—62no—
638C2—61no—
639C2—181no—
640C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC2×C3C22×D3×A4
commutator, cosocleC23×C3C23×C32
FrattiniC2C22×C3×D3×A4
FittingC22×C32×D4C2×C3
socleC23×C32C23×C3
hypercenterC3×D4D3×A4
hypocenterC22×C3C32×#41
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×D4×A4C2semidirect
2. C3×A4×#67C2semidirect
3. C2×C3×Dic3×A4C2semidirect
4. C3×C4×D3×A4C2semidirect
5. C3×A4×Dic6C2semidirect
6. C22×C3×#237C3semidirect
8. A4×#237C3direct (N × H)
15. C22×C32×D4C2×C3semidirect
16. C22×C3×#67C2×C3semidirect
17. C23×C3×Dic3C2×C3semidirect
24. C22×C3×C4×D3C2×C3semidirect
25. C22×C3×Dic6C2×C3semidirect
30. C3×D4×A4D3semidirect
31. C3×D4×A4C2×C3semidirect
34. A4×#67C2×C3semidirect
35. C2×Dic3×A4C2×C3semidirect
36. C4×D3×A4C2×C3semidirect
37. A4×Dic6C2×C3semidirect
47. C22×#237C32semidirect
83. C3×#237A4direct (N × H)
101. C32×A4#41semidirect
114. C22×C3×D4C3×D3semidirect
115. C22×C3×D4C2×C32semidirect
118. C22×#67C2×C32semidirect
119. C23×Dic3C2×C32semidirect
126. C22×C4×D3C2×C32semidirect
127. C22×Dic6C2×C32semidirect
131. D4×A4C3×D3semidirect
162. C32×D4C2×A4semidirect
172. C3×#67C2×A4semidirect
182. C2×C3×Dic3C2×A4semidirect
188. C3×C4×D3C2×A4semidirect
193. C3×Dic6C2×A4semidirect
267. #237C3×A4direct (N × H)
291. C22×C32C3×#41semidirect
302. C3×A4#237direct (N × H)
303. C3×A4C3×#41semidirect
326. C22×D4C32×D3semidirect
381. C3×D4D3×A4semidirect
382. C3×D4C2×C3×A4semidirect
409. #67C2×C3×A4semidirect
418. C2×Dic3C2×C3×A4semidirect
423. C4×D3C2×C3×A4semidirect
427. Dic6C2×C3×A4semidirect
511. C22×C3C3×#237semidirect
512. C22×C3C32×#41semidirect
522. A4C3×#237direct (N × H)
545. C32A4×#41semidirect
567. D4C3×D3×A4semidirect
621. C22C32×#237semidirect
632. C3A4×#237direct (N × H)
633. C3C3×A4×#41semidirect

Maximal subgroups

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

subgroupindex
C32×D4×A42
C3×A4×#672
C2×C3×Dic3×A42
C3×C4×D3×A42
C3×A4×Dic62
C22×C3×#2373
C3×A4×#413
A4×#2373
C32×#2374