Finite Group Databasebeta

← C3×A4×#235 #13181438

Subgroups — C3×A4×#235 #13181438

Order 1728.

1182 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×A4C211yesmaximal, characteristic, complemented
2C3×A4×#67C212yesmaximal, complemented
3C3×A4×D12C211yesmaximal, characteristic, complemented
4C3×C4×D3×A4C212yesmaximal, complemented
5C3×A4×Dic6C211yesmaximal, characteristic, complemented
6C22×C3×#235C311yesmaximal, characteristic, complemented
7C3×A4×#41—31nomaximal
8A4×#235C313yesmaximal, direct product factor
9C22×C32×A4C2211yescharacteristic
10C2×C3×D3×A4C2212yes—
11C32×#235—41nomaximal
12C32×C4×A4C2211yescharacteristic
13C32×C4×A4C2211yescharacteristic
14C3×Dic3×A4C2212yes—
15C23×C32×C4C2×C311yescharacteristic, complemented
16C22×C3×#67C2×C312yescomplemented
17C2×C3×#235—31no—
18C2×C3×#235—31no—
19C2×C3×#235—32no—
20C2×C3×#235—31no—
21C2×C3×#235—31no—
22C2×C3×#235—32no—
23C22×C3×D12C2×C311yescharacteristic, complemented
24C22×C3×C4×D3C2×C312yescomplemented
25C22×C3×Dic6C2×C311yescharacteristic, complemented
26C3×Q8×A4—31no—
27C3×D4×A4—31no—
28C3×D4×A4—32no—
29C2×C3×C4×A4—21no—
30C2×C3×C4×A4—23no—
31C2×C3×C4×A4—32no—
32C2×C3×C4×A4D311yescharacteristic, complemented
33C2×C3×C4×A4C2×C313yescomplemented
34A4×#67C2×C316yescomplemented
35A4×D12C2×C313yescomplemented
36C4×D3×A4C2×C316yescomplemented
37A4×Dic6C2×C313yescomplemented
38C2×C32×A4—21no—
39C2×C32×A4C2311yescharacteristic
40C3×D3×A4—22no—
41C2×C33×C4—41no—
42C32×#67—42no—
43C32×D12—41no—
44C32×C4×D3—42no—
45C32×Dic6—41no—
46C22×C3×#41—31no—
47C22×#235C3211yescharacteristic, complemented
48A4×#41—33no—
49C24×C32C22×C311yescharacteristic
50C23×C3×D3C22×C312yes—
51C22×C3×A4—21no—
52C22×C3×A4—23no—
53C22×C3×A4—32no—
54C22×C3×A4C2×D311yescharacteristic
55C22×C3×A4C22×C313yes—
56C2×D3×A4C22×C316yes—
57C32×#41—121no—
58C22×C32×C4—31no—
59C22×C32×C4—31no—
60C22×C32×C4—31no—
61C22×C32×C4—31no—
62C22×C32×C4C22×C311yescharacteristic
63C22×C32×C4C22×C311yescharacteristic
64C2×C3×#67—32no—
65C2×C3×#67—32no—
66C2×C3×#67—32no—
67C2×C3×#67—32no—
68C22×C3×Dic3C22×C312yes—
69C3×#235—32no—
70C3×#235—34no—
71C3×#235—32no—
72C3×#235—32no—
73C3×#235—34no—
74C3×#235—31no—
75C3×#235—32no—
76C3×#235—31no—
77C3×#235—32no—
78C3×#235—31no—
79C3×#235—43no—
80C3×#235A411yescharacteristic, direct product factor
81C2×C3×D12—31no—
82C2×C3×D12—32no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—32no—
87C2×C3×C4×D3—32no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—32no—
91C3×C4×A4—21no—
92C3×C4×A4—21no—
93C3×C4×A4—23no—
94C3×C4×A4—23no—
95C3×C4×A4—32no—
96C3×C4×A4C2×D311yescharacteristic
97C3×C4×A4C2×D311yescharacteristic
98C3×C4×A4C22×C313yes—
99C3×C4×A4C22×C313yes—
100Dic3×A4C22×C316yes—
101C22×C33—41no—
102C2×C32×D3—42no—
103C32×A4#4111yescharacteristic, complemented
104C33×C4—41no—
105C33×C4—41no—
106C32×Dic3—42no—
107C2×C3×#41—91no—
108C2×C3×#41—91no—
109C2×C3×#41—92no—
110C2×C3×#41—91no—
111C2×C3×#41—92no—
112C2×C3×#41—91no—
113C22×C3×Q8—31no—
114C22×C3×D4—32no—
115C22×C3×D4—31no—
116C23×C3×C4—21no—
117C23×C3×C4—32no—
118C23×C3×C4C3×D311yescharacteristic, complemented
119C23×C3×C4C2×C3211yescharacteristic, complemented
120C22×#67C2×C3212yescomplemented
121C2×#235—31no—
122C2×#235—31no—
123C2×#235—32no—
124C2×#235—31no—
125C2×#235—31no—
126C2×#235—32no—
127C22×D12C2×C3211yescharacteristic, complemented
128C22×C4×D3C2×C3212yescomplemented
129C22×Dic6C2×C3211yescharacteristic, complemented
130Q8×A4—33no—
131D4×A4—36no—
132D4×A4—33no—
133C2×C4×A4—23no—
134C2×C4×A4—36no—
135C2×C4×A4C3×D313yescomplemented
136C23×C32—21no—
137C23×C32—31no—
138C23×C32—31no—
139C23×C32—61no—
140C23×C32C23×C311yescharacteristic
141C22×C3×D3—22no—
142C22×C3×D3—32no—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C2×C3×A4—22no—
146C2×C3×A4—26no—
147C2×C3×A4—21no—
148C2×C3×A4—23no—
149C2×C3×A4—21no—
150C2×C3×A4—23no—
151C2×C3×A4—62no—
152C2×C3×A4C22×D311yescharacteristic
153C2×C3×A4C23×C313yes—
154D3×A4—26no—
155C32×Q8—121no—
156C32×D4—121no—
157C32×D4—122no—
158C2×C32×C4—32no—
159C2×C32×C4—31no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—31no—
164C2×C32×C4—31no—
165C2×C32×C4—31no—
166C2×C32×C4—41no—
167C2×C32×C4—43no—
168C2×C32×C4—83no—
169C2×C32×C4—81no—
170C2×C32×C4—122no—
171C2×C32×C4C2×A411yescharacteristic, complemented
172C3×#67—34no—
173C3×#67—32no—
174C3×#67—32no—
175C3×#67—32no—
176C3×#67—46no—
177C3×#67C2×A412yescomplemented
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C3×D12—32no—
181C3×D12—32no—
182C3×D12—31no—
183C3×D12—43no—
184C3×D12C2×A411yescharacteristic, complemented
185C3×C4×D3—34no—
186C3×C4×D3—32no—
187C3×C4×D3—32no—
188C3×C4×D3—32no—
189C3×C4×D3—46no—
190C3×C4×D3C2×A412yescomplemented
191C3×Dic6—32no—
192C3×Dic6—31no—
193C3×Dic6—32no—
194C3×Dic6—43no—
195C3×Dic6C2×A411yescharacteristic, complemented
196C22×#41—31noSylow
197C2×C33—41no—
198C2×C33—81no—
199C32×D3—82no—
200C24×C3—21no—
201C24×C3—32no—
202C24×C3C2×C3×D311yescharacteristic
203C24×C3C22×C3211yescharacteristic
204C23×D3C22×C3212yes—
205C22×A4—23no—
206C22×A4—36no—
207C22×A4C2×C3×D313yes—
208C3×#41—31no—
209C3×#41—92no—
210C3×#41—92no—
211C3×#41—91no—
212C3×#41—91no—
213C3×#41—94no—
214C3×#41—94no—
215C3×#41—92no—
216C3×#41—92no—
217C3×#41—92no—
218C3×#41—91no—
219C3×#41—123no—
220C2×C3×Q8—91no—
221C2×C3×Q8—91no—
222C2×C3×Q8—92no—
223C2×C3×D4—91no—
224C2×C3×D4—91no—
225C2×C3×D4—92no—
226C2×C3×D4—92no—
227C2×C3×D4—92no—
228C2×C3×D4—92no—
229C2×C3×D4—92no—
230C22×C3×C4—21no—
231C22×C3×C4—21no—
232C22×C3×C4—31no—
233C22×C3×C4—31no—
234C22×C3×C4—31no—
235C22×C3×C4—31no—
236C22×C3×C4—31no—
237C22×C3×C4—31no—
238C22×C3×C4—31no—
239C22×C3×C4—31no—
240C22×C3×C4—32no—
241C22×C3×C4—61no—
242C22×C3×C4—61no—
243C22×C3×C4—61no—
244C22×C3×C4—61no—
245C22×C3×C4—92no—
246C22×C3×C4—92no—
247C22×C3×C4—92no—
248C22×C3×C4—92no—
249C22×C3×C4C2×C3×D311yescharacteristic
250C22×C3×C4C2×C3×D311yescharacteristic
251C22×C3×C4C22×C3211yescharacteristic
252C22×C3×C4C22×C3211yescharacteristic
253C2×#67—32no—
254C2×#67—32no—
255C2×#67—32no—
256C2×#67—32no—
257C22×Dic3C22×C3212yes—
258#235—32no—
259#235—34no—
260#235—32no—
261#235—32no—
262#235—34no—
263#235—31no—
264#235—32no—
265#235—31no—
266#235—32no—
267#235—31no—
268#235C3×A411yescharacteristic, direct product factor
269C2×D12—31no—
270C2×D12—32no—
271C2×D12—31no—
272C2×C4×D3—32no—
273C2×C4×D3—32no—
274C2×C4×D3—32no—
275C2×C4×D3—32no—
276C2×Dic6—31no—
277C2×Dic6—31no—
278C2×Dic6—32no—
279C4×A4—23no—
280C4×A4—23no—
281C4×A4—36no—
282C4×A4C2×C3×D313yes—
283C4×A4C2×C3×D313yes—
284C22×C32—31no—
285C22×C32—31no—
286C22×C32—31no—
287C22×C32—41no—
288C22×C32—43no—
289C22×C32—61no—
290C22×C32—61no—
291C22×C32—61no—
292C22×C32—61no—
293C22×C32—83no—
294C22×C32—81no—
295C22×C32—122no—
296C22×C32C3×#4111yescharacteristic, complemented
297C22×C32C22×A411yescharacteristic
298C2×C3×D3—32no—
299C2×C3×D3—46no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3C22×A412yes—
305C3×A4—21no—
306C3×A4—23no—
307C3×A4#23511yescharacteristic, direct product factor
308C3×A4C3×#4113yescomplemented
309C32×C4—31no—
310C32×C4—31no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—43no—
314C32×C4—43no—
315C32×C4—83no—
316C32×C4—83no—
317C32×C4—81no—
318C32×C4—81no—
319C32×C4—122no—
320C32×C4C22×A411yescharacteristic
321C32×C4C22×A411yescharacteristic
322C3×Dic3—32no—
323C3×Dic3—46no—
324C3×Dic3C22×A412yes—
325C2×#41—91no—
326C2×#41—91no—
327C2×#41—92no—
328C2×#41—91no—
329C2×#41—92no—
330C2×#41—91no—
331C22×Q8—31no—
332C22×D4—32no—
333C22×D4—31no—
334C23×C4—32no—
335C23×C4C32×D311yescharacteristic, complemented
336C33—41noSylow
337C23×C3—22no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—31no—
342C23×C3—31no—
343C23×C3—31no—
344C23×C3—31no—
345C23×C3—61no—
346C23×C3—61no—
347C23×C3—61no—
348C23×C3—61no—
349C23×C3—62no—
350C23×C3—62no—
351C23×C3—92no—
352C23×C3—92no—
353C23×C3—182no—
354C23×C3C22×C3×D311yescharacteristic
355C23×C3C23×C3211yescharacteristic
356C22×D3—22no—
357C22×D3—32no—
358C22×D3—32no—
359C22×D3—62no—
360C2×A4—26no—
361C2×A4—23no—
362C2×A4—23no—
363C2×A4—66no—
364C2×A4C22×C3×D313yes—
365C3×Q8—31no—
366C3×Q8—92no—
367C3×Q8—92no—
368C3×Q8—91no—
369C3×Q8—123no—
370C3×D4—32no—
371C3×D4—31no—
372C3×D4—92no—
373C3×D4—92no—
374C3×D4—92no—
375C3×D4—92no—
376C3×D4—94no—
377C3×D4—92no—
378C3×D4—91no—
379C3×D4—123no—
380C3×D4—126no—
381C2×C3×C4—21no—
382C2×C3×C4—32no—
383C2×C3×C4—32no—
384C2×C3×C4—31no—
385C2×C3×C4—31no—
386C2×C3×C4—31no—
387C2×C3×C4—31no—
388C2×C3×C4—32no—
389C2×C3×C4—31no—
390C2×C3×C4—31no—
391C2×C3×C4—31no—
392C2×C3×C4—31no—
393C2×C3×C4—31no—
394C2×C3×C4—31no—
395C2×C3×C4—31no—
396C2×C3×C4—31no—
397C2×C3×C4—31no—
398C2×C3×C4—31no—
399C2×C3×C4—43no—
400C2×C3×C4—61no—
401C2×C3×C4—61no—
402C2×C3×C4—61no—
403C2×C3×C4—61no—
404C2×C3×C4—61no—
405C2×C3×C4—62no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—83no—
409C2×C3×C4—92no—
410C2×C3×C4—92no—
411C2×C3×C4—92no—
412C2×C3×C4—94no—
413C2×C3×C4—92no—
414C2×C3×C4—92no—
415C2×C3×C4—126no—
416C2×C3×C4D3×A411yescharacteristic, complemented
417C2×C3×C4C2×C3×A411yescharacteristic, complemented
418#67—34no—
419#67—32no—
420#67—32no—
421#67—32no—
422#67C2×C3×A412yescomplemented
423C2×Dic3—32no—
424C2×Dic3—32no—
425D12—32no—
426D12—32no—
427D12—31no—
428D12C2×C3×A411yescharacteristic, complemented
429C4×D3—34no—
430C4×D3—32no—
431C4×D3—32no—
432C4×D3—32no—
433C4×D3C2×C3×A412yescomplemented
434Dic6—32no—
435Dic6—31no—
436Dic6—32no—
437Dic6C2×C3×A411yescharacteristic, complemented
438C2×C32—21no—
439C2×C32—31no—
440C2×C32—31no—
441C2×C32—41no—
442C2×C32—43no—
443C2×C32—61no—
444C2×C32—83no—
445C2×C32—81no—
446C2×C32—83no—
447C2×C32—86no—
448C2×C32—82no—
449C2×C32—81no—
450C2×C32—242no—
451C2×C32C23×A411yescharacteristic
452C3×D3—22no—
453C3×D3—62no—
454C3×D3—86no—
455C24—32no—
456C24C2×C32×D311yescharacteristic
457#41—31no—
458#41—92no—
459#41—92no—
460#41—91no—
461#41—91no—
462#41—94no—
463#41—94no—
464#41—92no—
465#41—92no—
466#41—92no—
467#41—91no—
468C2×Q8—91no—
469C2×Q8—91no—
470C2×Q8—92no—
471C2×D4—91no—
472C2×D4—91no—
473C2×D4—92no—
474C2×D4—92no—
475C2×D4—92no—
476C2×D4—92no—
477C2×D4—92no—
478C22×C4—31no—
479C22×C4—31no—
480C22×C4—31no—
481C22×C4—31no—
482C22×C4—32no—
483C22×C4—92no—
484C22×C4—92no—
485C22×C4—92no—
486C22×C4—92no—
487C22×C4C2×C32×D311yescharacteristic
488C22×C4C2×C32×D311yescharacteristic
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—31no—
492C22×C3—31no—
493C22×C3—31no—
494C22×C3—31no—
495C22×C3—31no—
496C22×C3—32no—
497C22×C3—31no—
498C22×C3—43no—
499C22×C3—61no—
500C22×C3—61no—
501C22×C3—61no—
502C22×C3—61no—
503C22×C3—61no—
504C22×C3—61no—
505C22×C3—61no—
506C22×C3—62no—
507C22×C3—62no—
508C22×C3—61no—
509C22×C3—61no—
510C22×C3—61no—
511C22×C3—61no—
512C22×C3—62no—
513C22×C3—62no—
514C22×C3—83no—
515C22×C3—92no—
516C22×C3—126no—
517C22×C3—182no—
518C22×C3—182no—
519C22×C3—182no—
520C22×C3—182no—
521C22×C3C3×#23511yescharacteristic, complemented
522C22×C3C32×#4111yescharacteristic, complemented
523C22×C3C2×D3×A411yescharacteristic
524C22×C3C22×C3×A411yescharacteristic
525C2×D3—32no—
526C2×D3—62no—
527C2×D3—62no—
528C2×D3—62no—
529C2×D3—62no—
530C2×D3C22×C3×A412yes—
531A4—23no—
532A4C3×#23513yesdirect product factor
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—31no—
536C3×C4—31no—
537C3×C4—32no—
538C3×C4—31no—
539C3×C4—31no—
540C3×C4—43no—
541C3×C4—43no—
542C3×C4—61no—
543C3×C4—61no—
544C3×C4—83no—
545C3×C4—83no—
546C3×C4—92no—
547C3×C4—126no—
548C3×C4C2×D3×A411yescharacteristic
549C3×C4C2×D3×A411yescharacteristic
550C3×C4C22×C3×A411yescharacteristic
551C3×C4C22×C3×A411yescharacteristic
552Dic3—32no—
553Dic3C22×C3×A412yes—
554C32—41no—
555C32—43no—
556C32—83no—
557C32—81no—
558C32A4×#4111yescharacteristic, complemented
559C23—21no—
560C23—31no—
561C23—31no—
562C23—61no—
563C23—62no—
564C23—92no—
565C23—92no—
566C23—182no—
567C23C22×C32×D311yescharacteristic
568Q8—31no—
569Q8—92no—
570Q8—92no—
571Q8—91no—
572D4—32no—
573D4—31no—
574D4—92no—
575D4—92no—
576D4—92no—
577D4—92no—
578D4—94no—
579D4—92no—
580D4—91no—
581C2×C4—32no—
582C2×C4—31no—
583C2×C4—31no—
584C2×C4—31no—
585C2×C4—31no—
586C2×C4—31no—
587C2×C4—32no—
588C2×C4—31no—
589C2×C4—31no—
590C2×C4—92no—
591C2×C4—92no—
592C2×C4—92no—
593C2×C4—94no—
594C2×C4—92no—
595C2×C4—92no—
596C2×C4C3×D3×A411yescharacteristic, complemented
597C2×C3—22no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—31no—
602C2×C3—31no—
603C2×C3—31no—
604C2×C3—31no—
605C2×C3—43no—
606C2×C3—61no—
607C2×C3—61no—
608C2×C3—62no—
609C2×C3—61no—
610C2×C3—62no—
611C2×C3—61no—
612C2×C3—83no—
613C2×C3—86no—
614C2×C3—83no—
615C2×C3—182no—
616C2×C3—246no—
617C2×C3C22×D3×A411yescharacteristic
618C2×C3C23×C3×A411yescharacteristic
619D3—22no—
620D3—62no—
621C22—31no—
622C22—31no—
623C22—31no—
624C22—32no—
625C22—61no—
626C22—61no—
627C22—61no—
628C22—61no—
629C22—92no—
630C22—182no—
631C22—182no—
632C22—182no—
633C22—182no—
634C22C32×#23511yescharacteristic, complemented
635C22C2×C3×D3×A411yescharacteristic
636C4—31no—
637C4—31no—
638C4—32no—
639C4—92no—
640C4C2×C3×D3×A411yescharacteristic
641C4C2×C3×D3×A411yescharacteristic
642C3—21no—
643C3—43no—
644C3—83no—
645C3A4×#23511yescharacteristic, direct product factor
646C3C3×A4×#4111yescharacteristic, complemented
647C2—21no—
648C2—31no—
649C2—31no—
650C2—61no—
651C2—62no—
652C2—182no—
653C2C22×C3×D3×A411yescharacteristic
subgroupG/H
centerC3×C4C2×D3×A4
commutator, cosocleC23×C3C23×C32
FrattiniC2C22×C3×D3×A4
FittingC23×C32×C4C2×C3
socleC23×C32C23×C3
hypercenterC2×C3×C4D3×A4
hypocenterC22×C3C32×#41
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×A4C2semidirect
2. C3×A4×#67C2semidirect
3. C3×A4×D12C2semidirect
4. C3×C4×D3×A4C2semidirect
5. C3×A4×Dic6C2semidirect
6. C22×C3×#235C3semidirect
8. A4×#235C3direct (N × H)
15. C23×C32×C4C2×C3semidirect
16. C22×C3×#67C2×C3semidirect
23. C22×C3×D12C2×C3semidirect
24. C22×C3×C4×D3C2×C3semidirect
25. C22×C3×Dic6C2×C3semidirect
32. C2×C3×C4×A4D3semidirect
33. C2×C3×C4×A4C2×C3semidirect
34. A4×#67C2×C3semidirect
35. A4×D12C2×C3semidirect
36. C4×D3×A4C2×C3semidirect
37. A4×Dic6C2×C3semidirect
47. C22×#235C32semidirect
80. C3×#235A4direct (N × H)
103. C32×A4#41semidirect
118. C23×C3×C4C3×D3semidirect
119. C23×C3×C4C2×C32semidirect
120. C22×#67C2×C32semidirect
127. C22×D12C2×C32semidirect
128. C22×C4×D3C2×C32semidirect
129. C22×Dic6C2×C32semidirect
135. C2×C4×A4C3×D3semidirect
171. C2×C32×C4C2×A4semidirect
177. C3×#67C2×A4semidirect
184. C3×D12C2×A4semidirect
190. C3×C4×D3C2×A4semidirect
195. C3×Dic6C2×A4semidirect
268. #235C3×A4direct (N × H)
296. C22×C32C3×#41semidirect
307. C3×A4#235direct (N × H)
308. C3×A4C3×#41semidirect
335. C23×C4C32×D3semidirect
416. C2×C3×C4D3×A4semidirect
417. C2×C3×C4C2×C3×A4semidirect
422. #67C2×C3×A4semidirect
428. D12C2×C3×A4semidirect
433. C4×D3C2×C3×A4semidirect
437. Dic6C2×C3×A4semidirect
521. C22×C3C3×#235semidirect
522. C22×C3C32×#41semidirect
532. A4C3×#235direct (N × H)
558. C32A4×#41semidirect
596. C2×C4C3×D3×A4semidirect
634. C22C32×#235semidirect
645. C3A4×#235direct (N × H)
646. C3C3×A4×#41semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×A42
C3×A4×#672
C3×A4×D122
C3×C4×D3×A42
C3×A4×Dic62
C22×C3×#2353
C3×A4×#413
A4×#2353
C32×#2354