Finite Group Databasebeta

← C3×A4×#121 #11919694

Subgroups — C3×A4×#121 #11919694

Order 1152.

1344 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
1C2×C3×D4×A4C212yesmaximal, complemented
2C2×C3×D4×A4C211yesmaximal, characteristic, complemented
3C22×C3×C4×A4C211yesmaximal, characteristic, complemented
4C3×A4×C4⋊C4C211yesmaximal, characteristic, complemented
5C3×A4×C22⋊C4C212yesmaximal, complemented
6C22×C3×#121C311yesmaximal, characteristic, complemented
7A4×#121C313yesmaximal, direct product factor
8C23×C3×A4C2212yes—
9C23×C3×A4C2211yescharacteristic
10C3×D4×A4—22no—
11C3×D4×A4—24no—
12C2×C3×C4×A4—22no—
13C2×C3×C4×A4C2211yescharacteristic
14C2×C3×C4×A4C2212yes—
15C2×C3×C4×A4C2211yescharacteristic
16C32×#121—41nomaximal
17C23×C3×D4C2×C311yescharacteristic, complemented
18C23×C3×D4C2×C312yescomplemented
19C24×C3×C4C2×C311yescharacteristic, complemented
20C2×D4×A4C2×C316yescomplemented
21C2×D4×A4C2×C313yescomplemented
22C22×C4×A4C2×C313yescomplemented
23C2×C3×#121—31no—
24C2×C3×#121—32no—
25C2×C3×#121—31no—
26C2×C3×#121—31no—
27C2×C3×#121—32no—
28C2×C3×#121—31no—
29C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
30C22×C3×C22⋊C4C2×C312yescomplemented
31A4×C4⋊C4C2×C313yescomplemented
32A4×C22⋊C4C2×C316yescomplemented
33C22×C3×A4—21no—
34C22×C3×A4—21no—
35C22×C3×A4—22no—
36C22×C3×A4—22no—
37C22×C3×A4—22no—
38C22×C3×A4D412yescomplemented
39C22×C3×A4C2311yescharacteristic
40C2×C32×D4—41no—
41C2×C32×D4—42no—
42C22×C32×C4—41no—
43C3×C4×A4—21no—
44C3×C4×A4—22no—
45C3×C4×A4D412yescomplemented
46C32×C4⋊C4—41no—
47C32×C22⋊C4—42no—
48C22×#121C3211yescharacteristic, complemented, Sylow
49C25×C3C22×C312yes—
50C25×C3C22×C311yescharacteristic
51C23×A4C22×C316yes—
52C23×A4C22×C313yes—
53C22×C3×D4—24no—
54C22×C3×D4—22no—
55C22×C3×D4—31no—
56C22×C3×D4—32no—
57C22×C3×D4—31no—
58C22×C3×D4—32no—
59C22×C3×D4—32no—
60C22×C3×D4—32no—
61C22×C3×D4—32no—
62C22×C3×D4—62no—
63C22×C3×D4—64no—
64C23×C3×C4—22no—
65C23×C3×C4—31no—
66C23×C3×C4—31no—
67C23×C3×C4—31no—
68C23×C3×C4—31no—
69C23×C3×C4—62no—
70C23×C3×C4C22×C311yescharacteristic
71C23×C3×C4C22×C312yes—
72C23×C3×C4C22×C311yescharacteristic
73D4×A4—212no—
74D4×A4—26no—
75C2×C4×A4—26no—
76C2×C4×A4C22×C313yes—
77C2×C4×A4C22×C316yes—
78C2×C4×A4C22×C313yes—
79C3×#121—32no—
80C3×#121—32no—
81C3×#121—34no—
82C3×#121—34no—
83C3×#121—32no—
84C3×#121—31no—
85C3×#121—32no—
86C3×#121—31no—
87C3×#121—31no—
88C3×#121—32no—
89C3×#121—43no—
90C3×#121A411yescharacteristic, direct product factor
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4—32no—
94C2×C3×C22⋊C4—32no—
95C2×C3×C22⋊C4—32no—
96C2×C3×C22⋊C4—32no—
97C2×C3×C22⋊C4—32no—
98C23×C32—41no—
99C23×C32—42no—
100C2×C3×A4—22no—
101C2×C3×A4—42no—
102C2×C3×A4C2×D411yescharacteristic
103C2×C3×A4C2×D411yescharacteristic
104C2×C3×A4#4111yescharacteristic
105C32×D4—82no—
106C32×D4—84no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—42no—
110C2×C32×C4—82no—
111C23×D4C2×C3212yescomplemented
112C23×D4C2×C3211yescharacteristic, complemented
113C24×C4C2×C3211yescharacteristic, complemented
114C2×#121—31no—
115C2×#121—32no—
116C2×#121—31no—
117C2×#121—31no—
118C2×#121—32no—
119C2×#121—31no—
120C22×C4⋊C4C2×C3211yescharacteristic, complemented
121C22×C22⋊C4C2×C3212yescomplemented
122C24×C3—21no—
123C24×C3—21no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—31no—
128C24×C3—32no—
129C24×C3—31no—
130C24×C3—32no—
131C24×C3—32no—
132C24×C3—61no—
133C24×C3—61no—
134C24×C3—62no—
135C24×C3—62no—
136C24×C3—62no—
137C24×C3C3×D412yescomplemented
138C24×C3C23×C311yescharacteristic
139C22×A4—23no—
140C22×A4—23no—
141C22×A4—26no—
142C22×A4—26no—
143C22×A4—26no—
144C22×A4C3×D416yescomplemented
145C22×A4C23×C313yes—
146C2×C3×D4—32no—
147C2×C3×D4—34no—
148C2×C3×D4—32no—
149C2×C3×D4—32no—
150C2×C3×D4—32no—
151C2×C3×D4—32no—
152C2×C3×D4—31no—
153C2×C3×D4—43no—
154C2×C3×D4—46no—
155C2×C3×D4—62no—
156C2×C3×D4—62no—
157C2×C3×D4—64no—
158C2×C3×D4—62no—
159C2×C3×D4—64no—
160C2×C3×D4—62no—
161C2×C3×D4—64no—
162C2×C3×D4—64no—
163C2×C3×D4—64no—
164C2×C3×D4—64no—
165C2×C3×D4—64no—
166C2×C3×D4—64no—
167C2×C3×D4—64no—
168C2×C3×D4—64no—
169C2×C3×D4C2×A412yescomplemented
170C2×C3×D4C2×A411yescharacteristic, complemented
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—32no—
176C22×C3×C4—32no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—32no—
180C22×C3×C4—31no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—32no—
184C22×C3×C4—43no—
185C22×C3×C4—62no—
186C22×C3×C4—62no—
187C22×C3×C4—62no—
188C22×C3×C4—62no—
189C22×C3×C4—62no—
190C22×C3×C4—62no—
191C22×C3×C4—62no—
192C22×C3×C4—62no—
193C22×C3×C4—61no—
194C22×C3×C4—62no—
195C22×C3×C4C3×D412yescomplemented
196C22×C3×C4C2×A411yescharacteristic, complemented
197C4×A4—23no—
198C4×A4—26no—
199C4×A4C3×D416yescomplemented
200C3×C4⋊C4—32no—
201C3×C4⋊C4—32no—
202C3×C4⋊C4—31no—
203C3×C4⋊C4—43no—
204C3×C4⋊C4C2×A411yescharacteristic, complemented
205C3×C22⋊C4—34no—
206C3×C22⋊C4—32no—
207C3×C22⋊C4—32no—
208C3×C22⋊C4—32no—
209C3×C22⋊C4—46no—
210C3×C22⋊C4C2×A412yescomplemented
211C22×C32—42no—
212C22×C32—41no—
213C22×C32—81no—
214C22×C32—81no—
215C22×C32—82no—
216C22×C32—82no—
217C22×C32—82no—
218C3×A4#12111yescharacteristic, direct product factor
219C32×C4—42no—
220C32×C4—81no—
221C32×C4—82no—
222C25C22×C3212yes—
223C25C22×C3211yescharacteristic
224C22×D4—24no—
225C22×D4—22no—
226C22×D4—31no—
227C22×D4—32no—
228C22×D4—31no—
229C22×D4—32no—
230C22×D4—32no—
231C22×D4—32no—
232C22×D4—32no—
233C22×D4—62no—
234C22×D4—64no—
235C23×C4—22no—
236C23×C4—31no—
237C23×C4—31no—
238C23×C4—31no—
239C23×C4—31no—
240C23×C4—62no—
241C23×C4C22×C3211yescharacteristic
242C23×C4C22×C3212yes—
243C23×C4C22×C3211yescharacteristic
244#121—32no—
245#121—32no—
246#121—34no—
247#121—34no—
248#121—32no—
249#121—31no—
250#121—32no—
251#121—31no—
252#121—31no—
253#121—32no—
254#121C3×A411yescharacteristic, direct product factor
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—32no—
261C2×C22⋊C4—32no—
262C23×C3—22no—
263C23×C3—31no—
264C23×C3—31no—
265C23×C3—31no—
266C23×C3—31no—
267C23×C3—32no—
268C23×C3—32no—
269C23×C3—32no—
270C23×C3—32no—
271C23×C3—31no—
272C23×C3—32no—
273C23×C3—42no—
274C23×C3—43no—
275C23×C3—46no—
276C23×C3—61no—
277C23×C3—61no—
278C23×C3—61no—
279C23×C3—61no—
280C23×C3—62no—
281C23×C3—62no—
282C23×C3—62no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—61no—
288C23×C3—61no—
289C23×C3—62no—
290C23×C3—62no—
291C23×C3—61no—
292C23×C3—61no—
293C23×C3—64no—
294C23×C3—62no—
295C23×C3—62no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3—122no—
301C23×C3—122no—
302C23×C3—122no—
303C23×C3—124no—
304C23×C3C2×C3×D411yescharacteristic
305C23×C3C2×C3×D411yescharacteristic
306C23×C3C3×#4111yescharacteristic
307C23×C3C22×A412yes—
308C23×C3C22×A411yescharacteristic
309C2×A4—26no—
310C2×A4—46no—
311C2×A4C2×C3×D413yes—
312C2×A4C2×C3×D413yes—
313C2×A4C3×#4113yes—
314C3×D4—24no—
315C3×D4—22no—
316C3×D4—64no—
317C3×D4—64no—
318C3×D4—64no—
319C3×D4—64no—
320C3×D4—68no—
321C3×D4—64no—
322C3×D4—62no—
323C3×D4—86no—
324C3×D4—812no—
325C2×C3×C4—22no—
326C2×C3×C4—32no—
327C2×C3×C4—31no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—31no—
332C2×C3×C4—32no—
333C2×C3×C4—43no—
334C2×C3×C4—43no—
335C2×C3×C4—46no—
336C2×C3×C4—62no—
337C2×C3×C4—61no—
338C2×C3×C4—62no—
339C2×C3×C4—61no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—64no—
343C2×C3×C4—62no—
344C2×C3×C4—61no—
345C2×C3×C4—62no—
346C2×C3×C4—61no—
347C2×C3×C4—62no—
348C2×C3×C4—86no—
349C2×C3×C4C22×A411yescharacteristic
350C2×C3×C4C22×A412yes—
351C2×C3×C4C22×A411yescharacteristic
352C2×C32—41no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—82no—
356C2×C32—162no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—31no—
363C24—32no—
364C24—31no—
365C24—32no—
366C24—32no—
367C24—61no—
368C24—61no—
369C24—62no—
370C24—62no—
371C24—62no—
372C24C32×D412yescomplemented
373C24C23×C3211yescharacteristic
374C2×D4—32no—
375C2×D4—34no—
376C2×D4—32no—
377C2×D4—32no—
378C2×D4—32no—
379C2×D4—32no—
380C2×D4—31no—
381C2×D4—62no—
382C2×D4—62no—
383C2×D4—64no—
384C2×D4—62no—
385C2×D4—64no—
386C2×D4—62no—
387C2×D4—64no—
388C2×D4—64no—
389C2×D4—64no—
390C2×D4—64no—
391C2×D4—64no—
392C2×D4—64no—
393C2×D4—64no—
394C2×D4—64no—
395C2×D4C2×C3×A412yescomplemented
396C2×D4C2×C3×A411yescharacteristic, complemented
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—32no—
406C22×C4—31no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—32no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—62no—
420C22×C4C32×D412yescomplemented
421C22×C4C2×C3×A411yescharacteristic, complemented
422C4⋊C4—32no—
423C4⋊C4—32no—
424C4⋊C4—31no—
425C4⋊C4C2×C3×A411yescharacteristic, complemented
426C22⋊C4—34no—
427C22⋊C4—32no—
428C22⋊C4—32no—
429C22⋊C4—32no—
430C22⋊C4C2×C3×A412yescomplemented
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—32no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—31no—
444C22×C3—31no—
445C22×C3—31no—
446C22×C3—32no—
447C22×C3—46no—
448C22×C3—43no—
449C22×C3—62no—
450C22×C3—62no—
451C22×C3—62no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—62no—
457C22×C3—61no—
458C22×C3—61no—
459C22×C3—62no—
460C22×C3—62no—
461C22×C3—62no—
462C22×C3—83no—
463C22×C3—83no—
464C22×C3—86no—
465C22×C3—86no—
466C22×C3—86no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—122no—
471C22×C3—122no—
472C22×C3—122no—
473C22×C3—122no—
474C22×C3—122no—
475C22×C3C3×#12111yescharacteristic, complemented
476C22×C3D4×A412yescomplemented
477C22×C3C23×A411yescharacteristic
478A4C3×#12113yesdirect product factor
479C3×C4—21no—
480C3×C4—22no—
481C3×C4—32no—
482C3×C4—46no—
483C3×C4—61no—
484C3×C4—62no—
485C3×C4—83no—
486C3×C4—86no—
487C3×C4D4×A412yescomplemented
488C32—41noSylow
489C23—22no—
490C23—31no—
491C23—31no—
492C23—31no—
493C23—31no—
494C23—32no—
495C23—32no—
496C23—32no—
497C23—32no—
498C23—31no—
499C23—32no—
500C23—42no—
501C23—61no—
502C23—61no—
503C23—61no—
504C23—61no—
505C23—62no—
506C23—62no—
507C23—62no—
508C23—62no—
509C23—62no—
510C23—62no—
511C23—62no—
512C23—61no—
513C23—61no—
514C23—62no—
515C23—62no—
516C23—61no—
517C23—61no—
518C23—64no—
519C23—62no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—122no—
526C23—122no—
527C23—122no—
528C23—124no—
529C23C2×C32×D411yescharacteristic
530C23C2×C32×D411yescharacteristic
531C23C32×#4111yescharacteristic
532C23C22×C3×A412yes—
533C23C22×C3×A411yescharacteristic
534D4—24no—
535D4—22no—
536D4—64no—
537D4—64no—
538D4—64no—
539D4—64no—
540D4—68no—
541D4—64no—
542D4—62no—
543C2×C4—22no—
544C2×C4—32no—
545C2×C4—31no—
546C2×C4—32no—
547C2×C4—32no—
548C2×C4—32no—
549C2×C4—31no—
550C2×C4—32no—
551C2×C4—62no—
552C2×C4—61no—
553C2×C4—62no—
554C2×C4—61no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—61no—
560C2×C4—62no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4C22×C3×A411yescharacteristic
564C2×C4C22×C3×A412yes—
565C2×C4C22×C3×A411yescharacteristic
566C2×C3—22no—
567C2×C3—31no—
568C2×C3—31no—
569C2×C3—31no—
570C2×C3—31no—
571C2×C3—42no—
572C2×C3—43no—
573C2×C3—43no—
574C2×C3—43no—
575C2×C3—62no—
576C2×C3—86no—
577C2×C3—122no—
578C2×C3—166no—
579C2×C3C2×D4×A411yescharacteristic
580C2×C3C2×D4×A411yescharacteristic
581C2×C3A4×#4111yescharacteristic
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—31no—
588C22—31no—
589C22—31no—
590C22—32no—
591C22—31no—
592C22—31no—
593C22—31no—
594C22—31no—
595C22—31no—
596C22—31no—
597C22—32no—
598C22—62no—
599C22—62no—
600C22—62no—
601C22—62no—
602C22—62no—
603C22—62no—
604C22—62no—
605C22—62no—
606C22—61no—
607C22—61no—
608C22—62no—
609C22—62no—
610C22—62no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22—122no—
619C22C32×#12111yescharacteristic, complemented
620C22C3×D4×A412yescomplemented
621C22C23×C3×A411yescharacteristic
622C4—21no—
623C4—22no—
624C4—32no—
625C4—61no—
626C4—62no—
627C4C3×D4×A412yescomplemented
628C3—43no—
629C3A4×#12111yescharacteristic, direct product factor
630C2—22no—
631C2—31no—
632C2—31no—
633C2—31no—
634C2—31no—
635C2—42no—
636C2—62no—
637C2—122no—
638C2C2×C3×D4×A411yescharacteristic
639C2C2×C3×D4×A411yescharacteristic
640C2C3×A4×#4111yescharacteristic
subgroupG/H
centerC22×C3C23×A4
commutator, cosocleC24C23×C32
FrattiniC22C23×C3×A4
FittingC22×C3×#121C3
socleC24×C3C23×C3
hypercenterC3×#121A4
hypocenterC22C32×#121
LCSG ▷ C24 ▷ C22
UCSC1 ◁ C22×C3 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×A4C2semidirect
2. C2×C3×D4×A4C2semidirect
3. C22×C3×C4×A4C2semidirect
4. C3×A4×C4⋊C4C2semidirect
5. C3×A4×C22⋊C4C2semidirect
6. C22×C3×#121C3semidirect
7. A4×#121C3direct (N × H)
17. C23×C3×D4C2×C3semidirect
18. C23×C3×D4C2×C3semidirect
19. C24×C3×C4C2×C3semidirect
20. C2×D4×A4C2×C3semidirect
21. C2×D4×A4C2×C3semidirect
22. C22×C4×A4C2×C3semidirect
29. C22×C3×C4⋊C4C2×C3semidirect
30. C22×C3×C22⋊C4C2×C3semidirect
31. A4×C4⋊C4C2×C3semidirect
32. A4×C22⋊C4C2×C3semidirect
38. C22×C3×A4D4semidirect
45. C3×C4×A4D4semidirect
48. C22×#121C32semidirect
90. C3×#121A4direct (N × H)
111. C23×D4C2×C32semidirect
112. C23×D4C2×C32semidirect
113. C24×C4C2×C32semidirect
120. C22×C4⋊C4C2×C32semidirect
121. C22×C22⋊C4C2×C32semidirect
137. C24×C3C3×D4semidirect
144. C22×A4C3×D4semidirect
169. C2×C3×D4C2×A4semidirect
170. C2×C3×D4C2×A4semidirect
195. C22×C3×C4C3×D4semidirect
196. C22×C3×C4C2×A4semidirect
199. C4×A4C3×D4semidirect
204. C3×C4⋊C4C2×A4semidirect
210. C3×C22⋊C4C2×A4semidirect
218. C3×A4#121direct (N × H)
254. #121C3×A4direct (N × H)
372. C24C32×D4semidirect
395. C2×D4C2×C3×A4semidirect
396. C2×D4C2×C3×A4semidirect
420. C22×C4C32×D4semidirect
421. C22×C4C2×C3×A4semidirect
425. C4⋊C4C2×C3×A4semidirect
430. C22⋊C4C2×C3×A4semidirect
475. C22×C3C3×#121semidirect
476. C22×C3D4×A4semidirect
478. A4C3×#121direct (N × H)
487. C3×C4D4×A4semidirect
619. C22C32×#121semidirect
620. C22C3×D4×A4semidirect
627. C4C3×D4×A4semidirect
629. C3A4×#121direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×A42
C22×C3×C4×A42
C3×A4×C4⋊C42
C3×A4×C22⋊C42
C22×C3×#1213
A4×#1213
C32×#1214