Finite Group Databasebeta

← A42×#41 #72228097

Subgroups — A42×#41 #72228097

Order 2304.

5075 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 5075 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×A42C211yesmaximal, characteristic, complemented
2D4×A42C213yesmaximal, complemented
3C2×C4×A42C213yesmaximal, complemented
4#41×C22⋊A4C312yesmaximal, complemented
5C22×A4×#41C312yesmaximal, complemented
6C22×A42C2213yes—
7C3×A4×#41—42nomaximal
8C4×A42C2211yescharacteristic
9C4×A42C2213yes—
10Q8×C22⋊A4C2×C312yescomplemented
11D4×C22⋊A4C2×C316yescomplemented
12C2×C4×C22⋊A4C2×C316yescomplemented
13C2×A4×#41—32no—
14C2×A4×#41—36no—
15C2×A4×#41—36no—
16C2×A4×#41—32no—
17C22×Q8×A4C2×C312yescomplemented
18C22×D4×A4C2×C316yescomplemented
19C23×C4×A4C2×C316yescomplemented
20C2×A42—23no—
21C2×A42C2311yescharacteristic
22C3×Q8×A4—42no—
23C3×D4×A4—46no—
24C2×C3×C4×A4—46no—
25C24×#41C3211yescharacteristic, complemented, Sylow
26C22×C22⋊A4C22×C316yes—
27C24×A4C22×C316yes—
28C22×C3×#41—42no—
29C4×C22⋊A4C22×C312yes—
30C4×C22⋊A4C22×C316yes—
31A4×#41—312no—
32A4×#41—312no—
33A4×#41—34no—
34A4×#41—32no—
35A4×#41—36no—
36A4×#41—36no—
37A4×#41—44no—
38A4×#41—122no—
39A4×#41A412yesdirect product factor
40C2×Q8×A4—32no—
41C2×Q8×A4—36no—
42C2×D4×A4—36no—
43C2×D4×A4—36no—
44C2×D4×A4—312no—
45C22×C4×A4—36no—
46C22×C4×A4—36no—
47C22×C4×A4—36no—
48C22×C4×A4—36no—
49C22×C4×A4C22×C312yes—
50C22×C4×A4C22×C316yes—
51C22×C3×A4—46no—
52A42#4111yescharacteristic, direct product factor
53C32×#41—161no—
54C3×C4×A4—42no—
55C3×C4×A4—46no—
56C23×#41—36no—
57C23×#41—36no—
58C23×#41—32no—
59C23×#41—32no—
60C23×#41—91no—
61C23×#41—93no—
62C23×#41—93no—
63C23×#41—91no—
64C24×Q8C2×C3211yescharacteristic, complemented
65C24×D4C2×C3213yescomplemented
66C25×C4C2×C3213yescomplemented
67C2×C22⋊A4—26no—
68C2×C22⋊A4C23×C312yes—
69C23×A4—26no—
70C23×A4—36no—
71C23×A4—36no—
72C23×A4—66no—
73C23×A4C23×C312yes—
74C2×C3×#41—122no—
75C2×C3×#41—126no—
76C2×C3×#41—126no—
77C2×C3×#41—122no—
78C22×C3×Q8—42no—
79C22×C3×D4—46no—
80C23×C3×C4—46no—
81Q8×A4—34no—
82Q8×A4—36no—
83Q8×A4—44no—
84Q8×A4—122no—
85Q8×A4C2×A412yescomplemented
86D4×A4—312no—
87D4×A4—312no—
88D4×A4—36no—
89D4×A4—412no—
90D4×A4—126no—
91D4×A4C2×A416yescomplemented
92C2×C4×A4—312no—
93C2×C4×A4—36no—
94C2×C4×A4—32no—
95C2×C4×A4—36no—
96C2×C4×A4—36no—
97C2×C4×A4—36no—
98C2×C4×A4—32no—
99C2×C4×A4—36no—
100C2×C4×A4—412no—
101C2×C4×A4—126no—
102C2×C4×A4C2×A416yescomplemented
103C2×C3×A4—42no—
104C2×C3×A4—86no—
105C32×Q8—161no—
106C32×D4—163no—
107C2×C32×C4—163no—
108C26C22×C3213yes—
109C22×#41—312no—
110C22×#41—312no—
111C22×#41—34no—
112C22×#41—32no—
113C22×#41—32no—
114C22×#41—36no—
115C22×#41—36no—
116C22×#41—96no—
117C22×#41—96no—
118C22×#41—92no—
119C22×#41—91no—
120C22×#41—96no—
121C22×#41—96no—
122C22×#41—92no—
123C22×#41—96no—
124C22×#41—96no—
125C22×#41—92no—
126C22×#41—92no—
127C22×#41—93no—
128C22×#41—96no—
129C22×#41—96no—
130C22×#41—912no—
131C22×#41—96no—
132C22×#41—93no—
133C22×#41—96no—
134C22×#41—96no—
135C22×#41—91no—
136C22×#41—912no—
137C22×#41—912no—
138C22×#41—912no—
139C22×#41—912no—
140C22×#41—912no—
141C22×#41—912no—
142C22×#41—924no—
143C22×#41—94no—
144C22×#41—912no—
145C22×#41—94no—
146C22×#41—912no—
147C22×#41—912no—
148C22×#41C3×A412yescomplemented
149C23×Q8—36no—
150C23×Q8—32no—
151C23×Q8—91no—
152C23×Q8—93no—
153C23×D4—36no—
154C23×D4—36no—
155C23×D4—312no—
156C23×D4—93no—
157C23×D4—93no—
158C23×D4—96no—
159C24×C4—36no—
160C24×C4—36no—
161C24×C4—36no—
162C24×C4—36no—
163C24×C4—93no—
164C24×C4—93no—
165C24×C4—93no—
166C24×C4—93no—
167C24×C4C22×C3211yescharacteristic
168C24×C4C22×C3213yes—
169C24×C3—46no—
170C22⋊A4C3×#4112yescomplemented
171C22×A4—32no—
172C22×A4—32no—
173C22×A4—36no—
174C22×A4—412no—
175C22×A4—66no—
176C22×A4—66no—
177C22×A4—66no—
178C22×A4—66no—
179C22×A4—126no—
180C22×A4C3×#4112yescomplemented
181C22×A4C22×A416yes—
182C3×#41—42no—
183C3×#41—126no—
184C3×#41—126no—
185C3×#41—122no—
186C3×#41—124no—
187C3×#41—1212no—
188C3×#41—1212no—
189C3×#41—162no—
190C2×C3×Q8—122no—
191C2×C3×Q8—126no—
192C2×C3×D4—126no—
193C2×C3×D4—126no—
194C2×C3×D4—1212no—
195C22×C3×C4—42no—
196C22×C3×C4—46no—
197C22×C3×C4—126no—
198C22×C3×C4—126no—
199C22×C3×C4—126no—
200C22×C3×C4—126no—
201C4×A4—32no—
202C4×A4—36no—
203C4×A4—44no—
204C4×A4—412no—
205C4×A4—122no—
206C4×A4—126no—
207C4×A4C22×A412yes—
208C4×A4C22×A416yes—
209C22×C32—163no—
210C3×A4—42no—
211C32×C4—161no—
212C32×C4—163no—
213C25—23no—
214C25—36no—
215C25—36no—
216C25—66no—
217C25—93no—
218C25—93no—
219C25—183no—
220C25C23×C3211yescharacteristic
221C2×#41—36no—
222C2×#41—36no—
223C2×#41—32no—
224C2×#41—32no—
225C2×#41—96no—
226C2×#41—96no—
227C2×#41—912no—
228C2×#41—96no—
229C2×#41—912no—
230C2×#41—96no—
231C2×#41—96no—
232C2×#41—96no—
233C2×#41—912no—
234C2×#41—96no—
235C2×#41—912no—
236C2×#41—96no—
237C2×#41—92no—
238C2×#41—96no—
239C2×#41—96no—
240C2×#41—92no—
241C2×#41—91no—
242C2×#41—93no—
243C2×#41—93no—
244C2×#41—94no—
245C2×#41—912no—
246C2×#41—912no—
247C2×#41—912no—
248C2×#41—912no—
249C2×#41—912no—
250C2×#41—924no—
251C2×#41—912no—
252C2×#41—912no—
253C2×#41—94no—
254C2×#41—912no—
255C2×#41—912no—
256C2×#41—912no—
257C2×#41—912no—
258C2×#41—912no—
259C2×#41—924no—
260C2×#41—924no—
261C2×#41—912no—
262C2×#41—94no—
263C2×#41—912no—
264C2×#41—924no—
265C2×#41—912no—
266C2×#41—96no—
267C2×#41—924no—
268C2×#41—912no—
269C2×#41—912no—
270C2×#41—96no—
271C2×#41—92no—
272C2×#41—96no—
273C2×#41—96no—
274C2×#41—96no—
275C2×#41—924no—
276C2×#41—924no—
277C2×#41—924no—
278C2×#41—924no—
279C2×#41—948no—
280C2×#41—948no—
281C2×#41—924no—
282C2×#41—924no—
283C2×#41—924no—
284C2×#41—924no—
285C2×#41—96no—
286C2×#41—912no—
287C2×#41—924no—
288C2×#41—924no—
289C2×#41—96no—
290C2×#41—91no—
291C22×Q8—34no—
292C22×Q8—32no—
293C22×Q8—36no—
294C22×Q8—96no—
295C22×Q8—91no—
296C22×Q8—96no—
297C22×Q8—96no—
298C22×Q8—92no—
299C22×Q8—93no—
300C22×Q8—96no—
301C22×Q8—912no—
302C22×Q8—912no—
303C22×Q8—94no—
304C22×Q8C2×C3×A412yescomplemented
305C22×D4—312no—
306C22×D4—36no—
307C22×D4—312no—
308C22×D4—36no—
309C22×D4—96no—
310C22×D4—93no—
311C22×D4—96no—
312C22×D4—912no—
313C22×D4—96no—
314C22×D4—96no—
315C22×D4—912no—
316C22×D4—912no—
317C22×D4—912no—
318C22×D4—912no—
319C22×D4—912no—
320C22×D4—924no—
321C22×D4—924no—
322C22×D4—93no—
323C22×D4—912no—
324C22×D4—96no—
325C22×D4—96no—
326C22×D4C2×C3×A416yescomplemented
327C23×C4—312no—
328C23×C4—36no—
329C23×C4—36no—
330C23×C4—32no—
331C23×C4—36no—
332C23×C4—36no—
333C23×C4—36no—
334C23×C4—32no—
335C23×C4—36no—
336C23×C4—93no—
337C23×C4—91no—
338C23×C4—93no—
339C23×C4—96no—
340C23×C4—96no—
341C23×C4—96no—
342C23×C4—96no—
343C23×C4—96no—
344C23×C4—96no—
345C23×C4—96no—
346C23×C4—96no—
347C23×C4—96no—
348C23×C4—96no—
349C23×C4—912no—
350C23×C4—912no—
351C23×C4—912no—
352C23×C4—912no—
353C23×C4—912no—
354C23×C4—912no—
355C23×C4—912no—
356C23×C4—93no—
357C23×C4—93no—
358C23×C4—96no—
359C23×C4—96no—
360C23×C4—96no—
361C23×C4—93no—
362C23×C4—91no—
363C23×C4—93no—
364C23×C4C2×C3×A416yescomplemented
365C23×C3—42no—
366C23×C3—86no—
367C23×C3—126no—
368C23×C3—126no—
369C23×C3—246no—
370C2×A4—26no—
371C2×A4—32no—
372C2×A4—32no—
373C2×A4—44no—
374C2×A4—66no—
375C2×A4—812no—
376C2×A4—122no—
377C2×A4—246no—
378C2×A4C23×A412yes—
379C3×Q8—42no—
380C3×Q8—126no—
381C3×Q8—124no—
382C3×Q8—162no—
383C3×D4—46no—
384C3×D4—126no—
385C3×D4—1212no—
386C3×D4—1212no—
387C3×D4—166no—
388C2×C3×C4—46no—
389C2×C3×C4—122no—
390C2×C3×C4—126no—
391C2×C3×C4—126no—
392C2×C3×C4—126no—
393C2×C3×C4—126no—
394C2×C3×C4—1212no—
395C2×C3×C4—122no—
396C2×C3×C4—126no—
397C2×C3×C4—166no—
398C2×C32—161no—
399C2×C32—323no—
400C24—32no—
401C24—32no—
402C24—36no—
403C24—36no—
404C24—66no—
405C24—66no—
406C24—66no—
407C24—66no—
408C24—91no—
409C24—93no—
410C24—96no—
411C24—96no—
412C24—96no—
413C24—96no—
414C24—91no—
415C24—912no—
416C24—93no—
417C24—183no—
418C24—186no—
419C24—186no—
420C24—186no—
421C24—183no—
422C24—186no—
423C24—186no—
424C24—183no—
425C24—183no—
426C24—1812no—
427C24—1812no—
428C24—186no—
429C24C32×#4111yescharacteristic, complemented
430C24C22×C3×A416yes—
431#41—312no—
432#41—312no—
433#41—34no—
434#41—32no—
435#41—36no—
436#41—36no—
437#41—96no—
438#41—912no—
439#41—96no—
440#41—96no—
441#41—924no—
442#41—924no—
443#41—912no—
444#41—912no—
445#41—912no—
446#41—96no—
447#41—96no—
448#41—912no—
449#41—924no—
450#41—912no—
451#41—924no—
452#41—912no—
453#41—94no—
454#41—912no—
455#41—912no—
456#41—924no—
457#41—924no—
458#41—924no—
459#41—924no—
460#41—948no—
461#41—924no—
462#41—912no—
463#41—912no—
464#41—94no—
465#41—91no—
466#41—93no—
467#41—93no—
468#41A4211yescharacteristic, direct product factor
469C2×Q8—36no—
470C2×Q8—32no—
471C2×Q8—96no—
472C2×Q8—96no—
473C2×Q8—912no—
474C2×Q8—93no—
475C2×Q8—94no—
476C2×Q8—912no—
477C2×Q8—912no—
478C2×Q8—912no—
479C2×Q8—96no—
480C2×Q8—96no—
481C2×Q8—924no—
482C2×Q8—91no—
483C2×D4—36no—
484C2×D4—36no—
485C2×D4—312no—
486C2×D4—96no—
487C2×D4—96no—
488C2×D4—912no—
489C2×D4—96no—
490C2×D4—93no—
491C2×D4—912no—
492C2×D4—912no—
493C2×D4—912no—
494C2×D4—924no—
495C2×D4—924no—
496C2×D4—912no—
497C2×D4—96no—
498C2×D4—948no—
499C2×D4—924no—
500C2×D4—912no—
501C2×D4—93no—
502C2×D4—912no—
503C2×D4—912no—
504C2×D4—912no—
505C2×D4—912no—
506C2×D4—96no—
507C2×D4—924no—
508C2×D4—912no—
509C22×C4—36no—
510C22×C4—36no—
511C22×C4—36no—
512C22×C4—36no—
513C22×C4—32no—
514C22×C4—36no—
515C22×C4—32no—
516C22×C4—36no—
517C22×C4—91no—
518C22×C4—93no—
519C22×C4—96no—
520C22×C4—96no—
521C22×C4—93no—
522C22×C4—93no—
523C22×C4—93no—
524C22×C4—96no—
525C22×C4—912no—
526C22×C4—912no—
527C22×C4—912no—
528C22×C4—912no—
529C22×C4—912no—
530C22×C4—912no—
531C22×C4—912no—
532C22×C4—92no—
533C22×C4—96no—
534C22×C4—92no—
535C22×C4—92no—
536C22×C4—96no—
537C22×C4—96no—
538C22×C4—96no—
539C22×C4—96no—
540C22×C4—96no—
541C22×C4—96no—
542C22×C4—96no—
543C22×C4—96no—
544C22×C4—92no—
545C22×C4—96no—
546C22×C4—96no—
547C22×C4—96no—
548C22×C4—924no—
549C22×C4—924no—
550C22×C4—924no—
551C22×C4—96no—
552C22×C4—93no—
553C22×C4—96no—
554C22×C4—96no—
555C22×C4—96no—
556C22×C4—94no—
557C22×C4—912no—
558C22×C4—912no—
559C22×C4—912no—
560C22×C4—912no—
561C22×C4—96no—
562C22×C4—96no—
563C22×C4—96no—
564C22×C4—91no—
565C22×C4—93no—
566C22×C4C22×C3×A412yes—
567C22×C4C22×C3×A416yes—
568C22×C3—42no—
569C22×C3—46no—
570C22×C3—122no—
571C22×C3—126no—
572C22×C3—122no—
573C22×C3—166no—
574C22×C3—246no—
575C22×C3—246no—
576C22×C3—246no—
577C22×C3—246no—
578A4—44no—
579A4—122no—
580A4A4×#4112yesdirect product factor
581C3×C4—46no—
582C3×C4—42no—
583C3×C4—122no—
584C3×C4—126no—
585C3×C4—166no—
586C3×C4—162no—
587C32—161noSylow
588C23—26no—
589C23—36no—
590C23—36no—
591C23—32no—
592C23—32no—
593C23—32no—
594C23—66no—
595C23—66no—
596C23—66no—
597C23—91no—
598C23—91no—
599C23—96no—
600C23—93no—
601C23—92no—
602C23—92no—
603C23—92no—
604C23—92no—
605C23—912no—
606C23—96no—
607C23—91no—
608C23—94no—
609C23—96no—
610C23—93no—
611C23—186no—
612C23—186no—
613C23—183no—
614C23—186no—
615C23—183no—
616C23—186no—
617C23—186no—
618C23—186no—
619C23—186no—
620C23—186no—
621C23—186no—
622C23—186no—
623C23—186no—
624C23—1812no—
625C23—1812no—
626C23—186no—
627C23—183no—
628C23—186no—
629C23—183no—
630C23—1812no—
631C23—1812no—
632C23—1824no—
633C23—186no—
634C23—1812no—
635C23C23×C3×A412yes—
636Q8—34no—
637Q8—36no—
638Q8—96no—
639Q8—912no—
640Q8—94no—
641Q8—93no—
642Q8C2×A4211yescharacteristic, complemented
643D4—312no—
644D4—312no—
645D4—36no—
646D4—912no—
647D4—912no—
648D4—96no—
649D4—924no—
650D4—912no—
651D4—96no—
652D4—93no—
653D4C2×A4213yescomplemented
654C2×C4—312no—
655C2×C4—36no—
656C2×C4—36no—
657C2×C4—36no—
658C2×C4—32no—
659C2×C4—36no—
660C2×C4—32no—
661C2×C4—36no—
662C2×C4—91no—
663C2×C4—93no—
664C2×C4—96no—
665C2×C4—92no—
666C2×C4—96no—
667C2×C4—96no—
668C2×C4—96no—
669C2×C4—94no—
670C2×C4—912no—
671C2×C4—912no—
672C2×C4—912no—
673C2×C4—912no—
674C2×C4—912no—
675C2×C4—93no—
676C2×C4—93no—
677C2×C4—93no—
678C2×C4—91no—
679C2×C4—93no—
680C2×C4—92no—
681C2×C4—96no—
682C2×C4—92no—
683C2×C4—96no—
684C2×C4C2×A4213yescomplemented
685C2×C3—42no—
686C2×C3—86no—
687C2×C3—122no—
688C2×C3—122no—
689C2×C3—162no—
690C2×C3—246no—
691C2×C3—326no—
692C22—32no—
693C22—32no—
694C22—36no—
695C22—32no—
696C22—66no—
697C22—66no—
698C22—66no—
699C22—66no—
700C22—91no—
701C22—92no—
702C22—91no—
703C22—92no—
704C22—92no—
705C22—94no—
706C22—92no—
707C22—91no—
708C22—93no—
709C22—183no—
710C22—183no—
711C22—186no—
712C22—186no—
713C22—186no—
714C22—186no—
715C22—183no—
716C22—183no—
717C22—1812no—
718C22—1812no—
719C22—186no—
720C22—186no—
721C22C3×A4×#4112yescomplemented
722C22C22×A4213yes—
723C4—32no—
724C4—36no—
725C4—91no—
726C4—93no—
727C4C22×A4213yes—
728C4C22×A4211yescharacteristic
729C3—42no—
730C3—162no—
731C2—23no—
732C2—32no—
733C2—32no—
734C2—66no—
735C2—91no—
736C2—91no—
737C2—183no—
738C2C23×A4211yescharacteristic
subgroupG/H
centerC4C22×A42
commutator, socle, cosocleC25C23×C32
FrattiniC2C23×A42
FittingC24×#41C32
hypercenter#41A42
hypocenterC24C32×#41
LCSG ▷ C25 ▷ C24
UCSC1 ◁ C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. Q8×A42C2semidirect
2. D4×A42C2semidirect
3. C2×C4×A42C2semidirect
4. #41×C22⋊A4C3semidirect
5. C22×A4×#41C3semidirect
10. Q8×C22⋊A4C2×C3semidirect
11. D4×C22⋊A4C2×C3semidirect
12. C2×C4×C22⋊A4C2×C3semidirect
17. C22×Q8×A4C2×C3semidirect
18. C22×D4×A4C2×C3semidirect
19. C23×C4×A4C2×C3semidirect
25. C24×#41C32semidirect
39. A4×#41A4direct (N × H)
52. A42#41direct (N × H)
64. C24×Q8C2×C32semidirect
65. C24×D4C2×C32semidirect
66. C25×C4C2×C32semidirect
85. Q8×A4C2×A4semidirect
91. D4×A4C2×A4semidirect
102. C2×C4×A4C2×A4semidirect
148. C22×#41C3×A4semidirect
170. C22⋊A4C3×#41semidirect
180. C22×A4C3×#41semidirect
304. C22×Q8C2×C3×A4semidirect
326. C22×D4C2×C3×A4semidirect
364. C23×C4C2×C3×A4semidirect
429. C24C32×#41semidirect
468. #41A42direct (N × H)
580. A4A4×#41direct (N × H)
642. Q8C2×A42semidirect
653. D4C2×A42semidirect
684. C2×C4C2×A42semidirect
721. C22C3×A4×#41semidirect

Maximal subgroups

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

subgroupindex
Q8×A422
D4×A422
C2×C4×A422
#41×C22⋊A43
C22×A4×#413
C3×A4×#414