Finite Group Databasebeta

← #72341821

Subgroups — #72341821

Order 1536.

1270 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 42 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
1#11710604C211yesmaximal, characteristic
2C2×#1791⋊C3C211yesmaximal, characteristic
3#20610⋊C3C211yesmaximal, characteristic
4#392611C311yesmaximal, characteristic, complemented, Sylow
5C2×#5950C2211yescharacteristic
6C4×#997—21no—
7A4×#106—41nomaximal
8#1791⋊C3C412yescomplemented
9#1790⋊C3—21no—
10#62654C2×C311yescharacteristic
11#20858—31no—
12#20848—31no—
13C2×#1791C2×C311yescharacteristic
14#20610C2×C311yescharacteristic
15#10765—31no—
16#10752—31no—
17#5950C2×C411yescharacteristic
18#5950Q811yescharacteristic
19C2×#997D411yescharacteristic
20C2×C8×A4—41no—
21#520⋊C3—81no—
22A4×C4⋊C4—41no—
23A4×C4⋊C4—41no—
24A4×C4⋊C4—81no—
25C2×#585C22×C311yescharacteristic
26C4×2−1+4—62no—
27C4×H(5,𝔽2)—21no—
28C4×H(5,𝔽2)—61no—
29C4×H(5,𝔽2)—62no—
30#3428—31no—
31#3428—61no—
32#3428—61no—
33C2×#520—31no—
34C2×#520—61no—
35C2×#520—61no—
36C2×#428—31no—
37C22×#106A411yescharacteristic, complemented
38C2×#413—31no—
39C2×#410—31no—
40C2×#408—31no—
41#2311—31no—
42#1857—62no—
43#1856—62no—
44#1854—61no—
45#1854—61no—
46#1853—61no—
47#1853—61no—
48#1791—32no—
49#1791C3×C412yescomplemented
50#1790—21no—
51#1790—61no—
52#1736—31no—
53#1735—31no—
54#1733—32no—
55#997C4⋊C411yescharacteristic
56C2×#41⋊C3—81no—
57C2×C4×A4—41no—
58C2×C4×A4—81no—
59C2×C4×A4—81no—
60C2×C4×A4—81no—
61C2×C4×A4—82no—
62C8×A4—42no—
63C4×SL(2,𝔽3)—82no—
64C3×#106—161no—
65#585—31no—
66#585—31no—
67#585—61no—
68#585—61no—
69#585—61no—
70#585—61no—
71#585C2×C3×C411yescharacteristic
72#585C3×Q811yescharacteristic
73C2×2−1+4—61no—
74C2×H(5,𝔽2)—31no—
75C2×H(5,𝔽2)—61no—
76C2×H(5,𝔽2)C3×D411yescharacteristic
77C22×#41—31no—
78C22×#41—61no—
79C22×#41—61no—
80C22×M16—31no—
81C23×C8C2×A411yescharacteristic
82#520—21no—
83#520—62no—
84#520—62no—
85#520—62no—
86#520—62no—
87#520—64no—
88#520—61no—
89#520—62no—
90#520—64no—
91#520—64no—
92#520—62no—
93#520—64no—
94#520—62no—
95#520—61no—
96#520—61no—
97#519—32no—
98#519—62no—
99#519—62no—
100#518—32no—
101#518—62no—
102#518—62no—
103#518—64no—
104#518—62no—
105#518—61no—
106#518—61no—
107#518—62no—
108C4×#41—64no—
109C4×#41—62no—
110C4×#41—64no—
111C4×#41—62no—
112C4×#41—64no—
113C4×#41—64no—
114C2×C4×Q8—61no—
115C2×C4×Q8—62no—
116C2×C4×Q8—62no—
117C2×C4×D4—61no—
118C2×C4×D4—61no—
119C2×C4×D4—62no—
120C2×C4×D4—62no—
121C2×C4×D4—61no—
122C2×C4×D4—64no—
123C2×C4×D4—62no—
124C2×C4×D4—62no—
125C2×#117—31no—
126C2×#117—31no—
127C2×#117—31no—
128C2×#117—31no—
129C2×#117—61no—
130C2×#117—61no—
131C2×#117—61no—
132C2×#117—62no—
133C2×#117—61no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—31no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—61no—
138C22×C4⋊C4C2×A411yescharacteristic
139C22×C4⋊C4C2×A411yescharacteristic
140#428—64no—
141C2×#107—61no—
142C2×#106—61no—
143C2×#106—61no—
144C2×#106—61no—
145#413—32no—
146#413—32no—
147#413—61no—
148#413—61no—
149C2×#101—61no—
150C2×#100—61no—
151#410—61no—
152#410—61no—
153#410—61no—
154#410—61no—
155C2×#99—61no—
156C2×#99—61no—
157#408—32no—
158#408—32no—
159#408—61no—
160#408—61no—
161C2×C22⋊C8—61no—
162C2×C22⋊C8—61no—
163#391—61no—
164#390—61no—
165#386—61no—
166#386—61no—
167#380—61no—
168#380—61no—
169#340—61no—
170#340—61no—
171#340—61no—
172#340—61no—
173#339—64no—
174C22×A4—41no—
175C22×A4—81no—
176#41⋊C3—81no—
177#41⋊C3—81no—
178C2×SL(2,𝔽3)—81no—
179C4×A4—41no—
180C4×A4—41no—
181C4×A4—81no—
182C4×A4—81no—
183C4×A4—161no—
184C4×A4—161no—
185C4×A4—162no—
186C2×C3×C8—161no—
187C3×C4⋊C4—161no—
188C3×C4⋊C4—161no—
1892−1+4—61no—
1902−1+4—62no—
1912−1+4—62no—
1922−1+4—61no—
1932−1+4—61no—
1942−1+4—61no—
1952−1+4—62no—
1962−1+4—62no—
1972−1+4—62no—
1982−1+4—62no—
199H(5,𝔽2)—32no—
200H(5,𝔽2)—31no—
201H(5,𝔽2)—31no—
202H(5,𝔽2)—31no—
203H(5,𝔽2)—61no—
204H(5,𝔽2)—61no—
205H(5,𝔽2)—62no—
206H(5,𝔽2)—62no—
207H(5,𝔽2)—61no—
208H(5,𝔽2)—62no—
209H(5,𝔽2)—62no—
210H(5,𝔽2)—61no—
211H(5,𝔽2)—61no—
212H(5,𝔽2)—61no—
213H(5,𝔽2)—61no—
214H(5,𝔽2)—61no—
215H(5,𝔽2)—62no—
216H(5,𝔽2)—62no—
217H(5,𝔽2)—62no—
218H(5,𝔽2)—62no—
219H(5,𝔽2)C3×C4⋊C411yescharacteristic
220C2×#41—21no—
221C2×#41—31no—
222C2×#41—31no—
223C2×#41—31no—
224C2×#41—31no—
225C2×#41—31no—
226C2×#41—31no—
227C2×#41—31no—
228C2×#41—31no—
229C2×#41—61no—
230C2×#41—61no—
231C2×#41—61no—
232C2×#41—61no—
233C2×#41—62no—
234C2×#41—61no—
235C2×#41—62no—
236C2×#41—62no—
237C2×#41—62no—
238C2×#41—62no—
239C2×#41—61no—
240C2×#41—62no—
241C2×#41—62no—
242C2×#41—61no—
243C2×#41—61no—
244C2×#41—62no—
245C2×#41—62no—
246C2×#41—61no—
247C2×#41—61no—
248C2×#41—61no—
249C2×#41—61no—
250C2×#41—61no—
251C2×#41—61no—
252C2×#41—61no—
253C2×#41—61no—
254C2×#41—62no—
255C2×#41—62no—
256C2×#41—62no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—61no—
261C2×#41—61no—
262C2×#41—61no—
263C22×Q8—31no—
264C22×Q8—61no—
265C22×Q8—61no—
266C22×D4—31no—
267C22×D4—31no—
268C22×D4—31no—
269C22×D4—61no—
270C22×D4—62no—
271C22×D4—61no—
272C22×D4—61no—
273C22×D4—61no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—31no—
279C23×C4—31no—
280C23×C4—61no—
281C23×C4—61no—
282C23×C4—61no—
283C23×C4—62no—
284C23×C4C22×A411yescharacteristic
285C2×M16—32no—
286C2×M16—61no—
287C2×M16—62no—
288C2×M16—62no—
289C22×C8—61no—
290C22×C8—61no—
291C22×C8—61no—
292C22×C8C4×A412yescomplemented
293C4×Q8—22no—
294C4×Q8—64no—
295C4×Q8—64no—
296C4×Q8—62no—
297C4×Q8—64no—
298C4×Q8—62no—
299C4×Q8—64no—
300C4×Q8—64no—
301C4×Q8—62no—
302C4×D4—64no—
303C4×D4—64no—
304C4×D4—64no—
305C4×D4—62no—
306C4×D4—64no—
307C4×D4—64no—
308C4×D4—64no—
309C4×D4—64no—
310C4×D4—64no—
311C4×D4—64no—
312C4×D4—64no—
313C4×D4—64no—
314C4×D4—62no—
315C4×D4—64no—
316C4×D4—64no—
317C4×D4—64no—
318C4×D4—64no—
319C4×D4—64no—
320C4×D4—64no—
321C4×D4—62no—
322C4×D4—64no—
323C4×D4—62no—
324#117—34no—
325#117—62no—
326#117—61no—
327#117—62no—
328#117—62no—
329#117—62no—
330#117—62no—
331#117—62no—
332#117—62no—
333#117—62no—
334#117—64no—
335#117—62no—
336#117—62no—
337#117—64no—
338#117—64no—
339#117—64no—
340#117—61no—
341#117—62no—
342#117—62no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—61no—
348C2×C4⋊C4—62no—
349C2×C4⋊C4—62no—
350C2×C4⋊C4—62no—
351C2×C4⋊C4—61no—
352C2×C4⋊C4—61no—
353C2×C4⋊C4—61no—
354C2×C4⋊C4—64no—
355C2×C4⋊C4—62no—
356C2×C4⋊C4—61no—
357C2×C4⋊C4—61no—
358C2×C4⋊C4—62no—
359C2×C4⋊C4—61no—
360C2×C4⋊C4—61no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—61no—
363C2×C4⋊C4—61no—
364C2×C4⋊C4—61no—
365C2×C22⋊C4—62no—
366C2×C22⋊C4—61no—
367C2×C22⋊C4—61no—
368C2×C22⋊C4—62no—
369C2×C22⋊C4—62no—
370C2×C22⋊C4—62no—
371C2×C22⋊C4—64no—
372C2×C22⋊C4—61no—
373C2×C22⋊C4—61no—
374C2×C22⋊C4—121no—
375C2×C22⋊C4—121no—
376C2×C42—61no—
377C2×C42—61no—
378C2×C42—61no—
379C2×C42—62no—
380C2×C42—62no—
381C2×C42—62no—
382C2×C42—64no—
383C2×C42—62no—
384C2×C42—61no—
385#107—122no—
386#106—41no—
387#106—121no—
388#106—122no—
389#101—62no—
390#101—62no—
391#100—62no—
392#100—62no—
393#99—121no—
394#99—121no—
395#99—121no—
396#99—121no—
397C22⋊C8—62no—
398C22⋊C8—62no—
399C22⋊C8—62no—
400C22⋊C8—62no—
401#95—121no—
402#95—121no—
403C2×A4—41no—
404C2×A4—41no—
405C2×A4—41no—
406C2×A4—81no—
407C2×A4—81no—
408C2×A4—81no—
409C2×C3×C4—161no—
410C2×C3×C4—321no—
411C2×C3×C4—321no—
412SL(2,𝔽3)—81no—
413C3×C8—162no—
414C24—21no—
415C24—31no—
416C24—31no—
417C24—61no—
418C24D4×A411yescharacteristic
419#41—21no—
420#41—21no—
421#41—62no—
422#41—62no—
423#41—62no—
424#41—62no—
425#41—62no—
426#41—62no—
427#41—62no—
428#41—62no—
429#41—62no—
430#41—62no—
431#41—62no—
432#41—62no—
433#41—62no—
434#41—62no—
435#41—62no—
436#41—62no—
437#41—61no—
438#41—61no—
439#41—61no—
440#41—62no—
441#41—62no—
442#41—62no—
443#41—62no—
444#41—62no—
445#41—62no—
446#41—61no—
447#41—61no—
448#41—62no—
449#41—62no—
450#41—62no—
451#41—62no—
452#41—62no—
453#41—62no—
454#41—62no—
455#41—62no—
456#41—62no—
457#41—62no—
458#41—62no—
459#41—62no—
460#41—62no—
461#41—62no—
462#41—62no—
463#41—62no—
464#41—62no—
465#41—62no—
466#41—61no—
467#41—61no—
468#41—62no—
469#41—62no—
470#41—62no—
471#41—62no—
472#41—62no—
473#41—62no—
474#41—61no—
475#41—61no—
476#41—61no—
477#41—61no—
478#41—61no—
479#41—61no—
480#41—61no—
481C2×Q8—21no—
482C2×Q8—31no—
483C2×Q8—31no—
484C2×Q8—31no—
485C2×Q8—31no—
486C2×Q8—62no—
487C2×Q8—62no—
488C2×Q8—61no—
489C2×Q8—61no—
490C2×Q8—61no—
491C2×Q8—61no—
492C2×Q8—61no—
493C2×Q8—61no—
494C2×Q8—62no—
495C2×Q8—62no—
496C2×Q8—62no—
497C2×Q8—62no—
498C2×Q8—61no—
499C2×Q8—61no—
500C2×Q8—61no—
501C2×D4—31no—
502C2×D4—31no—
503C2×D4—31no—
504C2×D4—31no—
505C2×D4—31no—
506C2×D4—31no—
507C2×D4—31no—
508C2×D4—31no—
509C2×D4—31no—
510C2×D4—31no—
511C2×D4—31no—
512C2×D4—31no—
513C2×D4—62no—
514C2×D4—62no—
515C2×D4—61no—
516C2×D4—61no—
517C2×D4—61no—
518C2×D4—62no—
519C2×D4—62no—
520C2×D4—62no—
521C2×D4—62no—
522C2×D4—62no—
523C2×D4—62no—
524C2×D4—62no—
525C2×D4—62no—
526C2×D4—62no—
527C2×D4—62no—
528C2×D4—61no—
529C2×D4—61no—
530C2×D4—61no—
531C2×D4—61no—
532C2×D4—61no—
533C2×D4—61no—
534C2×D4—61no—
535C2×D4—62no—
536C2×D4—61no—
537C2×D4—61no—
538C2×D4—62no—
539C2×D4—62no—
540C2×D4—62no—
541C2×D4—62no—
542C2×D4—61no—
543C2×D4—61no—
544C2×D4—62no—
545C2×D4—62no—
546C2×D4—62no—
547C2×D4—62no—
548C2×D4—62no—
549C2×D4—62no—
550C2×D4—61no—
551C2×D4—61no—
552C2×D4—61no—
553C2×D4—61no—
554C22×C4—21no—
555C22×C4—21no—
556C22×C4—31no—
557C22×C4—31no—
558C22×C4—31no—
559C22×C4—31no—
560C22×C4—31no—
561C22×C4—31no—
562C22×C4—31no—
563C22×C4—31no—
564C22×C4—31no—
565C22×C4—31no—
566C22×C4—31no—
567C22×C4—31no—
568C22×C4—31no—
569C22×C4—31no—
570C22×C4—61no—
571C22×C4—62no—
572C22×C4—61no—
573C22×C4—61no—
574C22×C4—61no—
575C22×C4—61no—
576C22×C4—61no—
577C22×C4—61no—
578C22×C4—62no—
579C22×C4—62no—
580C22×C4—62no—
581C22×C4—61no—
582C22×C4—61no—
583C22×C4—61no—
584C22×C4—61no—
585C22×C4—61no—
586C22×C4—61no—
587C22×C4—61no—
588C22×C4—61no—
589C22×C4—61no—
590C22×C4—61no—
591C22×C4—61no—
592C22×C4—61no—
593C22×C4—62no—
594C22×C4—62no—
595C22×C4—62no—
596C22×C4—64no—
597C22×C4—61no—
598C22×C4—62no—
599C22×C4—62no—
600C22×C4—121no—
601C22×C4—121no—
602C22×C4—161no—
603C22×C4—161no—
604C22×C4—162no—
605C22×C4—481no—
606C22×C4—481no—
607C22×C4—482no—
608C22×C4C2×C4×A411yescharacteristic
609C22×C4Q8×A411yescharacteristic
610M16—64no—
611M16—122no—
612C2×C8—41no—
613C2×C8—62no—
614C2×C8—62no—
615C2×C8—121no—
616C2×C8—121no—
617C2×C8—121no—
618C4⋊C4—32no—
619C4⋊C4—32no—
620C4⋊C4—31no—
621C4⋊C4—41no—
622C4⋊C4—61no—
623C4⋊C4—62no—
624C4⋊C4—61no—
625C4⋊C4—62no—
626C4⋊C4—62no—
627C4⋊C4—62no—
628C4⋊C4—64no—
629C4⋊C4—62no—
630C4⋊C4—62no—
631C4⋊C4—61no—
632C4⋊C4—61no—
633C4⋊C4—121no—
634C4⋊C4—121no—
635C4⋊C4—121no—
636C4⋊C4#120⋊C311yescharacteristic
637C22⋊C4—62no—
638C22⋊C4—64no—
639C22⋊C4—64no—
640C22⋊C4—64no—
641C22⋊C4—64no—
642C22⋊C4—62no—
643C22⋊C4—122no—
644C22⋊C4—481no—
645C22⋊C4—481no—
646C42—62no—
647C42—64no—
648C42—64no—
649C42—62no—
650C42—64no—
651C42—64no—
652C42—122no—
653C22×C3—161no—
654A4—41no—
655A4—81no—
656C3×C4—161no—
657C3×C4—161no—
658C3×C4—641no—
659C3×C4—641no—
660C23—21no—
661C23—31no—
662C23—31no—
663C23—31no—
664C23—31no—
665C23—31no—
666C23—31no—
667C23—41no—
668C23—41no—
669C23—61no—
670C23—61no—
671C23—62no—
672C23—61no—
673C23—61no—
674C23—61no—
675C23—61no—
676C23—61no—
677C23—81no—
678C23—81no—
679C23—121no—
680C23—121no—
681C23—121no—
682C23—121no—
683C23—241no—
684C23—241no—
685C23A4×C4⋊C411yescharacteristic
686Q8—21no—
687Q8—62no—
688Q8—62no—
689Q8—62no—
690Q8—62no—
691Q8—61no—
692Q8—61no—
693Q8—61no—
694Q8—62no—
695D4—62no—
696D4—62no—
697D4—62no—
698D4—62no—
699D4—62no—
700D4—62no—
701D4—62no—
702D4—62no—
703D4—62no—
704D4—62no—
705D4—62no—
706D4—62no—
707D4—61no—
708D4—61no—
709D4—61no—
710D4—61no—
711D4—62no—
712D4—62no—
713D4—62no—
714D4—62no—
715D4—62no—
716D4—62no—
717C2×C4—21no—
718C2×C4—32no—
719C2×C4—31no—
720C2×C4—31no—
721C2×C4—31no—
722C2×C4—31no—
723C2×C4—31no—
724C2×C4—31no—
725C2×C4—31no—
726C2×C4—31no—
727C2×C4—31no—
728C2×C4—31no—
729C2×C4—31no—
730C2×C4—61no—
731C2×C4—61no—
732C2×C4—61no—
733C2×C4—61no—
734C2×C4—61no—
735C2×C4—61no—
736C2×C4—61no—
737C2×C4—61no—
738C2×C4—61no—
739C2×C4—61no—
740C2×C4—62no—
741C2×C4—62no—
742C2×C4—62no—
743C2×C4—62no—
744C2×C4—62no—
745C2×C4—62no—
746C2×C4—61no—
747C2×C4—61no—
748C2×C4—61no—
749C2×C4—61no—
750C2×C4—61no—
751C2×C4—61no—
752C2×C4—61no—
753C2×C4—61no—
754C2×C4—61no—
755C2×C4—61no—
756C2×C4—61no—
757C2×C4—61no—
758C2×C4—61no—
759C2×C4—61no—
760C2×C4—62no—
761C2×C4—62no—
762C2×C4—61no—
763C2×C4—81no—
764C2×C4—241no—
765C2×C4—242no—
766C2×C4—241no—
767C2×C4—241no—
768C2×C4—242no—
769C2×C4—242no—
770C2×C4—242no—
771C2×C4—244no—
772C2×C4—241no—
773C2×C4—242no—
774C2×C4—242no—
775C2×C4—241no—
776C2×C4—241no—
777C2×C4—241no—
778C2×C4—241no—
779C2×C4—241no—
780C2×C4C2×#120⋊C311yescharacteristic
781C8—42no—
782C8—122no—
783C2×C3—161no—
784C2×C3—161no—
785C2×C3—161no—
786C22—31no—
787C22—31no—
788C22—41no—
789C22—61no—
790C22—61no—
791C22—61no—
792C22—61no—
793C22—61no—
794C22—61no—
795C22—81no—
796C22—121no—
797C22—121no—
798C22—121no—
799C22—121no—
800C22—121no—
801C22—121no—
802C22—121no—
803C22—121no—
804C22—121no—
805C22—241no—
806C22—241no—
807C22—241no—
808C22—241no—
809C22—241no—
810C22—242no—
811C22AΣL(1,𝔽64)11yescharacteristic
812C4—31no—
813C4—31no—
814C4—41no—
815C4—61no—
816C4—61no—
817C4—61no—
818C4—61no—
819C4—122no—
820C4—122no—
821C4—121no—
822C4—161no—
823C4—481no—
824C4C4×#120⋊C311yescharacteristic
825C4#2848⋊C311yescharacteristic
826C3—161noSylow
827C2—61no—
828C2—61no—
829C2—121no—
830C2—121no—
831C2—121no—
832C2—121no—
833C2#14693⋊C311yescharacteristic
834C2#32895⋊C311yescharacteristic
835C2#32896⋊C311yescharacteristic
subgroupG/H
center, socleC22AΣL(1,𝔽64)
commutator#585C2×C3×C4
FrattiniC23×C4C22×A4
Fitting#392611C3
prime radicalC2×#5950C22
cosocleC2×#585C22×C3
hypercenterC4⋊C4#120⋊C3
hypocenterH(5,𝔽2)C3×C4⋊C4
LCSG ▷ #585 ▷ H(5,𝔽2)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #392611C3semidirect
8. #1791⋊C3C4semidirect
37. C22×#106A4semidirect
49. #1791C3×C4semidirect
292. C22×C8C4×A4semidirect

Maximal subgroups

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

subgroupindex
#117106042
C2×#1791⋊C32
#20610⋊C32
#3926113
A4×#1064