Finite Group Databasebeta

← #7487728

Subgroups — #7487728

Order 512.

2743 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59816C211yesmaximal, characteristic, complemented
2#39362C211yesmaximal, characteristic, complemented
3#31382C214yesmaximal, complemented
4#31260C212yesmaximal, complemented
5#30499C212yesmaximal, complemented
6#30396C214yesmaximal, complemented
7#30038C212yesmaximal, complemented
8#29364C212yesmaximal, complemented
9#24424C214yesmaximal, complemented
10#23960C212yesmaximal, complemented
11#23848C212yesmaximal, complemented
12#23848C214yesmaximal, complemented
13#23355C211yesmaximal, characteristic, complemented
14C23×D8C2212yescomplemented
15C2×#547C2211yescharacteristic, complemented
16C2×D42—21no—
17C2×D42C2211yescharacteristic, complemented
18C2×D42C2214yescomplemented
19C2×#535—21no—
20C2×#466C2214yescomplemented
21C2×#447C2214yescomplemented
22C2×#447C2212yescomplemented
23C2×#447C2214yescomplemented
24C2×#447C2212yescomplemented
25C22×#102C2212yescomplemented
26#2858C2211yescharacteristic, complemented
27#2679—21no—
28#2679—21no—
29#2678—24no—
30#2620—21no—
31#2547C2211yescharacteristic, complemented
32#2541C2211yescharacteristic, complemented
33#2537—21no—
34#2537C2214yescomplemented
35#2435—21no—
36#2431—21no—
37#2416C2214yescomplemented
38#2408—24no—
39#2408C2212yes—
40#2408C2212yescomplemented
41#2408C2214yescomplemented
42C4×#127C2211yescharacteristic
43C4×#127C2212yescomplemented
44#2293C2211yescharacteristic, complemented
45#2099C2212yescomplemented
46#2093C2212yescomplemented
47#2075C2214yescomplemented
48#2071C2214yescomplemented
49#2041C2212yescomplemented
50#2033C2214yescomplemented
51#2033C2214yescomplemented
52#2023C2214yescomplemented
53#2011C2212yes—
54#2011C2214yescomplemented
55#2011C2214yescomplemented
56#1999C2214yescomplemented
57#1999C2214yescomplemented
58#1999C2214yescomplemented
59#1925C2214yescomplemented
60#1893C2214yescomplemented
61#1893C2212yescomplemented
62#1893C2212yescomplemented
63#1879C2214yescomplemented
64#1879C2214yescomplemented
65#1879C2214yescomplemented
66#1879C2214yescomplemented
67#1874C2214yescomplemented
68#1869C2212yescomplemented
69#1869C2212yescomplemented
70#1869C2214yescomplemented
71#1869C2214yescomplemented
72#1862C2214yescomplemented
73#1839C2212yescomplemented
74#1823C2212yescomplemented
75#1786C2214yescomplemented
76#1786C2212yescomplemented
77C22×#41—21no—
78C23×D4—24no—
79C23×D4—21no—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C23×D4C2312yes—
83C22×D8—48no—
84C22×D8—44no—
85C22×D8C2314yes—
86C22×D8C2314yes—
87C22×D8C2314yes—
88C23×C8C2312yes—
89C23×C8C2312yes—
90#547—42no—
91#547—42no—
92D42—42no—
93D42—42no—
94D42—48no—
95D42—48no—
96D42—48no—
97#535—48no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127—21no—
100C2×#127D411yescharacteristic
101C2×#127C2312yes—
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×#124—21no—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—21no—
108C2×#121—24no—
109C2×#121—24no—
110C2×#121—21no—
111C2×#121—24no—
112C2×#121C2314yes—
113C2×#121C2314yes—
114C2×#121C2312yes—
115C2×#121C2312yes—
116C2×#121C2314yes—
117C2×#120—22no—
118C2×#120—24no—
119C2×#120—22no—
120C2×#120—24no—
121C2×#120—24no—
122C2×#120—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2314yes—
130C2×#117—21no—
131C22×C4⋊C4C2312yes—
132#466—48no—
133#447—44no—
134#447—48no—
135#447—48no—
136#447—44no—
137C2×#107C2314yes—
138C2×#102—44no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2312yes—
151#397—22no—
152#396—24no—
153#394—24no—
154#394—22no—
155#394—22no—
156#394—24no—
157#394—24no—
158#394—24no—
159#392—22no—
160#392—22no—
161#392—24no—
162#392—22no—
163#391—21no—
164#390—21no—
165#390—22no—
166#390—22no—
167#390—22no—
168#390D414yes—
169#390D411yescharacteristic
170#390D411yescharacteristic
171#390C2314yes—
172#390C2314yes—
173#390C2312yes—
174#389C2312yes—
175#388—22no—
176#386—24no—
177#384D411yescharacteristic
178#384C2312yes—
179#380—24no—
180#380C2312yes—
181#380C2314yes—
182#380C2312yes—
183#379—22no—
184#379—22no—
185#379—24no—
186C4×C4⋊C4C2311yescharacteristic
187C4×C22⋊C4—21no—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2312yes—
195#336C2312yes—
196#336C2314yes—
197C25—22no—
198C25—22no—
199C25—22no—
200C2×#41—44no—
201C2×#41—44no—
202C2×#41—82no—
203C22×Q8—21no—
204C22×D4—28no—
205C22×D4—24no—
206C22×D4—28no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4—42no—
222C22×D4—44no—
223C22×D4—42no—
224C22×D4—44no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—416no—
230C22×D4—44no—
231C22×D4—416no—
232C22×D4—82no—
233C22×D4—88no—
234C22×D4C2×D411yescharacteristic
235C22×D4C2×D411yescharacteristic
236C22×D4C2×D414yes—
237C22×D4C2412yes—
238C22×D4C2414yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—24no—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C2×D8—28no—
246C2×D8—28no—
247C2×D8—28no—
248C2×D8—28no—
249C2×D8—44no—
250C2×D8—44no—
251C2×D8—44no—
252C2×D8—44no—
253C2×D8—44no—
254C2×D8—432no—
255C22×C8—42no—
256C22×C8—44no—
257C22×C8—42no—
258C22×C8—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—41no—
262C4⋊Q8—41no—
263#127—28no—
264#127—41no—
265#127—41no—
266#127—44no—
267#127D818yescomplemented
268#124—44no—
269#124—44no—
270#124—82no—
271#123—48no—
272C22⋊Q8—44no—
273#121—28no—
274#121—28no—
275#121—24no—
276#121—24no—
277#121—42no—
278#121—44no—
279#121—44no—
280#121—42no—
281#121—44no—
282#121—416no—
283#121—416no—
284#121—416no—
285#121—44no—
286#121—44no—
287#120—48no—
288#120—416no—
289#120—416no—
290#120—48no—
291#120—416no—
292#120—48no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—41no—
302C4×D4—41no—
303C4×D4—48no—
304C4×D4—84no—
305#117—44no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346#107—44no—
347#102—28no—
348#102—28no—
349#102—44no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354#102—44no—
355C22⋊C8—42no—
356C22⋊C8—44no—
357C22⋊C8—44no—
358C22⋊C8—42no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—24no—
364C24—21no—
365C24—24no—
366C24—22no—
367C24—48no—
368C24—44no—
369C24—48no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—82no—
374C24—82no—
375C24—84no—
376C24—84no—
377C24—84no—
378C24—82no—
379C24—82no—
380C24—84no—
381C24—82no—
382C24C22×D412yes—
383C24C22×D412yes—
384#41—816no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—21no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—41no—
402C2×D4—44no—
403C2×D4—41no—
404C2×D4—44no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—82no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—84no—
433C2×D4—82no—
434C2×D4—88no—
435C2×D4—84no—
436C2×D4—88no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—88no—
440C2×D4—816no—
441C2×D4—82no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—82no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4—864no—
448C2×D4C2×D814yes—
449C2×D4C2×D814yescomplemented
450C2×D4C2×D8116yescomplemented
451C2×D4H(5,𝔽2)18yes—
452C2×D4H(5,𝔽2)12yes—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—82no—
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C2511yescharacteristic
481D8—416no—
482D8—416no—
483D8—88no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—48no—
491C2×C8—84no—
492C4⋊C4—21no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—41no—
499C4⋊C4—44no—
500C4⋊C4—41no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—82no—
506C4⋊C4H(5,𝔽2)12yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C42—21no—
528C42—21no—
529C42—24no—
530C42—42no—
531C42—41no—
532C42—42no—
533C42C2×D814yescomplemented
534C23—22no—
535C23—24no—
536C23—22no—
537C23—24no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—28no—
544C23—44no—
545C23—42no—
546C23—41no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—41no—
553C23—44no—
554C23—42no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—816no—
560C23—88no—
561C23—816no—
562C23—88no—
563C23—88no—
564C23—88no—
565C23—164no—
566C23—164no—
567C23—168no—
568C23—168no—
569C23—168no—
570C23—164no—
571C23—168no—
572C23—164no—
573C23—164no—
574C23C22×D814yes—
575C23C22×D814yes—
576C23C23×D411yescharacteristic
577Q8—84no—
578D4—24no—
579D4—28no—
580D4—28no—
581D4—44no—
582D4—416no—
583D4—44no—
584D4—416no—
585D4—44no—
586D4—42no—
587D4—42no—
588D4—44no—
589D4—42no—
590D4—44no—
591D4—84no—
592D4—88no—
593D4—88no—
594D4—88no—
595D4—88no—
596D4—88no—
597D4—816no—
598D4—816no—
599D4—88no—
600D4—162no—
601D4—168no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—816no—
635C2×C4—88no—
636C2×C4C22×D812yes—
637C2×C4C22×D814yes—
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×H(5,𝔽2)14yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C8—44no—
642C8—44no—
643C22—24no—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—48no—
655C22—42no—
656C22—48no—
657C22—48no—
658C22—42no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22—82no—
666C22—82no—
667C22—84no—
668C22—84no—
669C22—82no—
670C22—1616no—
671C22—1616no—
672C22—1616no—
673C22#348412yes—
674C22#348411yescharacteristic
675C22C23×D811yescharacteristic
676C22#358511yescharacteristic
677C22#358512yes—
678C4—22no—
679C4—24no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—41no—
685C4—84no—
686C4—82no—
687C4—82no—
688C4#353212yes—
689C4#353712yes—
690C2—44no—
691C2—44no—
692C2—82no—
693C2—88no—
694C2—162no—
695C2#5200411yescharacteristic
696C2#6155911yescharacteristic
697C2#6155912yes—
698C2#6166912yes—
699C2#6167311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59816C2semidirect
2. #39362C2semidirect
3. #31382C2semidirect
4. #31260C2semidirect
5. #30499C2semidirect
6. #30396C2semidirect
7. #30038C2semidirect
8. #29364C2semidirect
9. #24424C2semidirect
10. #23960C2semidirect
11. #23848C2semidirect
12. #23848C2semidirect
13. #23355C2semidirect
14. C23×D8C22semidirect
15. C2×#547C22semidirect
17. C2×D42C22semidirect
18. C2×D42C22semidirect
20. C2×#466C22semidirect
21. C2×#447C22semidirect
22. C2×#447C22semidirect
23. C2×#447C22semidirect
24. C2×#447C22semidirect
25. C22×#102C22semidirect
26. #2858C22semidirect
31. #2547C22semidirect
32. #2541C22semidirect
34. #2537C22semidirect
37. #2416C22semidirect
40. #2408C22semidirect
41. #2408C22semidirect
43. C4×#127C22semidirect
44. #2293C22semidirect
45. #2099C22semidirect
46. #2093C22semidirect
47. #2075C22semidirect
48. #2071C22semidirect
49. #2041C22semidirect
50. #2033C22semidirect
51. #2033C22semidirect
52. #2023C22semidirect
54. #2011C22semidirect
55. #2011C22semidirect
56. #1999C22semidirect
57. #1999C22semidirect
58. #1999C22semidirect
59. #1925C22semidirect
60. #1893C22semidirect
61. #1893C22semidirect
62. #1893C22semidirect
63. #1879C22semidirect
64. #1879C22semidirect
65. #1879C22semidirect
66. #1879C22semidirect
67. #1874C22semidirect
68. #1869C22semidirect
69. #1869C22semidirect
70. #1869C22semidirect
71. #1869C22semidirect
72. #1862C22semidirect
73. #1839C22semidirect
74. #1823C22semidirect
75. #1786C22semidirect
76. #1786C22semidirect
267. #127D8semidirect
449. C2×D4C2×D8semidirect
450. C2×D4C2×D8semidirect
533. C42C2×D8semidirect

Maximal subgroups

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

subgroupindex
#598162
#393622
#313822
#312602
#304992
#303962
#300382
#293642
#244242
#239602
#238482
#233552