Finite Group Databasebeta

← #6632013

Subgroups — #6632013

Order 512.

2475 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35102C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic, complemented
3#32882C211yesmaximal, characteristic, complemented
4#32879C211yesmaximal, characteristic, complemented
5#31267C214yesmaximal, complemented
6#30508C214yesmaximal, complemented
7#30411C214yesmaximal, complemented
8#30403C214yesmaximal, complemented
9#29893C211yesmaximal, characteristic, complemented
10#29890C211yesmaximal, characteristic, complemented
11#27589C212yesmaximal, complemented
12#24381C212yesmaximal, complemented
13#24380C212yesmaximal, complemented
14#23937C212yesmaximal, complemented
15#21738C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C2×#529C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C2×#467C2214yescomplemented
21C2×#465C2214yescomplemented
22C2×#451C2212yes—
23C2×#451C2214yescomplemented
24C2×#448C2212yes—
25C2×#448C2214yescomplemented
26C22×C22⋊C8C2212yes—
27#2848C2211yescharacteristic
28#2848C2211yescharacteristic
29#2847C2211yescharacteristic
30#2847C2211yescharacteristic
31#2623—21no—
32#2613—21no—
33#2603—22no—
34#2598—22no—
35#2562C2212yes—
36#2540C2212yes—
37#2536C2212yes—
38#2535C2212yes—
39#2409—22no—
40#2409—22no—
41#2409C2214yes—
42#2409C2214yes—
43C2×#395—21no—
44C2×#393—21no—
45#2321—21no—
46#2321—21no—
47C2×#391C2212yes—
48C2×#386—21no—
49C2×#385—21no—
50#2283C2211yescharacteristic
51C2×#380C2211yescharacteristic
52#2097C2214yescomplemented
53#2077C2214yescomplemented
54#2075C2214yescomplemented
55#2074C2214yescomplemented
56#2036C2214yescomplemented
57#2035C2214yescomplemented
58#2035C2214yescomplemented
59#2033C2214yescomplemented
60#2026C2214yescomplemented
61#2015C2214yescomplemented
62#2001C2214yescomplemented
63#2000C2214yescomplemented
64#1928C2214yescomplemented
65#1927C2214yescomplemented
66#1892C2214yescomplemented
67#1892C2214yescomplemented
68#1878C2214yescomplemented
69#1877C2214yescomplemented
70#1874C2212yes—
71#1872C2212yes—
72#1869C2214yes—
73#1869C2214yes—
74#1853C2212yes—
75#1852C2212yes—
76#1817C2212yes—
77#1787C2212yes—
78#1787C2212yes—
79#1787C2214yescomplemented
80#1787C2214yescomplemented
81#1779C2211yescharacteristic
82#1779C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8—22no—
85C23×Q8—44no—
86C23×Q8—48no—
87C23×Q8C2312yes—
88C23×Q8C2314yes—
89C23×D4D411yescharacteristic
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×Q16C2314yes—
94C22×SD16C2314yes—
95C23×C8C2314yes—
96#529—44no—
97C2×#126—21no—
98C2×#125C2311yescharacteristic
99C2×#124C2311yescharacteristic
100C2×#123—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—48no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C22⋊Q8C2314yes—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×#121C2311yescharacteristic
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#120—21no—
113C2×#120—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—44no—
117C2×C4×D4D412yes—
118C2×C4×D4D412yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42C2311yescharacteristic
131#467—48no—
132#465—48no—
133#451—44no—
134#451—48no—
135#448—44no—
136#448—48no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8—44no—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153#400—22no—
154#399—22no—
155#398—22no—
156#397—22no—
157#395—24no—
158#395—24no—
159#394—22no—
160#394—22no—
161#393—24no—
162#393—24no—
163#392—22no—
164#392—22no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#391C2314yes—
169#390—22no—
170#390—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—24no—
176#386—24no—
177#386—21no—
178#386—21no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—21no—
184#385—21no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#383—21no—
190#382C2312yes—
191#381—21no—
192#380—22no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2314yes—
198#380C2311yescharacteristic
199#380C2311yescharacteristic
200#379—21no—
201#379—21no—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—21no—
204#376—21no—
205C2×#95—21no—
206C2×#95—22no—
207C2×#95—22no—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C25C2×D411yescharacteristic
215C25C2×D411yescharacteristic
216C22×Q8—28no—
217C22×Q8—22no—
218C22×Q8—24no—
219C22×Q8—28no—
220C22×Q8—28no—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—48no—
229C22×Q8—44no—
230C22×Q8—48no—
231C22×Q8—48no—
232C22×Q8—48no—
233C22×Q8—464no—
234C22×Q8C2411yescharacteristic
235C22×Q8C2414yes—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4C2×D412yes—
243C22×D4C2×D412yes—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—48no—
254C23×C4—44no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×Q16—28no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×SD16—28no—
268C2×SD16—44no—
269C2×SD16—44no—
270C22×C8—44no—
271C22×C8—44no—
272C22×C8—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275#126—44no—
276#125—42no—
277#124—42no—
278#123—44no—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C22⋊Q8—24no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C22⋊Q8—48no—
285C22⋊Q8—48no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288#121—24no—
289#121—44no—
290#121—44no—
291#120—44no—
292#120—44no—
293C4×Q8—42no—
294C4×D4—42no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—44no—
302#117—44no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4—48no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#106—44no—
357#103—28no—
358#103—28no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#102—44no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C22⋊C8—44no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—21no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24#13514yes—
394C24#13714yes—
395C24C22×D412yes—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398C2×Q8—28no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—416no—
409C2×Q8—416no—
410C2×Q8—44no—
411C2×Q8—48no—
412C2×Q8—416no—
413C2×Q8—416no—
414C2×Q8—416no—
415C2×Q8—464no—
416C2×Q8—432no—
417C2×Q8—416no—
418C2×Q8—44no—
419C2×Q8H(5,𝔽2)18yes—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—416no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—82no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—82no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—416no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—432no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—88no—
482C22×C4—82no—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4H(5,𝔽2)14yes—
491C22×C4C2511yescharacteristic
492Q16—88no—
493SD16—88no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—88no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—48no—
529C22⋊C4—44no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—42no—
545C42—82no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—28no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—28no—
562C23—48no—
563C23—48no—
564C23—44no—
565C23—44no—
566C23—48no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—82no—
573C23—82no—
574C23—84no—
575C23#53412yes—
576C23C2×#13514yes—
577C23C2×#13512yes—
578C23C2×#13714yes—
579C23C2×#13712yes—
580C23C23×D411yescharacteristic
581Q8—28no—
582Q8—28no—
583Q8—416no—
584Q8—432no—
585Q8—44no—
586Q8—44no—
587Q8—44no—
588D4—48no—
589D4—48no—
590D4—44no—
591D4—84no—
592D4—164no—
593D4—164no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—416no—
616C2×C4—432no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—84no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—84no—
644C2×C4—88no—
645C2×C4#53412yes—
646C2×C4C2×H(5,𝔽2)12yes—
647C2×C4C2×H(5,𝔽2)14yes—
648C8—44no—
649C8—44no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—416no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—88no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—82no—
674C22—82no—
675C22#319014yes—
676C22C2×#53411yescharacteristic
677C22#348412yes—
678C22C22×#13511yescharacteristic
679C22C22×#13711yescharacteristic
680C22#358612yes—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—42no—
687C4—42no—
688C4—48no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—84no—
693C4—82no—
694C4—82no—
695C4#353214yes—
696C2—24no—
697C2—44no—
698C2—44no—
699C2—161no—
700C2#3670211yescharacteristic
701C2C2×#319011yescharacteristic
702C2#6075411yescharacteristic
703C2#6156412yes—
704C2#6157412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35102C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2409C2semidirect
3. #32882C2semidirect
4. #32879C2semidirect
5. #31267C2semidirect
6. #30508C2semidirect
7. #30411C2semidirect
8. #30403C2semidirect
9. #29893C2semidirect
10. #29890C2semidirect
11. #27589C2semidirect
12. #24381C2semidirect
13. #24380C2semidirect
14. #23937C2semidirect
15. #21738C2semidirect
20. C2×#467C22semidirect
21. C2×#465C22semidirect
23. C2×#451C22semidirect
25. C2×#448C22semidirect
52. #2097C22semidirect
53. #2077C22semidirect
54. #2075C22semidirect
55. #2074C22semidirect
56. #2036C22semidirect
57. #2035C22semidirect
58. #2035C22semidirect
59. #2033C22semidirect
60. #2026C22semidirect
61. #2015C22semidirect
62. #2001C22semidirect
63. #2000C22semidirect
64. #1928C22semidirect
65. #1927C22semidirect
66. #1892C22semidirect
67. #1892C22semidirect
68. #1878C22semidirect
69. #1877C22semidirect
79. #1787C22semidirect
80. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#351022
C2×#24092
#328822
#328792
#312672
#305082
#304112
#304032
#298932
#298902
#275892
#243812
#243802
#239372
#217382