Finite Group Databasebeta

← #6660840

Subgroups — #6660840

Order 512.

2051 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35272C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic, complemented
3#32730C214yesmaximal, complemented
4#30324C212yesmaximal, complemented
5#30316C212yesmaximal, complemented
6#29910C212yesmaximal, complemented
7#29758C212yesmaximal, complemented
8#29712C212yesmaximal, complemented
9#29622C212yesmaximal, complemented
10#29340C212yesmaximal, complemented
11#28319C212yesmaximal, complemented
12#24512C214yesmaximal, complemented
13#24015C212yesmaximal, complemented
14#23925C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#535C2212yes—
18C22×#127C2211yescharacteristic, complemented
19C2×C4×#41C2211yescharacteristic
20C2×#482C2214yescomplemented
21C2×#447C2214yes—
22C2×#447C2214yescomplemented
23C2×#433C2212yescomplemented
24#2856C2212yes—
25#2856C2212yescomplemented
26#2854C2212yes—
27#2854C2212yescomplemented
28#2624C2212yes—
29#2624C2212yescomplemented
30#2619C2212yes—
31#2619C2212yescomplemented
32#2548—22no—
33#2541—22no—
34#2448—22no—
35#2445—22no—
36#2418—22no—
37#2418—22no—
38#2408—22no—
39#2408—22no—
40#2316C2211yescharacteristic, complemented
41#2311C2211yescharacteristic
42#2309C2211yescharacteristic, complemented
43#2309C2212yescomplemented
44#2296C2211yescharacteristic, complemented
45C2×#390C2211yescharacteristic, complemented
46#2285C2211yescharacteristic, complemented
47#2101C2214yescomplemented
48#2092C2214yescomplemented
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2071C2212yescomplemented
52#2071C2212yescomplemented
53#2067C2214yescomplemented
54#2065C2214yescomplemented
55#2054C2214yescomplemented
56#2054C2214yescomplemented
57#2041C2212yescomplemented
58#2041C2212yescomplemented
59#2041C2214yescomplemented
60#2034C2214yescomplemented
61#2033C2214yescomplemented
62#2011C2212yes—
63#2011C2212yes—
64#2011C2214yes—
65#1999C2214yescomplemented
66#1999C2214yescomplemented
67#1972C2214yescomplemented
68#1970C2214yescomplemented
69#1936C2214yescomplemented
70#1934C2214yescomplemented
71#1875C2214yes—
72#1875C2214yescomplemented
73#1874C2214yescomplemented
74#1854C2214yescomplemented
75#1817C2212yescomplemented
76#1798C2212yescomplemented
77#1797C2212yescomplemented
78#1786C2214yescomplemented
79#1780C2212yes—
80#1750C2212yescomplemented
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×D4—22no—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C22×D8C2314yes—
88C22×D8C2314yes—
89#535—44no—
90#535—48no—
91C2×#127—21no—
92C2×#127—22no—
93C2×#127—22no—
94C2×#127—44no—
95C2×#127C2312yes—
96C2×#127C2312yes—
97C2×#124—21no—
98C2×#124—22no—
99C2×#123—22no—
100C2×#123—22no—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121—22no—
105C2×#121C2314yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#121C2314yes—
109C2×#120—22no—
110C2×#120—22no—
111C2×#120—21no—
112C2×#120—21no—
113C4×#41—44no—
114C2×C4×Q8D412yes—
115C2×C4×D4D412yes—
116C2×C4×D4D412yes—
117C2×C4×D4D412yes—
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#482—48no—
130#447—48no—
131#447—48no—
132#433—44no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#399—22no—
151#399—22no—
152#397—22no—
153#397—22no—
154#394—24no—
155#394—24no—
156#394—22no—
157#394—22no—
158#394—22no—
159#394—22no—
160#392—24no—
161#392—24no—
162#391—21no—
163#390—21no—
164#390—22no—
165#390—24no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#390C2312yes—
173#390C2312yes—
174#390C2314yes—
175#390C2312yes—
176#390C2312yes—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#386—21no—
187#386—21no—
188#386—22no—
189#386—22no—
190#385—22no—
191#384C2312yes—
192#382C2312yes—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#379—24no—
198#379—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2312yes—
206#336C2312yes—
207#336C2314yes—
208C25—21no—
209C25—22no—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41—42no—
213C22×Q8C2×D411yescharacteristic
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—24no—
223C22×D4—28no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4—44no—
227C22×D4—42no—
228C22×D4—42no—
229C22×D4—42no—
230C22×D4—48no—
231C22×D4—84no—
232C22×D4—84no—
233C22×D4C2×D411yescharacteristic
234C22×D4C2×D411yescharacteristic
235C22×D4C2×D411yescharacteristic
236C22×D4C2414yes—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×D8—44no—
246C2×D8—44no—
247C2×D8—44no—
248C2×D8—44no—
249C2×D8—44no—
250C2×D8—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253#127—44no—
254#127—42no—
255#127—42no—
256#127—48no—
257#127—44no—
258#127—48no—
259#124—44no—
260#124—48no—
261#123—84no—
262#123—84no—
263#121—28no—
264#121—28no—
265#121—28no—
266#121—44no—
267#121—44no—
268#121—44no—
269#121—84no—
270#121—84no—
271#121—84no—
272#121—84no—
273#120—44no—
274#120—48no—
275#120—44no—
276#120—48no—
277C4×Q8—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281#117—24no—
282#117—24no—
283#117—42no—
284#117—42no—
285#117—42no—
286#117—44no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—84no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—44no—
341#106—44no—
342C4⋊C8—44no—
343#102—44no—
344#102—44no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350#102—44no—
351C22⋊C8—28no—
352C22⋊C8—28no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355#97—42no—
356C4×C8—42no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364C24—22no—
365C24—24no—
366C24—42no—
367C24—44no—
368C24—42no—
369C24—44no—
370C24—44no—
371C24—42no—
372C24—82no—
373C24—84no—
374C24—84no—
375C24—82no—
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378C24C22×D411yescharacteristic
379#41—416no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—41no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—88no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—84no—
423C2×D4—84no—
424C2×D4—832no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4—82no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—82no—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C4C2511yescharacteristic
465D8—88no—
466D8—88no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—42no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—88no—
551C23—84no—
552C23—816no—
553C23—88no—
554C23—162no—
555C23—164no—
556C23—164no—
557C23—162no—
558C23—164no—
559C23—168no—
560C23#53412yes—
561C23#53512yes—
562C23C23×D411yescharacteristic
563Q8—44no—
564D4—44no—
565D4—44no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—44no—
570D4—44no—
571D4—44no—
572D4—44no—
573D4—88no—
574D4—84no—
575D4—88no—
576D4—88no—
577D4—816no—
578D4—164no—
579D4—168no—
580D4—164no—
581D4—168no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—162no—
619C2×C4#53412yes—
620C2×C4#53512yes—
621C2×C4#57614yes—
622C2×C4C2×H(5,𝔽2)12yes—
623C2×C4C2×H(5,𝔽2)12yes—
624C2×C4#58512yes—
625C8—44no—
626C8—44no—
627C22—22no—
628C22—22no—
629C22—24no—
630C22—24no—
631C22—41no—
632C22—41no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—82no—
643C22—84no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—84no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—168no—
657C22—1616no—
658C22C2×#53411yescharacteristic
659C22C2×#53511yescharacteristic
660C22#344911yescharacteristic
661C22C2×#57612yes—
662C22#358512yes—
663C4—22no—
664C4—22no—
665C4—24no—
666C4—41no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4#353414yes—
675C2—42no—
676C2—42no—
677C2—81no—
678C2—164no—
679C2—162no—
680C2#3744011yescharacteristic
681C2#6075612yes—
682C2#6083212yes—
683C2#6101111yescharacteristic
684C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35272C2semidirect
2. #35254C2semidirect
3. #32730C2semidirect
4. #30324C2semidirect
5. #30316C2semidirect
6. #29910C2semidirect
7. #29758C2semidirect
8. #29712C2semidirect
9. #29622C2semidirect
10. #29340C2semidirect
11. #28319C2semidirect
12. #24512C2semidirect
13. #24015C2semidirect
14. #23925C2semidirect
15. #20720C2semidirect
18. C22×#127C22semidirect
20. C2×#482C22semidirect
22. C2×#447C22semidirect
23. C2×#433C22semidirect
25. #2856C22semidirect
27. #2854C22semidirect
29. #2624C22semidirect
31. #2619C22semidirect
40. #2316C22semidirect
42. #2309C22semidirect
43. #2309C22semidirect
44. #2296C22semidirect
45. C2×#390C22semidirect
46. #2285C22semidirect
47. #2101C22semidirect
48. #2092C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
51. #2071C22semidirect
52. #2071C22semidirect
53. #2067C22semidirect
54. #2065C22semidirect
55. #2054C22semidirect
56. #2054C22semidirect
57. #2041C22semidirect
58. #2041C22semidirect
59. #2041C22semidirect
60. #2034C22semidirect
61. #2033C22semidirect
65. #1999C22semidirect
66. #1999C22semidirect
67. #1972C22semidirect
68. #1970C22semidirect
69. #1936C22semidirect
70. #1934C22semidirect
72. #1875C22semidirect
73. #1874C22semidirect
74. #1854C22semidirect
75. #1817C22semidirect
76. #1798C22semidirect
77. #1797C22semidirect
78. #1786C22semidirect
80. #1750C22semidirect
81. #1749C22semidirect
82. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#352722
#352542
#327302
#303242
#303162
#299102
#297582
#297122
#296222
#293402
#283192
#245122
#240152
#239252
#207202