Finite Group Databasebeta

← #6506482

Subgroups — #6506482

Order 512.

2560 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#37307C211yesmaximal, characteristic, complemented
2#35527C211yesmaximal, characteristic
3#29115C212yesmaximal, complemented
4#29107C212yesmaximal, complemented
5#27279C214yesmaximal, complemented
6#27270C214yesmaximal, complemented
7#27259C214yesmaximal, complemented
8#27254C214yesmaximal, complemented
9#25867C212yesmaximal, complemented
10#25855C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20602C212yesmaximal, complemented
13#20530C211yesmaximal, characteristic, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C2×#490C2212yescomplemented
19C2×#488C2212yescomplemented
20C2×#486C2214yescomplemented
21C2×#417C2212yescomplemented
22C2×#418C2212yescomplemented
23C2×#416C2212yescomplemented
24C22×#102C2212yescomplemented
25C2×#408C2212yes—
26#2857C2212yescomplemented
27#2806—22no—
28#2748—22no—
29#2732—22no—
30#2683—22no—
31#2629C2212yescomplemented
32#2441C2214yescomplemented
33#2440C2214yescomplemented
34#2433—22no—
35#2433C2214yes—
36#2431—22no—
37#2431C2214yes—
38#2403—22no—
39#2403—22no—
40#2311C2212yes—
41#2309C2211yescharacteristic
42#2297C2211yescharacteristic
43#2297C2211yescharacteristic
44#2072C2212yescomplemented
45#2071C2214yescomplemented
46#2071C2212yescomplemented
47#2042C2214yescomplemented
48#2041C2214yescomplemented
49#2041C2214yescomplemented
50#2041C2214yescomplemented
51#2025C2214yescomplemented
52#2024C2212yescomplemented
53#2023C2212yescomplemented
54#1959C2212yescomplemented
55#1957C2212yescomplemented
56#1957C2214yescomplemented
57#1936C2214yescomplemented
58#1934C2214yescomplemented
59#1934C2214yescomplemented
60#1933C2214yescomplemented
61#1895C2214yescomplemented
62#1893C2214yescomplemented
63#1875C2214yescomplemented
64#1874C2214yescomplemented
65#1825C2214yescomplemented
66#1788C2214yes—
67#1788C2214yescomplemented
68#1786C2214yes—
69#1786C2214yescomplemented
70#1786C2214yescomplemented
71#1753C2214yescomplemented
72C8×C22⋊C4C2214yescomplemented
73#1750C2212yescomplemented
74C4×C22⋊C8C2212yescomplemented
75C2×H(5,𝔽2)—28no—
76C2×H(5,𝔽2)—44no—
77C2×H(5,𝔽2)—42no—
78C2×H(5,𝔽2)—42no—
79C22×#41—21no—
80C22×#41C2312yes—
81C22×#41C2311yescharacteristic
82C22×#41C2311yescharacteristic
83C23×D4—22no—
84C23×D4—21no—
85C23×D4—21no—
86C23×D4—21no—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C23×D4C2311yescharacteristic
90C22×M16C2312yes—
91C23×C8C2312yes—
92#535—28no—
93#535—42no—
94#535—42no—
95C2×C4⋊Q8C2312yes—
96C2×#127—21no—
97C2×#127—22no—
98C2×#127—44no—
99C2×#127C2314yes—
100C2×#127C2312yes—
101C2×#125C2312yes—
102C2×#124—21no—
103C2×#124C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×#121—22no—
107C2×#121C2314yes—
108C2×#120—22no—
109C2×#120—22no—
110C2×#120—24no—
111C2×#120—24no—
112C2×#117C2311yescharacteristic
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117#490—44no—
118#488—44no—
119#486—48no—
120#418—44no—
121#417—44no—
122#416—44no—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102—44no—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136#408—28no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2312yes—
140C2×C4×C8C2314yes—
141C2×C4×C8C2312yes—
142#399—24no—
143#398—24no—
144#398—24no—
145#396—24no—
146#396—24no—
147#395—24no—
148#392—24no—
149#392—24no—
150#392—24no—
151#392—24no—
152#391—22no—
153#391—22no—
154#391D412yes—
155#391C2312yes—
156#390—22no—
157#390—22no—
158#390D412yes—
159#390C2314yes—
160#390C2312yes—
161#389—22no—
162#389—24no—
163#389C2314yes—
164#389C2314yes—
165#387—24no—
166#386—22no—
167#386—22no—
168#386—21no—
169#386—21no—
170#386—24no—
171#386—24no—
172#386D412yes—
173#386D412yes—
174#385—24no—
175#384C2312yes—
176#384C2312yes—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180#379—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2314yes—
189#336C2312yes—
190C25—21no—
191C25—22no—
192C25—21no—
193H(5,𝔽2)—432no—
194H(5,𝔽2)—432no—
195C2×#41—22no—
196C2×#41—22no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—24no—
202C2×#41—42no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41—44no—
211C2×#41#4118yes—
212C22×Q8—22no—
213C22×Q8C2×D411yescharacteristic
214C22×Q8C2411yescharacteristic
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—24no—
219C22×D4—22no—
220C22×D4—24no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—41no—
234C22×D4—41no—
235C22×D4—42no—
236C22×D4—41no—
237C22×D4—41no—
238C22×D4—44no—
239C22×D4—48no—
240C22×D4—44no—
241C22×D4—48no—
242C22×D4—48no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—84no—
247C22×D4—84no—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C22×D4C2412yes—
251C22×D4C2411yescharacteristic
252C22×D4C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×M16—24no—
261C2×M16—42no—
262C2×M16—42no—
263C22×C8—24no—
264C22×C8—42no—
265C22×C8—42no—
266C22×C8C2414yes—
267C22×C8C2414yes—
268C4⋊Q8—44no—
269#127—28no—
270#127—24no—
271#127—44no—
272#127—42no—
273#127—48no—
274#127—44no—
275#125—42no—
276#125—42no—
277#124—24no—
278#124—42no—
279#124—44no—
280C22⋊Q8—88no—
281C22⋊Q8—84no—
282#121—28no—
283#121—28no—
284#121—48no—
285#120—48no—
286#120—48no—
287#120—88no—
288#120—88no—
289#117—24no—
290#117—41no—
291#117—41no—
292#117—42no—
293#117—42no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—42no—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327#103—28no—
328#103—44no—
329#102—28no—
330#102—28no—
331#102—28no—
332#102—44no—
333#102—44no—
334#102—44no—
335#102—44no—
336#102—44no—
337#102—44no—
338#102—44no—
339C22⋊C8—28no—
340C22⋊C8—28no—
341#97—42no—
342C4×C8—44no—
343C4×C8—42no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95C2×D414yes—
351#95C2×D414yes—
352C24—24no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—44no—
357C24—44no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—44no—
362C24—42no—
363C24—42no—
364C24—82no—
365C24—82no—
366C24—82no—
367C24—82no—
368C24—84no—
369C24—84no—
370C24C22×D411yescharacteristic
371C24C22×D412yes—
372#41—216no—
373#41—48no—
374#41—48no—
375#41—48no—
376#41—48no—
377#41—48no—
378#41—48no—
379#41—48no—
380#41—432no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—44no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8#12414yes—
390C2×Q8H(5,𝔽2)12yes—
391C2×D4—216no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—41no—
411C2×D4—42no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—432no—
427C2×D4—464no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—42no—
434C2×D4—48no—
435C2×D4—48no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—816no—
443C2×D4—816no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4#12418yes—
447C2×D4#12414yes—
448C2×D4H(5,𝔽2)12yes—
449C2×D4H(5,𝔽2)14yes—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—82no—
472C22×C4—84no—
473C22×C4C22×D414yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2511yescharacteristic
482M16—84no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—84no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—42no—
507C22⋊C4—48no—
508C22⋊C4—42no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—82no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—42no—
522C42—82no—
523C23—21no—
524C23—22no—
525C23—21no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—416no—
534C23—48no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—168no—
554C23—164no—
555C23—164no—
556C23—164no—
557C23—164no—
558C23—168no—
559C23—164no—
560C23—164no—
561C23C2×#12412yes—
562C23C2×#12414yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—44no—
566Q8—42no—
567Q8—164no—
568D4—24no—
569D4—24no—
570D4—24no—
571D4—44no—
572D4—48no—
573D4—48no—
574D4—416no—
575D4—48no—
576D4—416no—
577D4—42no—
578D4—42no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—816no—
583D4—816no—
584D4—84no—
585D4—168no—
586D4—168no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—416no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—164no—
621C2×C4C2×#12412yes—
622C2×C4C2×#12414yes—
623C2×C4#57614yes—
624C2×C4#57714yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×H(5,𝔽2)12yes—
629C8—44no—
630C8—44no—
631C22—22no—
632C22—24no—
633C22—28no—
634C22—24no—
635C22—41no—
636C22—42no—
637C22—41no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—81no—
647C22—81no—
648C22—82no—
649C22—84no—
650C22—84no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22—84no—
656C22—168no—
657C22—1616no—
658C22—168no—
659C22C22×#12411yescharacteristic
660C22#348411yescharacteristic
661C22#348411yescharacteristic
662C22C2×#57612yes—
663C22C2×#57712yes—
664C4—22no—
665C4—24no—
666C4—24no—
667C4—24no—
668C4—42no—
669C4—44no—
670C4—84no—
671C4—82no—
672C4—84no—
673C4#349012yes—
674C4#349012yes—
675C2—44no—
676C2—42no—
677C2—84no—
678C2—82no—
679C2—162no—
680C2#3536711yescharacteristic
681C2#6062212yes—
682C2#6163011yescharacteristic
683C2#6163111yescharacteristic
684C2#6166812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35527C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37307C2semidirect
3. #29115C2semidirect
4. #29107C2semidirect
5. #27279C2semidirect
6. #27270C2semidirect
7. #27259C2semidirect
8. #27254C2semidirect
9. #25867C2semidirect
10. #25855C2semidirect
11. #21770C2semidirect
12. #20602C2semidirect
13. #20530C2semidirect
18. C2×#490C22semidirect
19. C2×#488C22semidirect
20. C2×#486C22semidirect
21. C2×#417C22semidirect
22. C2×#418C22semidirect
23. C2×#416C22semidirect
24. C22×#102C22semidirect
26. #2857C22semidirect
31. #2629C22semidirect
32. #2441C22semidirect
33. #2440C22semidirect
44. #2072C22semidirect
45. #2071C22semidirect
46. #2071C22semidirect
47. #2042C22semidirect
48. #2041C22semidirect
49. #2041C22semidirect
50. #2041C22semidirect
51. #2025C22semidirect
52. #2024C22semidirect
53. #2023C22semidirect
54. #1959C22semidirect
55. #1957C22semidirect
56. #1957C22semidirect
57. #1936C22semidirect
58. #1934C22semidirect
59. #1934C22semidirect
60. #1933C22semidirect
61. #1895C22semidirect
62. #1893C22semidirect
63. #1875C22semidirect
64. #1874C22semidirect
65. #1825C22semidirect
67. #1788C22semidirect
69. #1786C22semidirect
70. #1786C22semidirect
71. #1753C22semidirect
72. C8×C22⋊C4C22semidirect
73. #1750C22semidirect
74. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#373072
#355272
#291152
#291072
#272792
#272702
#272592
#272542
#258672
#258552
#217702
#206022
#205302