Finite Group Databasebeta

← #7487746

Subgroups — #7487746

Order 512.

2505 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59816C211yesmaximal, characteristic, complemented
2#38951C211yesmaximal, characteristic, complemented
3#31402C214yesmaximal, complemented
4#31271C212yesmaximal, complemented
5#30500C212yesmaximal, complemented
6#30401C214yesmaximal, complemented
7#30039C212yesmaximal, complemented
8#29365C212yesmaximal, complemented
9#24432C214yesmaximal, complemented
10#23960C212yesmaximal, complemented
11#23864C212yesmaximal, complemented
12#23860C214yesmaximal, complemented
13#23356C211yesmaximal, characteristic, complemented
14C22×AGL(1,ℤ/8)C2212yescomplemented
15C2×#547C2211yescharacteristic, complemented
16C2×#547C2211yescharacteristic, complemented
17C2×D42—21no—
18C2×D42C2214yes—
19C2×#465C2214yescomplemented
20C2×#449C2214yescomplemented
21C2×#450C2214yescomplemented
22C2×#450C2212yescomplemented
23C2×#447C2212yescomplemented
24C22×#102C2212yescomplemented
25#2862C2211yescharacteristic, complemented
26#2706—24no—
27#2679—21no—
28#2679—21no—
29#2620—21no—
30#2547C2211yescharacteristic, complemented
31#2542C2211yescharacteristic, complemented
32#2537—21no—
33#2537C2214yes—
34#2502—21no—
35#2431—21no—
36#2417C2214yescomplemented
37#2408—24no—
38#2408C2212yes—
39#2408C2212yes—
40#2407C2214yescomplemented
41#2311—21no—
42C4×#127C2212yes—
43C4×#127C2211yescharacteristic
44#2293C2211yescharacteristic, complemented
45#2100C2212yescomplemented
46#2094C2212yescomplemented
47#2077C2214yescomplemented
48#2072C2214yescomplemented
49#2041C2212yes—
50#2035C2214yescomplemented
51#2034C2214yescomplemented
52#2023C2214yescomplemented
53#2014C2214yes—
54#2013C2214yescomplemented
55#2011C2212yes—
56#2000C2214yescomplemented
57#2000C2214yescomplemented
58#1999C2214yescomplemented
59#1926C2214yescomplemented
60#1895C2214yescomplemented
61#1894C2212yescomplemented
62#1893C2212yescomplemented
63#1879C2214yescomplemented
64#1879C2214yescomplemented
65#1877C2214yescomplemented
66#1877C2214yescomplemented
67#1874C2214yescomplemented
68#1870C2212yescomplemented
69#1870C2214yescomplemented
70#1869C2212yescomplemented
71#1869C2214yescomplemented
72#1864C2214yescomplemented
73#1839C2212yescomplemented
74#1823C2212yescomplemented
75#1788C2212yescomplemented
76#1786C2214yescomplemented
77C22×#41—21no—
78C22×#41C2312yes—
79C23×D4—24no—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C2×AGL(1,ℤ/8)—44no—
83C2×AGL(1,ℤ/8)—48no—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×D8C2314yes—
87C22×M16C2312yes—
88C23×C8C2312yes—
89#547—42no—
90#547—42no—
91#547—42no—
92#547—42no—
93D42—48no—
94D42—48no—
95D42—48no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127D411yescharacteristic
99C2×#127C2312yes—
100C2×#124—21no—
101C2×#123—22no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2314yes—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121C2314yes—
110C2×#121C2312yes—
111C2×#121C2314yes—
112C2×#120—22no—
113C2×#120—22no—
114C2×#120—24no—
115C2×#120—24no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2314yes—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2314yes—
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126#465—48no—
127#450—44no—
128#450—48no—
129#449—48no—
130#447—44no—
131C2×#106C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102—44no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2312yes—
145#399—22no—
146#398—24no—
147#397—22no—
148#396—24no—
149#394—22no—
150#394—22no—
151#394—24no—
152#394—24no—
153#394—24no—
154#392—24no—
155#392—22no—
156#392—22no—
157#391—21no—
158#391—21no—
159#391—21no—
160#391C2314yes—
161#390—21no—
162#390—22no—
163#390—22no—
164#390D411yescharacteristic
165#390D411yescharacteristic
166#390D414yes—
167#390C2314yes—
168#390C2312yes—
169#389C2312yes—
170#388—22no—
171#386—24no—
172#386—24no—
173#385—24no—
174#384—21no—
175#384D411yescharacteristic
176#384C2312yes—
177#382—22no—
178#380—24no—
179#380—24no—
180#380C2312yes—
181#380C2314yes—
182#380C2312yes—
183#379—22no—
184#379—22no—
185C4×C4⋊C4C2311yescharacteristic
186C4×C22⋊C4—21no—
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2312yes—
194#336C2312yes—
195#336C2314yes—
196C25—22no—
197C25—22no—
198C2×#41—24no—
199C2×#41—42no—
200C2×#41—44no—
201C2×#41—44no—
202C2×#41—44no—
203C2×#41—44no—
204C2×#41—82no—
205C22×Q8—21no—
206C22×Q8C2412yes—
207C22×D4—28no—
208C22×D4—28no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—44no—
220C22×D4—44no—
221C22×D4—42no—
222C22×D4—42no—
223C22×D4—416no—
224C22×D4—416no—
225C22×D4—88no—
226C22×D4C2×D414yes—
227C22×D4C2×D411yescharacteristic
228C22×D4C2×D411yescharacteristic
229C22×D4C2414yes—
230C22×D4C2411yescharacteristic
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—22no—
235C23×C4C2412yes—
236C23×C4C2412yes—
237AGL(1,ℤ/8)—432no—
238C2×SD16—28no—
239C2×SD16—28no—
240C2×SD16—44no—
241C2×SD16—44no—
242C2×SD16—44no—
243C2×SD16—44no—
244C2×D8—28no—
245C2×D8—28no—
246C2×D8—44no—
247C2×M16—42no—
248C2×M16—44no—
249C22×C8—42no—
250C22×C8—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—41no—
254C4⋊Q8—41no—
255C4⋊Q8—41no—
256C4⋊Q8—41no—
257#127—24no—
258#127—28no—
259#124—82no—
260#123—48no—
261#123—48no—
262C22⋊Q8—24no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267#121—28no—
268#121—28no—
269#121—24no—
270#121—44no—
271#121—44no—
272#121—42no—
273#121—44no—
274#121—416no—
275#121—416no—
276#120—416no—
277#120—416no—
278#120—48no—
279#120—48no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—41no—
287C4×D4—41no—
288C4×D4—41no—
289C4×D4—41no—
290C4×D4—48no—
291C4×D4—84no—
292#117—82no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D414yes—
327C2×C22⋊C4C2×D414yes—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—21no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336#106—44no—
337#103—44no—
338#103—44no—
339#102—28no—
340#102—28no—
341#102—44no—
342#102—44no—
343#102—44no—
344#102—44no—
345C22⋊C8—42no—
346C22⋊C8—44no—
347C22⋊C8—42no—
348C22⋊C8—44no—
349#95—22no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356C24—21no—
357C24—24no—
358C24—48no—
359C24—48no—
360C24—44no—
361C24—44no—
362C24—82no—
363C24—82no—
364C24—84no—
365C24—84no—
366C24—82no—
367C24—82no—
368C24C22×D412yes—
369C24C22×D412yes—
370#41—48no—
371#41—84no—
372#41—88no—
373#41—816no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—21no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—44no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—82no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—88no—
416C2×D4—88no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4—82no—
423C2×D4—864no—
424C2×D4AGL(1,ℤ/8)14yescomplemented
425C2×D4H(5,𝔽2)18yes—
426C2×D4H(5,𝔽2)12yes—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C2511yescharacteristic
458SD16—416no—
459SD16—88no—
460D8—416no—
461M16—48no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—84no—
469C4⋊C4—21no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—41no—
476C4⋊C4—44no—
477C4⋊C4—41no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4H(5,𝔽2)12yes—
488C22⋊C4—24no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—22no—
507C42—24no—
508C42—21no—
509C42—21no—
510C42—24no—
511C42—41no—
512C42—42no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—41no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—41no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—816no—
535C23—88no—
536C23—88no—
537C23—816no—
538C23—164no—
539C23—168no—
540C23—168no—
541C23—164no—
542C23—164no—
543C23—164no—
544C23C2×AGL(1,ℤ/8)14yes—
545C23C23×D411yescharacteristic
546Q8—42no—
547Q8—42no—
548Q8—44no—
549Q8—84no—
550D4—24no—
551D4—28no—
552D4—28no—
553D4—44no—
554D4—416no—
555D4—416no—
556D4—44no—
557D4—44no—
558D4—42no—
559D4—82no—
560D4—88no—
561D4—88no—
562D4—88no—
563D4—816no—
564D4—88no—
565D4—816no—
566D4—88no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4#57714yes—
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×H(5,𝔽2)14yes—
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C8—44no—
610C8—44no—
611C22—24no—
612C22—28no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22—48no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—48no—
623C22—42no—
624C22—42no—
625C22—48no—
626C22—44no—
627C22—84no—
628C22—84no—
629C22—82no—
630C22—84no—
631C22—82no—
632C22—84no—
633C22—1616no—
634C22—1616no—
635C22#348411yescharacteristic
636C22#348412yes—
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C2×#57712yes—
639C22#358511yescharacteristic
640C4—22no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—82no—
651C4#353212yes—
652C4#353712yes—
653C2—44no—
654C2—82no—
655C2—82no—
656C2—88no—
657C2#5200411yescharacteristic
658C2#6156911yescharacteristic
659C2#6160012yes—
660C2#6166411yescharacteristic
661C2#6168512yes—
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. #38951C2semidirect
3. #31402C2semidirect
4. #31271C2semidirect
5. #30500C2semidirect
6. #30401C2semidirect
7. #30039C2semidirect
8. #29365C2semidirect
9. #24432C2semidirect
10. #23960C2semidirect
11. #23864C2semidirect
12. #23860C2semidirect
13. #23356C2semidirect
14. C22×AGL(1,ℤ/8)C22semidirect
15. C2×#547C22semidirect
16. C2×#547C22semidirect
19. C2×#465C22semidirect
20. C2×#449C22semidirect
21. C2×#450C22semidirect
22. C2×#450C22semidirect
23. C2×#447C22semidirect
24. C22×#102C22semidirect
25. #2862C22semidirect
30. #2547C22semidirect
31. #2542C22semidirect
36. #2417C22semidirect
40. #2407C22semidirect
44. #2293C22semidirect
45. #2100C22semidirect
46. #2094C22semidirect
47. #2077C22semidirect
48. #2072C22semidirect
50. #2035C22semidirect
51. #2034C22semidirect
52. #2023C22semidirect
54. #2013C22semidirect
56. #2000C22semidirect
57. #2000C22semidirect
58. #1999C22semidirect
59. #1926C22semidirect
60. #1895C22semidirect
61. #1894C22semidirect
62. #1893C22semidirect
63. #1879C22semidirect
64. #1879C22semidirect
65. #1877C22semidirect
66. #1877C22semidirect
67. #1874C22semidirect
68. #1870C22semidirect
69. #1870C22semidirect
70. #1869C22semidirect
71. #1869C22semidirect
72. #1864C22semidirect
73. #1839C22semidirect
74. #1823C22semidirect
75. #1788C22semidirect
76. #1786C22semidirect
424. C2×D4AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#598162
#389512
#314022
#312712
#305002
#304012
#300392
#293652
#244322
#239602
#238642
#238602
#233562