Finite Group Databasebeta

← #6479664

Subgroups — #6479664

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35296C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic, complemented
3#32962C212yesmaximal, complemented
4#32911C212yesmaximal, complemented
5#29914C212yesmaximal, complemented
6#29622C212yesmaximal, complemented
7#29505C214yesmaximal, complemented
8#29365C212yesmaximal, complemented
9#29364C212yesmaximal, complemented
10#29127C211yesmaximal, characteristic, complemented
11#29124C211yesmaximal, characteristic, complemented
12#28381C211yesmaximal, characteristic, complemented
13#28369C211yesmaximal, characteristic, complemented
14#28210C212yesmaximal, complemented
15#28165C212yesmaximal, complemented
16#22056C214yesmaximal, complemented
17#20781C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518C2212yes—
20C2×#476C2212yescomplemented
21C2×#474C2212yescomplemented
22C2×#459C2212yescomplemented
23C2×#459C2212yescomplemented
24C2×#447C2214yes—
25#2864C2211yescharacteristic, complemented
26#2858C2211yescharacteristic, complemented
27#2854C2212yes—
28#2644C2212yescomplemented
29#2628C2211yescharacteristic
30#2619C2211yescharacteristic
31#2595—22no—
32#2557—22no—
33#2474—22no—
34#2471—22no—
35#2366—22no—
36#2366—22no—
37D4×C22⋊C4—22no—
38D4×C22⋊C4—22no—
39#2331C2211yescharacteristic, complemented
40#2328C2212yes—
41#2328C2212yes—
42#2328C2211yescharacteristic, complemented
43#2324C2212yes—
44#2309C2212yes—
45C4×#127C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#127C2211yescharacteristic
49#2293C2212yescomplemented
50#2285C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2212yescomplemented
53#2099C2212yescomplemented
54#2094C2212yescomplemented
55#2093C2212yescomplemented
56#2085C2214yescomplemented
57#2073C2214yescomplemented
58#2041C2214yes—
59#2041C2212yes—
60#2041C2212yes—
61#2011C2214yes—
62#2011C2212yes—
63#2011C2212yes—
64#1972C2214yescomplemented
65#1970C2214yescomplemented
66#1936C2214yescomplemented
67#1934C2214yescomplemented
68#1926C2214yescomplemented
69#1925C2214yescomplemented
70#1879C2214yescomplemented
71#1879C2214yescomplemented
72#1879C2214yescomplemented
73#1879C2214yescomplemented
74#1875C2214yes—
75#1864C2214yescomplemented
76#1862C2214yescomplemented
77#1854C2214yes—
78#1839C2212yescomplemented
79#1832C2212yescomplemented
80#1817C2212yes—
81#1804C2212yescomplemented
82#1804C2212yescomplemented
83#1804C2212yescomplemented
84#1804C2212yescomplemented
85#1798C2212yes—
86#1786C2214yes—
87#1767C2211yescharacteristic, complemented
88C4×C4⋊C8C2211yescharacteristic, complemented
89#1750C2212yes—
90#1746C2212yescomplemented
91C23×D4—22no—
92C23×D4C2312yes—
93C22×D8C2314yes—
94C22×D8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127—28no—
97C2×#127—22no—
98C2×#127C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#127C2312yes—
101C2×#127C2311yescharacteristic
102C2×#124—22no—
103C2×#123—22no—
104C2×#121C2312yes—
105#518—44no—
106#518—44no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42C2311yescharacteristic
125#476—44no—
126#474—44no—
127#459—44no—
128#459—44no—
129#447—216no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2312yes—
144C2×C4×C8C2312yes—
145#400—21no—
146#399—21no—
147#397—22no—
148#397—22no—
149#397—22no—
150#396—22no—
151#395—21no—
152#394—21no—
153#394—21no—
154#393—21no—
155#392—21no—
156#392—21no—
157#391—22no—
158#390—24no—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#390C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#386—22no—
186#386—22no—
187#386—24no—
188#386—24no—
189#386—24no—
190#386—24no—
191#386—22no—
192#386—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2311yescharacteristic
197#382C2311yescharacteristic
198#382C2312yes—
199#380—22no—
200#380C2312yes—
201#379—24no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C25—22no—
223C22×Q8—21no—
224C22×D4—216no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—24no—
233C22×D4—42no—
234C22×D4—42no—
235C22×D4—42no—
236C22×D4—42no—
237C22×D4—84no—
238C22×D4—84no—
239C22×D4#4118yes—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2411yescharacteristic
250C2×D8—28no—
251C2×D8—28no—
252C2×D8—44no—
253C2×D8—44no—
254C2×D8—44no—
255C2×D8—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—24no—
259#127—216no—
260#127—24no—
261#127—24no—
262#127—24no—
263#127—24no—
264#127—24no—
265#124—84no—
266#123—84no—
267#121—28no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C4×D4—42no—
278C4×D4—48no—
279C4×D4—48no—
280C4×D4—84no—
281C4×D4—84no—
282C4×D4—84no—
283C4×D4—84no—
284#117—24no—
285#117—41no—
286#117—41no—
287#117—42no—
288#117—42no—
289#117—84no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
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—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4C2×D411yescharacteristic
328C2×C22⋊C4C2×D411yescharacteristic
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—44no—
345#106—44no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348#102—28no—
349#102—28no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356C22⋊C8—28no—
357#97—42no—
358C4×C8—42no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—22no—
375C24—22no—
376C24—28no—
377C24—22no—
378C24—44no—
379C24—44no—
380C24—84no—
381C24—84no—
382C24C22×D411yescharacteristic
383C2×Q8—42no—
384C2×Q8—82no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—216no—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—832no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4C2×#4114yes—
424C2×D4C2×#4114yes—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2511yescharacteristic
458D8—88no—
459D8—88no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—82no—
521C42H(5,𝔽2)14yes—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—24no—
527C23—24no—
528C23—416no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—816no—
545C23—168no—
546C23—164no—
547C23—164no—
548C23#52512yes—
549C23C23×D411yescharacteristic
550Q8—162no—
551D4—24no—
552D4—24no—
553D4—24no—
554D4—24no—
555D4—416no—
556D4—416no—
557D4—416no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—44no—
563D4—44no—
564D4—816no—
565D4—162no—
566D4—162no—
567D4—164no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—164no—
605C2×C4—164no—
606C2×C4—164no—
607C2×C4—164no—
608C2×C4#52512yes—
609C2×C4#53512yes—
610C2×C4#53512yes—
611C2×C4#57614yes—
612C2×C4C22×#4112yes—
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4#58511yescharacteristic
615C2×C4#58511yescharacteristic
616C8—44no—
617C8—44no—
618C22—22no—
619C22—24no—
620C22—42no—
621C22—41no—
622C22—41no—
623C22—44no—
624C22—44no—
625C22—48no—
626C22—48no—
627C22—84no—
628C22—84no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—1616no—
641C22C2×#52511yescharacteristic
642C22C2×#53511yescharacteristic
643C22#344911yescharacteristic
644C22C2×#57612yes—
645C22#358512yes—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—41no—
654C4—42no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#350512yes—
661C4#350512yes—
662C2—42no—
663C2—84no—
664C2—84no—
665C2—162no—
666C2#3510211yescharacteristic
667C2#6057812yes—
668C2#6081711yescharacteristic
669C2#6082711yescharacteristic
670C2#6099012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35296C2semidirect
2. #34210C2semidirect
3. #32962C2semidirect
4. #32911C2semidirect
5. #29914C2semidirect
6. #29622C2semidirect
7. #29505C2semidirect
8. #29365C2semidirect
9. #29364C2semidirect
10. #29127C2semidirect
11. #29124C2semidirect
12. #28381C2semidirect
13. #28369C2semidirect
14. #28210C2semidirect
15. #28165C2semidirect
16. #22056C2semidirect
17. #20781C2semidirect
20. C2×#476C22semidirect
21. C2×#474C22semidirect
22. C2×#459C22semidirect
23. C2×#459C22semidirect
25. #2864C22semidirect
26. #2858C22semidirect
28. #2644C22semidirect
39. #2331C22semidirect
42. #2328C22semidirect
49. #2293C22semidirect
52. #2100C22semidirect
53. #2099C22semidirect
54. #2094C22semidirect
55. #2093C22semidirect
56. #2085C22semidirect
57. #2073C22semidirect
64. #1972C22semidirect
65. #1970C22semidirect
66. #1936C22semidirect
67. #1934C22semidirect
68. #1926C22semidirect
69. #1925C22semidirect
70. #1879C22semidirect
71. #1879C22semidirect
72. #1879C22semidirect
73. #1879C22semidirect
75. #1864C22semidirect
76. #1862C22semidirect
78. #1839C22semidirect
79. #1832C22semidirect
81. #1804C22semidirect
82. #1804C22semidirect
83. #1804C22semidirect
84. #1804C22semidirect
87. #1767C22semidirect
88. C4×C4⋊C8C22semidirect
90. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#352962
#342102
#329622
#329112
#299142
#296222
#295052
#293652
#293642
#291272
#291242
#283812
#283692
#282102
#281652
#220562
#207812