Finite Group Databasebeta

← #6986700

Subgroups — #6986700

Order 512.

1537 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic, complemented
2#35289C211yesmaximal, characteristic
3#32842C212yesmaximal, complemented
4#32294C214yesmaximal, complemented
5#32066C214yesmaximal, complemented
6#30343C212yesmaximal, complemented
7#30059C212yesmaximal, complemented
8#29690C211yesmaximal, characteristic
9#29211C212yesmaximal, complemented
10#28539C212yesmaximal, complemented
11#27821C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25450C212yesmaximal, complemented
14#24679C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic, complemented
16#23570C211yesmaximal, characteristic
17#22643C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#500C2212yes—
21C2×#477C2212yes—
22C2×#476C2212yes—
23C2×C4×SD16C2212yes—
24#2863C2212yes—
25#2862C2212yes—
26#2858C2212yes—
27#2857C2211yescharacteristic
28#2640—22no—
29#2605—22no—
30#2567C2211yescharacteristic
31#2563C2212yes—
32#2563C2212yes—
33#2551C2211yescharacteristic
34#2547—21no—
35#2547—21no—
36#2509—21no—
37#2504C2214yes—
38#2497C2212yes—
39#2492C2212yes—
40#2465—22no—
41#2439—21no—
42#2436—22no—
43#2369C2211yescharacteristic
44#2365—21no—
45#2355C2211yescharacteristic
46#2347—21no—
47#2311—21no—
48#2311—21no—
49C4×C4⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2093C2214yes—
55#2085C2214yes—
56#2074C2214yes—
57#2067C2214yes—
58#2058C2211yescharacteristic
59#2058C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2055C2214yes—
64#2041C2211yescharacteristic
65#2041C2212yes—
66#2034C2212yes—
67#2034C2214yes—
68#2024C2211yescharacteristic
69#2024C2212yes—
70#2023C2214yes—
71#2014C2211yescharacteristic
72#2000C2212yes—
73#1993C2214yes—
74#1990C2212yes—
75#1987C2212yes—
76#1973C2214yes—
77#1949C2214yes—
78#1949C2214yes—
79#1939C2212yes—
80#1936C2214yes—
81#1925C2214yes—
82#1925C2214yes—
83#1924C2214yes—
84#1872C2212yes—
85#1852C2212yes—
86#1852C2214yes—
87#1847C2214yes—
88#1839C2212yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92#1767C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—21no—
96C23×Q8—21no—
97C22×SD16C2312yes—
98#532—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#124—21no—
110C2×#124—22no—
111C2×#124—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—22no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#500—44no—
126#477—44no—
127#476—44no—
128C4×SD16—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#400—22no—
149#397—24no—
150#397—24no—
151#396—22no—
152#395—22no—
153#395—22no—
154#393—22no—
155#393—24no—
156#393—24no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—21no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#391C2312yes—
167#391C2311yescharacteristic
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—21no—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#386—24no—
191#386—24no—
192#385—21no—
193#384C2312yes—
194#384C2311yescharacteristic
195#383—22no—
196#382—21no—
197#382C2314yes—
198#382C2314yes—
199#380—21no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—44no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2411yescharacteristic
246C22×D4—22no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—24no—
256C2×SD16—42no—
257C2×SD16—42no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—44no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271#127—42no—
272#124—44no—
273#124—84no—
274#124—84no—
275C22⋊Q8—22no—
276C22⋊Q8—41no—
277C22⋊Q8—82no—
278#121—42no—
279#121—42no—
280#121—44no—
281C4×Q8—24no—
282C4×Q8—84no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—44no—
286#117—24no—
287#117—42no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C42—24no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343#107—44no—
344#107—44no—
345#107—44no—
346#106—42no—
347#106—44no—
348C4⋊C8—28no—
349C4⋊C8—24no—
350#103—42no—
351#103—44no—
352#102—24no—
353#102—24no—
354#102—44no—
355#102—42no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8—42no—
359#97—44no—
360C4×C8—24no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—22no—
385C24C22×D411yescharacteristic
386#41—816no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—21no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—82no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×Q8—816no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—41no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4#13714yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C82−1+414yes—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C42−1+412yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—81no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—82no—
521C42—82no—
522C42H(5,𝔽2)14yes—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C2×#13712yes—
537C23C23×D411yescharacteristic
538Q8—42no—
539Q8—42no—
540Q8—88no—
541Q8—84no—
542Q8—88no—
543Q8—88no—
544Q8—162no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—84no—
550D4—84no—
551D4—88no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—88no—
602C2×C4—88no—
603C2×C4#53512yes—
604C2×C4#53512yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4#57512yes—
608C2×C4#57612yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+412yes—
611C2×C4C2×2−1+411yescharacteristic
612C2×C4#58511yescharacteristic
613C8—24no—
614C8—44no—
615C8—42no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×#53511yescharacteristic
631C22#348911yescharacteristic
632C22#349911yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57511yescharacteristic
635C22C2×#57611yescharacteristic
636C22#358511yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—84no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4#321514yes—
654C4#354112yes—
655C4#355012yes—
656C2—42no—
657C2—162no—
658C2#4291111yescharacteristic
659C2C2×#321511yescharacteristic
660C2#6081111yescharacteristic
661C2#6211411yescharacteristic
662C2#6214311yescharacteristic
663C2#6248011yescharacteristic
664C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #36480C2semidirect
3. #32842C2semidirect
4. #32294C2semidirect
5. #32066C2semidirect
6. #30343C2semidirect
7. #30059C2semidirect
9. #29211C2semidirect
10. #28539C2semidirect
11. #27821C2semidirect
12. #25813C2semidirect
13. #25450C2semidirect
14. #24679C2semidirect
15. #24302C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#364802
#352892
#328422
#322942
#320662
#303432
#300592
#296902
#292112
#285392
#278212
#258132
#254502
#246792
#243022
#235702
#226432