Finite Group Databasebeta

← #6573182

Subgroups — #6573182

Order 512.

1733 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30147C212yesmaximal, complemented
4#29690C211yesmaximal, characteristic
5#29492C212yesmaximal, complemented
6#28202C212yesmaximal, complemented
7#27948C211yesmaximal, characteristic, complemented
8#27647C211yesmaximal, characteristic, complemented
9#27096C214yesmaximal, complemented
10#26786C214yesmaximal, complemented
11#26044C212yesmaximal, complemented
12#24815C212yesmaximal, complemented
13#23837C212yesmaximal, complemented
14#23570C211yesmaximal, characteristic
15#23061C212yesmaximal, complemented
16#22805C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#483C2212yes—
21C2×#476C2212yes—
22C2×#476C2214yes—
23C2×C4×SD16C2212yes—
24#2863C2212yes—
25#2857C2211yescharacteristic
26#2605—22no—
27#2567C2211yescharacteristic
28#2551C2211yescharacteristic
29#2509—21no—
30#2497C2212yes—
31#2492C2212yes—
32#2465—22no—
33#2439—21no—
34#2369C2211yescharacteristic
35#2365—21no—
36#2361—22no—
37#2361C2214yes—
38#2355C2211yescharacteristic
39#2354—21no—
40#2354C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43#2347—21no—
44#2326—22no—
45#2311—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#127C2212yes—
49C4×C22⋊Q8C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2088C2214yes—
53#2084C2212yes—
54#2084C2212yes—
55#2084C2214yes—
56#2084C2212yes—
57#2072C2212yes—
58#2066C2214yes—
59#2058C2211yescharacteristic
60#2058C2214yes—
61#2058C2212yes—
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2034C2212yes—
65#2034C2214yes—
66#2024C2211yescharacteristic
67#2014C2211yescharacteristic
68#2000C2212yes—
69#1990C2212yes—
70#1968C2214yes—
71#1948C2212yes—
72#1939C2212yes—
73#1934C2214yes—
74#1927C2214yes—
75#1926C2214yes—
76#1926C2214yes—
77#1923C2214yes—
78#1920C2214yes—
79#1877C2214yes—
80#1864C2214yes—
81#1853C2214yes—
82#1844C2212yes—
83#1831C2212yes—
84#1823C2212yes—
85#1803C2212yes—
86#1802C2212yes—
87#1802C2214yes—
88#1797C2211yescharacteristic
89C4×#106C2214yes—
90C4×#106C2214yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97#532—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121—21no—
107C2×#121C2312yes—
108#520—48no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—24no—
113C2×C4×D4—21no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2314yes—
118C2×#117—22no—
119C2×#117D411yescharacteristic
120C2×#117D411yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126#483—44no—
127#476—28no—
128#476—48no—
129C4×SD16—28no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#400—22no—
150#396—22no—
151#395—22no—
152#395—22no—
153#393—22no—
154#391—22no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2311yescharacteristic
161#391C2312yes—
162#390—22no—
163#390—22no—
164#390C2314yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389C2311yescharacteristic
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—24no—
180#388—24no—
181#387—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—24no—
188#385—21no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2312yes—
197#383—22no—
198#382—21no—
199#380—21no—
200#380—22no—
201#380—22no—
202#380C2311yescharacteristic
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
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#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2411yescharacteristic
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—24no—
252C2×SD16—24no—
253C2×SD16—42no—
254C22×C8C2412yes—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262#127—24no—
263#124—28no—
264C22⋊Q8—22no—
265C22⋊Q8—41no—
266C22⋊Q8—82no—
267#121—28no—
268#121—24no—
269#121—24no—
270C4×Q8—24no—
271C4×Q8—48no—
272C4×Q8—84no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—416no—
278C4×D4—48no—
279C4×D4—84no—
280C4×D4#4118yes—
281#117—42no—
282#117—48no—
283#117SD1618yescomplemented
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—48no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#106—24no—
346#106—28no—
347#106—28no—
348#106—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—24no—
353#103—44no—
354#102—24no—
355#102—28no—
356#102—24no—
357#102—24no—
358#102—44no—
359C22⋊C8—24no—
360C22⋊C8—42no—
361C4×C8—28no—
362C4×C8—24no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
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—432no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—82no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4C2×#4114yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—41no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4C2×SD1614yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C82−1+414yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4C2×SD1614yes—
493C4⋊C4C2×SD1614yes—
494C4⋊C4C2×#4114yes—
495C4⋊C4H(5,𝔽2)12yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—48no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—81no—
504C22⋊C4—82no—
505C22⋊C4C2×SD1618yes—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—42no—
514C42—42no—
515C42—42no—
516C42—48no—
517C42—42no—
518C42—82no—
519C42—82no—
520C42C2×SD1618yes—
521C42C2×#4114yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—24no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C22×SD1612yes—
536C23C23×D411yescharacteristic
537Q8—48no—
538Q8—42no—
539Q8—42no—
540Q8—88no—
541Q8—162no—
542D4—24no—
543D4—24no—
544D4—48no—
545D4—48no—
546D4—416no—
547D4—42no—
548D4—42no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—416no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—168no—
598C2×C4—164no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4C22×SD1614yes—
602C2×C4C22×SD1612yes—
603C2×C4C22×SD1614yes—
604C2×C4C22×SD1612yes—
605C2×C4#57512yes—
606C2×C4#57712yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×2−1+412yes—
610C2×C4#58511yescharacteristic
611C8—24no—
612C8—42no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C2×#52511yescharacteristic
629C22#348811yescharacteristic
630C22#349911yescharacteristic
631C22C23×SD1611yescharacteristic
632C22C2×#57511yescharacteristic
633C22C2×#57711yescharacteristic
634C22#358511yescharacteristic
635C4—24no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—24no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—41no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—84no—
651C4#310114yes—
652C4#349512yes—
653C4#351112yes—
654C2—42no—
655C2—84no—
656C2#3608911yescharacteristic
657C2C2×#310111yescharacteristic
658C2#6058811yescharacteristic
659C2#6195111yescharacteristic
660C2#6199611yescharacteristic
661C2#6246911yescharacteristic
662C2#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. #30147C2semidirect
5. #29492C2semidirect
6. #28202C2semidirect
7. #27948C2semidirect
8. #27647C2semidirect
9. #27096C2semidirect
10. #26786C2semidirect
11. #26044C2semidirect
12. #24815C2semidirect
13. #23837C2semidirect
15. #23061C2semidirect
16. #22805C2semidirect
17. #22632C2semidirect
283. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#364802
#342072
#301472
#296902
#294922
#282022
#279482
#276472
#270962
#267862
#260442
#248152
#238372
#235702
#230612
#228052
#226322