Finite Group Databasebeta

← #6565047

Subgroups — #6565047

Order 512.

1667 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#36194C211yesmaximal, characteristic, complemented
2#34308C211yesmaximal, characteristic
3#30113C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#29335C212yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#28205C211yesmaximal, characteristic, complemented
8#27611C212yesmaximal, complemented
9#27349C214yesmaximal, complemented
10#26739C214yesmaximal, complemented
11#24971C212yesmaximal, complemented
12#24120C212yesmaximal
13#23761C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23068C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22762C212yesmaximal, complemented
18C2×#547C2211yescharacteristic
19C2×#520—21no—
20C2×#475C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2571—21no—
25#2563—21no—
26#2538C2212yes—
27#2497—22no—
28#2492C2214yes—
29#2439—22no—
30#2369—21no—
31#2365C2211yescharacteristic
32#2361C2212yes—
33#2360—21no—
34#2359—22no—
35#2355C2212yes—
36#2354—21no—
37D4×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2211yescharacteristic
39#2344C2212yes—
40#2339—21no—
41#2323—21no—
42#2322—22no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#121C2212yes—
46#2293C2211yescharacteristic
47C2×#389C2212yes—
48#2285C2212yes—
49C42×D4C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2099C2212yes—
53#2093C2212yes—
54#2084C2212yes—
55#2084C2211yescharacteristic
56#2076C2214yes—
57#2071C2211yescharacteristic
58#2066C2212yes—
59#2057C2212yes—
60#2055C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2036C2214yes—
64#2034C2214yes—
65#2023C2212yes—
66#2013C2211yescharacteristic
67#2000C2214yes—
68#1990C2212yes—
69#1988C2214yes—
70#1954C2212yes—
71#1947C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1924C2214yes—
77#1920C2214yes—
78#1920C2214yes—
79#1877C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1853C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1824C2214yes—
86#1817C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97#547—24no—
98#547—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#123—22no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2314yes—
107C2×#121C2312yes—
108#520—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—22no—
122C2×#117—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#475—28no—
129C4×SD16—28no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—24no—
147#396—24no—
148#393—24no—
149#391—21no—
150#391—21no—
151#391—24no—
152#391C2312yes—
153#391C2314yes—
154#390—22no—
155#390C2311yescharacteristic
156#390C2311yescharacteristic
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#389C2312yes—
171#388—24no—
172#388—21no—
173#388—22no—
174#388—21no—
175#388—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—24no—
180#386—22no—
181#386—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#381—22no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—21no—
198#380C2312yes—
199#380C2314yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C2×#95—22no—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233C2×#41—28no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—24no—
250C2×SD16—42no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—44no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#123—216no—
262C22⋊Q8—28no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—24no—
266#121—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—44no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—44no—
280C4×D4#4114yes—
281#117—48no—
282#117—44no—
283#117—84no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—48no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#106—24no—
351#106—28no—
352C4⋊C8—24no—
353C4⋊C8—42no—
354#103—24no—
355#103—28no—
356#103—42no—
357#103—42no—
358#103—44no—
359#102—24no—
360#102—44no—
361C22⋊C8—44no—
362C22⋊C8—44no—
363#97—44no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—21no—
379#95—24no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D412yes—
385#41—432no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×D4—28no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4#13714yes—
407C2×D4C2×#4112yes—
408C22×C4—28no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448SD16—48no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4C2×#4112yes—
492C4⋊C4C2×#4114yes—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—81no—
503C22⋊C4—82no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—42no—
512C42—42no—
513C42—41no—
514C42—44no—
515C42—44no—
516C42—44no—
517C42—42no—
518C42—84no—
519C42AGL(1,ℤ/8)14yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23C2×#13714yes—
536C23C23×D411yescharacteristic
537Q8—48no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—164no—
543D4—28no—
544D4—24no—
545D4—416no—
546D4—42no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—48no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—164no—
598C2×C4—164no—
599C2×C4#52512yes—
600C2×C4#52512yes—
601C2×C4C2×AGL(1,ℤ/8)12yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×#13712yes—
605C2×C4#57712yes—
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4#58512yes—
609C2×C4#58512yes—
610C8—42no—
611C8—42no—
612C8—44no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22C2×#52511yescharacteristic
628C22#348511yescharacteristic
629C22#350011yescharacteristic
630C22C22×AGL(1,ℤ/8)11yescharacteristic
631C22C22×#13711yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—42no—
640C4—41no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—81no—
645C4—82no—
646C4—81no—
647C4—84no—
648C4—82no—
649C4—82no—
650C4#311014yes—
651C4#349312yes—
652C4#352412yes—
653C2—44no—
654C2—82no—
655C2#3601111yescharacteristic
656C2C2×#311011yescharacteristic
657C2#6059911yescharacteristic
658C2#6172211yescharacteristic
659C2#6173511yescharacteristic
660C2#6254111yescharacteristic
661C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #30113C2semidirect
4. #29980C2semidirect
5. #29335C2semidirect
6. #29219C2semidirect
7. #28205C2semidirect
8. #27611C2semidirect
9. #27349C2semidirect
10. #26739C2semidirect
11. #24971C2semidirect
13. #23761C2semidirect
14. #23677C2semidirect
15. #23068C2semidirect
16. #22839C2semidirect
17. #22762C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343082
#301132
#299802
#293352
#292192
#282052
#276112
#273492
#267392
#249712
#241202
#237612
#236772
#230682
#228392
#227622