Finite Group Databasebeta

← #6508433

Subgroups — #6508433

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3#29103C211yesmaximal, characteristic
4#29075C214yesmaximal, complemented
5#28949C214yesmaximal, complemented
6#28302C211yesmaximal, characteristic
7#26820C214yesmaximal, complemented
8#26480C214yesmaximal, complemented
9#25902C211yesmaximal, characteristic, complemented
10#25811C211yesmaximal, characteristic, complemented
11#24601C212yesmaximal, complemented
12#24501C212yesmaximal
13#22078C212yesmaximal, complemented
14C4×#450C212yesmaximal
15#20533C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C22×C4×D4C2211yescharacteristic
19C2×#498C2212yescomplemented
20C2×#487C2212yescomplemented
21C2×#483C2212yes—
22C2×#450C2212yes—
23C2×C4×SD16C2214yes—
24C2×#431C2212yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2646—21no—
29#2627—21no—
30#2590—22no—
31#2588—22no—
32#2511C2212yescomplemented
33#2507C2212yescomplemented
34#2452—22no—
35#2452—22no—
36#2452C2214yes—
37#2452C2214yes—
38#2451C2212yescomplemented
39#2446C2212yescomplemented
40C2×#396—21no—
41C2×#393—21no—
42#2329—21no—
43#2313—21no—
44C4×C22⋊Q8C2212yes—
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#385—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2094C2212yes—
52#2088C2214yescomplemented
53#2071C2212yes—
54#2066C2214yescomplemented
55#2065C2214yes—
56#2058C2214yescomplemented
57#2055C2214yes—
58#2024C2214yescomplemented
59#2023C2212yes—
60#2013C2212yes—
61#1990C2212yescomplemented
62#1990C2212yescomplemented
63#1968C2214yes—
64#1968C2214yes—
65#1941C2214yescomplemented
66#1941C2214yescomplemented
67#1934C2214yes—
68#1934C2214yes—
69#1933C2214yescomplemented
70#1933C2214yescomplemented
71#1919C2214yescomplemented
72#1877C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1865C2214yescomplemented
76#1864C2214yes—
77#1853C2212yes—
78#1853C2212yes—
79#1823C2212yes—
80#1786C2212yes—
81C8×C4⋊C4C2214yescomplemented
82C8×C4⋊C4C2214yescomplemented
83C4×#102C2214yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2212yes—
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
102C2×C22⋊Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4—44no—
106C2×C4×D4C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#498—44no—
124#487—44no—
125#483—28no—
126#450—28no—
127C4×SD16—48no—
128#431—28no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—22no—
146#400—22no—
147#400—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#396—24no—
153#396—24no—
154#395—22no—
155#393—24no—
156#393—24no—
157#391—21no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390—21no—
165#390—21no—
166#390C2314yes—
167#390C2312yes—
168#390C2312yes—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#389C2312yes—
179#389C2312yes—
180#389C2314yes—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—21no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—82no—
228C22×Q8C2412yes—
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—24no—
232C22×D4—42no—
233C22×D4C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—24no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—82no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—28no—
249C2×SD16—28no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—44no—
264C22⋊Q8—416no—
265C22⋊Q8—416no—
266C4×Q8—44no—
267C4×D4—28no—
268C4×D4—44no—
269C4×D4—48no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4SD1618yescomplemented
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—82no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335#106—28no—
336#106—28no—
337#106—44no—
338C4⋊C8—28no—
339#103—28no—
340#103—28no—
341#103—44no—
342#102—28no—
343#102—28no—
344#102—44no—
345C22⋊C8—24no—
346C22⋊C8#4118yes—
347C4×C8—28no—
348C4×C8—44no—
349C4×C8—42no—
350C4×C8—42no—
351#95—24no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—24no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24C22×D411yescharacteristic
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8—816no—
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—416no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D42−1+414yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—48no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—48no—
436C22×C4—84no—
437C22×C4—816no—
438C22×C4C2×SD1614yes—
439C22×C4C2×SD1614yes—
440C22×C4C2×SD1614yes—
441C22×C4#13514yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4114yes—
452C22×C4C2511yescharacteristic
453SD16—416no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8C2×#4118yes—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4C2×SD1618yes—
491C4⋊C4C2×SD1618yes—
492C4⋊C4H(5,𝔽2)14yes—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—84no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—42no—
509C42—41no—
510C42—41no—
511C42—82no—
512C42—82no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23#53212yes—
532C23C22×SD1612yes—
533C23C2×#13512yes—
534C23C23×D411yescharacteristic
535Q8—44no—
536Q8—44no—
537Q8—88no—
538Q8—88no—
539Q8—162no—
540D4—28no—
541D4—48no—
542D4—44no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4#53212yes—
596C2×C4C22×SD1614yes—
597C2×C4C22×SD1612yes—
598C2×C4C22×SD1612yes—
599C2×C4C22×SD1614yes—
600C2×C4C2×#13512yes—
601C2×C4C2×#13512yes—
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4C2×2−1+412yes—
605C8—28no—
606C8—44no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—816no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22#316714yes—
622C22C2×#53211yescharacteristic
623C22#348411yescharacteristic
624C22#348611yescharacteristic
625C22C23×SD1611yescharacteristic
626C22C22×#13511yescharacteristic
627C22#358511yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—24no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—24no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#352014yes—
646C2—24no—
647C2—84no—
648C2#3539711yescharacteristic
649C2C2×#316711yescharacteristic
650C2#6070311yescharacteristic
651C2#6156011yescharacteristic
652C2#6156411yescharacteristic
653C2#6181811yescharacteristic
654C2#6181911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35015C2semidirect
4. #29075C2semidirect
5. #28949C2semidirect
7. #26820C2semidirect
8. #26480C2semidirect
9. #25902C2semidirect
10. #25811C2semidirect
11. #24601C2semidirect
13. #22078C2semidirect
15. #20533C2semidirect
19. C2×#498C22semidirect
20. C2×#487C22semidirect
32. #2511C22semidirect
33. #2507C22semidirect
38. #2451C22semidirect
39. #2446C22semidirect
52. #2088C22semidirect
54. #2066C22semidirect
56. #2058C22semidirect
58. #2024C22semidirect
61. #1990C22semidirect
62. #1990C22semidirect
65. #1941C22semidirect
66. #1941C22semidirect
69. #1933C22semidirect
70. #1933C22semidirect
71. #1919C22semidirect
75. #1865C22semidirect
81. C8×C4⋊C4C22semidirect
82. C8×C4⋊C4C22semidirect
297. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24522
#291032
#290752
#289492
#283022
#268202
#264802
#259022
#258112
#246012
#245012
#220782
C4×#4502
#205332