Finite Group Databasebeta

← #6495094

Subgroups — #6495094

Order 512.

2244 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35304C211yesmaximal, characteristic, complemented
2C2×#2657C211yesmaximal, characteristic
3#33228C214yesmaximal, complemented
4#30222C214yesmaximal, complemented
5#28761C214yesmaximal, complemented
6#28759C214yesmaximal, complemented
7#25864C212yesmaximal, complemented
8#25860C212yesmaximal, complemented
9#25859C212yesmaximal, complemented
10#25855C212yesmaximal, complemented
11#21810C211yesmaximal, characteristic, complemented
12#21808C211yesmaximal, characteristic, complemented
13#21806C211yesmaximal, characteristic, complemented
14#21803C211yesmaximal, characteristic, complemented
15#20469C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#127C2212yes—
18C22×#124C2211yescharacteristic
19C22×#124C2212yes—
20C2×#490C2214yescomplemented
21C2×#489C2214yescomplemented
22C2×#488C2214yescomplemented
23C2×#486C2214yescomplemented
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2857C2212yes—
27#2851C2212yes—
28#2660C2212yescomplemented
29#2657—21no—
30#2657—21no—
31#2657—22no—
32#2657C2212yes—
33#2657C2214yes—
34#2657C2212yes—
35#2647C2212yescomplemented
36#2612—22no—
37#2606—21no—
38#2605—21no—
39#2569—21no—
40#2549—21no—
41#2438—22no—
42C2×#392—21no—
43C2×#392—22no—
44C2×#392—21no—
45#2331C2212yescomplemented
46#2330C2212yescomplemented
47#2285C2211yescharacteristic
48#2283C2211yescharacteristic
49C2×#376C2211yescharacteristic
50#2042C2214yescomplemented
51#2042C2214yescomplemented
52#2042C2214yescomplemented
53#2042C2214yescomplemented
54#2041C2214yescomplemented
55#2041C2214yescomplemented
56#2041C2214yescomplemented
57#2041C2214yescomplemented
58#1983C2214yescomplemented
59#1983C2212yescomplemented
60#1983C2212yescomplemented
61#1982C2212yescomplemented
62#1982C2212yescomplemented
63#1982C2214yescomplemented
64#1961C2214yescomplemented
65#1961C2214yescomplemented
66#1959C2214yescomplemented
67#1959C2214yescomplemented
68#1958C2214yescomplemented
69#1958C2214yescomplemented
70#1957C2214yescomplemented
71#1957C2214yescomplemented
72#1787C2211yescharacteristic
73#1787C2212yes—
74#1787C2211yescharacteristic
75#1786C2212yes—
76#1786C2212yes—
77#1786C2211yescharacteristic
78#1786C2212yes—
79#1786C2212yes—
80#1786C2212yes—
81#1786C2211yescharacteristic
82#1768C2214yescomplemented
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1746C2212yescomplemented
86#1726C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4—22no—
89C23×D4—22no—
90C23×D4—21no—
91C23×D4—21no—
92C23×D4C2311yescharacteristic
93C23×D4C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×#127—22no—
96C2×#127—22no—
97C2×#127—21no—
98C2×#127—21no—
99C2×#127—24no—
100C2×#127—24no—
101C2×#127—48no—
102C2×#127C2314yes—
103C2×#127C2312yes—
104C2×#127C2312yes—
105C2×#127C2314yes—
106C2×#125C2312yes—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124—21no—
110C2×#124—21no—
111C2×#124—24no—
112C2×#124—24no—
113C2×#124—48no—
114C2×#124C2314yes—
115C2×#124C2312yes—
116C2×#124C2312yes—
117C2×#124C2314yes—
118C2×C22⋊Q8—22no—
119C2×#117—21no—
120C22×C22⋊C4—22no—
121C22×C22⋊C4—22no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#490—48no—
127#489—48no—
128#488—48no—
129#486—48no—
130#432—24no—
131#431—24no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2311yescharacteristic
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154C2×#97C2312yes—
155C2×C4×C8C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#395—22no—
164#395—22no—
165#392—24no—
166#392—24no—
167#392—24no—
168#392—24no—
169#392—22no—
170#392—24no—
171#392—22no—
172#392—22no—
173#392—24no—
174#392—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—22no—
192#385—22no—
193#384C2312yes—
194#384C2312yes—
195#383—21no—
196#382—21no—
197#382C2312yes—
198#382C2312yes—
199#381—21no—
200#380—22no—
201#380—21no—
202#380C2311yescharacteristic
203#380C2312yes—
204#380C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—21no—
210#376—21no—
211#376D411yescharacteristic
212#376D411yescharacteristic
213#376D412yes—
214#376C2312yes—
215#376C2311yescharacteristic
216#376C2311yescharacteristic
217C2×#95D411yescharacteristic
218C2×#95D412yes—
219C2×#95D411yescharacteristic
220#336C2314yes—
221#336C2314yes—
222C25—22no—
223C25—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—44no—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—24no—
233C22×D4—24no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4—24no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—24no—
243C22×D4—24no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—48no—
248C22×D4—48no—
249C22×D4—44no—
250C22×D4—44no—
251C22×D4—82no—
252C22×D4—82no—
253C22×D4—84no—
254C22×D4—84no—
255C22×D4—84no—
256C22×D4—84no—
257C22×D4—84no—
258C22×D4C2412yes—
259C22×D4C2414yes—
260C23×C4—21no—
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C22×C8C2412yes—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C4⋊Q8—28no—
270#127—24no—
271#127—24no—
272#127—24no—
273#127—24no—
274#127—28no—
275#127—28no—
276#127—44no—
277#127—44no—
278#127—48no—
279#127—48no—
280#127—44no—
281#127—44no—
282#125—28no—
283#124—24no—
284#124—24no—
285#124—24no—
286#124—24no—
287#124—28no—
288#124—28no—
289#124—44no—
290#124—44no—
291#124—48no—
292#124—48no—
293#124—44no—
294#124—44no—
295C22⋊Q8—84no—
296#117—82no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C42—28no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42#4114yes—
336C2×C42#4114yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C4⋊C8—24no—
342C4⋊C8—24no—
343#103—44no—
344#103—44no—
345#103—44no—
346#103—44no—
347#102—44no—
348#102—44no—
349#102—44no—
350#102—44no—
351#102—44no—
352#102—44no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358#102—44no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—24no—
362#97—24no—
363#97—28no—
364C4×C8—24no—
365C4×C8—28no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—22no—
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—
384#95C2×D411yescharacteristic
385#95C2×D412yes—
386#95C2×D411yescharacteristic
387#95C2×D411yescharacteristic
388#95C2×D412yes—
389#95C2×D411yescharacteristic
390C24—22no—
391C24—24no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24—44no—
396C24—44no—
397C24—44no—
398C24—42no—
399C24—44no—
400C24—42no—
401C24—44no—
402C24—42no—
403C24—44no—
404C24C22×D411yescharacteristic
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—84no—
415C2×Q8—84no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—88no—
442C2×D4—88no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4—82no—
448C2×D4—82no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—82no—
452C2×D4—82no—
453C2×D4—84no—
454C2×D4—84no—
455C2×D4—84no—
456C2×D4—84no—
457C2×D4—816no—
458C2×D4—816no—
459C2×D4—832no—
460C2×D4—832no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4#12414yes—
479C22×C4#12414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2511yescharacteristic
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C42—24no—
526C42—24no—
527C42—24no—
528C42—82no—
529C42—81no—
530C42—81no—
531C42#12414yes—
532C42#12414yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42H(5,𝔽2)18yes—
538C23—22no—
539C23—22no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—88no—
553C23—88no—
554C23—88no—
555C23—88no—
556C23—88no—
557C23—88no—
558C23—84no—
559C23—88no—
560C23—88no—
561C23—84no—
562C23—88no—
563C23—84no—
564C23C2×#12412yes—
565C23#57612yes—
566C23#57712yes—
567C23C23×D411yescharacteristic
568Q8—42no—
569Q8—44no—
570Q8—42no—
571Q8—84no—
572Q8—164no—
573D4—42no—
574D4—44no—
575D4—42no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—88no—
581D4—84no—
582D4—816no—
583D4—816no—
584D4—88no—
585D4—88no—
586D4—88no—
587D4—88no—
588D4—816no—
589D4—816no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—84no—
628C2×C4C2×#12412yes—
629C2×C4C2×#12412yes—
630C2×C4C2×#12412yes—
631C2×C4C2×#12412yes—
632C2×C4C2×#12412yes—
633C2×C4#53514yes—
634C2×C4#53514yes—
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)14yes—
638C8—42no—
639C8—42no—
640C8—44no—
641C22—24no—
642C22—24no—
643C22—22no—
644C22—22no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—82no—
653C22—84no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—168no—
658C22—1616no—
659C22#313614yes—
660C22C22×#12411yescharacteristic
661C22C2×#53512yes—
662C22C2×#57611yescharacteristic
663C22C2×#57711yescharacteristic
664C22#358512yes—
665C4—24no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#350812yes—
680C4#350812yes—
681C2—24no—
682C2—162no—
683C2—164no—
684C2#3528511yescharacteristic
685C2C2×#313611yescharacteristic
686C2#6061911yescharacteristic
687C2#6081112yes—
688C2#6081212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2657C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35304C2semidirect
3. #33228C2semidirect
4. #30222C2semidirect
5. #28761C2semidirect
6. #28759C2semidirect
7. #25864C2semidirect
8. #25860C2semidirect
9. #25859C2semidirect
10. #25855C2semidirect
11. #21810C2semidirect
12. #21808C2semidirect
13. #21806C2semidirect
14. #21803C2semidirect
15. #20469C2semidirect
20. C2×#490C22semidirect
21. C2×#489C22semidirect
22. C2×#488C22semidirect
23. C2×#486C22semidirect
28. #2660C22semidirect
35. #2647C22semidirect
45. #2331C22semidirect
46. #2330C22semidirect
50. #2042C22semidirect
51. #2042C22semidirect
52. #2042C22semidirect
53. #2042C22semidirect
54. #2041C22semidirect
55. #2041C22semidirect
56. #2041C22semidirect
57. #2041C22semidirect
58. #1983C22semidirect
59. #1983C22semidirect
60. #1983C22semidirect
61. #1982C22semidirect
62. #1982C22semidirect
63. #1982C22semidirect
64. #1961C22semidirect
65. #1961C22semidirect
66. #1959C22semidirect
67. #1959C22semidirect
68. #1958C22semidirect
69. #1958C22semidirect
70. #1957C22semidirect
71. #1957C22semidirect
82. #1768C22semidirect
85. #1746C22semidirect
86. #1726C22semidirect

Maximal subgroups

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

subgroupindex
#353042
C2×#26572
#332282
#302222
#287612
#287592
#258642
#258602
#258592
#258552
#218102
#218082
#218062
#218032
#204692