Finite Group Databasebeta

← #6495097

Subgroups — #6495097

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35274C211yesmaximal, characteristic
2#34625C211yesmaximal, characteristic, complemented
3#33228C214yesmaximal, complemented
4#30222C214yesmaximal, complemented
5#28756C212yesmaximal, complemented
6#28754C212yesmaximal, complemented
7#28752C212yesmaximal, complemented
8#28749C212yesmaximal, complemented
9#25867C214yesmaximal, complemented
10#25865C214yesmaximal, complemented
11#21810C211yesmaximal, characteristic, complemented
12#21809C211yesmaximal, characteristic, complemented
13#21807C211yesmaximal, characteristic, complemented
14#21803C211yesmaximal, characteristic, complemented
15#20470C211yesmaximal, characteristic, complemented
16C2×#535C2214yes—
17C2×#532C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C2×#490C2214yescomplemented
20C2×#489C2214yescomplemented
21C2×#488C2214yescomplemented
22C2×#486C2214yescomplemented
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2867C2212yes—
26#2858C2212yes—
27#2667—21no—
28#2666—22no—
29#2665—21no—
30#2660C2212yescomplemented
31#2657C2212yes—
32#2657C2212yes—
33#2647C2212yescomplemented
34#2629C2212yes—
35#2628C2212yes—
36#2541—21no—
37#2541—21no—
38#2541—22no—
39#2433—22no—
40#2433—21no—
41#2433—21no—
42#2402—21no—
43#2399—21no—
44#2396—22no—
45C4×C4⋊Q8C2212yescomplemented
46C4×#125C2212yescomplemented
47#2285C2211yescharacteristic
48#2283C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2042C2214yescomplemented
51#2042C2214yescomplemented
52#2042C2214yescomplemented
53#2042C2214yescomplemented
54#2041C2214yescomplemented
55#2041C2214yescomplemented
56#2041C2214yescomplemented
57#2041C2214yescomplemented
58#1982C2212yes—
59#1982C2212yes—
60#1982C2212yes—
61#1982C2212yes—
62#1982C2211yescharacteristic, complemented
63#1982C2211yescharacteristic, complemented
64#1982C2212yescomplemented
65#1981C2211yescharacteristic, complemented
66#1981C2211yescharacteristic, complemented
67#1981C2212yescomplemented
68#1961C2214yes—
69#1961C2214yescomplemented
70#1959C2214yescomplemented
71#1959C2214yescomplemented
72#1958C2214yescomplemented
73#1958C2214yescomplemented
74#1957C2214yes—
75#1957C2214yescomplemented
76#1788C2212yes—
77#1788C2214yes—
78#1788C2212yes—
79#1786C2212yes—
80#1786C2214yes—
81#1786C2212yes—
82#1767C2212yescomplemented
83C4×C4⋊C8C2212yescomplemented
84#1750C2214yes—
85#1746C2212yescomplemented
86#1726C2212yescomplemented
87C22×#41C2312yes—
88C23×D4—21no—
89C23×D4—21no—
90C23×D4C2312yes—
91#535—48no—
92#535—48no—
93#532—24no—
94#532—24no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×#127—21no—
98C2×#127—21no—
99C2×#127—22no—
100C2×#127—22no—
101C2×#127—22no—
102C2×#127—22no—
103C2×#127—24no—
104C2×#127—24no—
105C2×#127C2312yes—
106C2×#127C2311yescharacteristic
107C2×#127C2314yes—
108C2×#127C2312yes—
109C2×#127C2314yes—
110C2×#125C2312yes—
111C2×#124C2314yes—
112C2×#124C2312yes—
113C2×#124C2314yes—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121—24no—
117C2×#121—24no—
118C2×#120—24no—
119C2×#120—24no—
120C2×C4×Q8—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×#117—21no—
124C2×#117C2312yes—
125C22×C42C2311yescharacteristic
126#490—48no—
127#489—48no—
128#488—48no—
129#486—48no—
130#432—24no—
131#431—24no—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154C2×#97C2312yes—
155C2×C4×C8C2314yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#400—22no—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#396—22no—
168#396—22no—
169#395—22no—
170#395—22no—
171#395—22no—
172#393—22no—
173#392—22no—
174#392—24no—
175#392—22no—
176#392—22no—
177#392—24no—
178#392—22no—
179#391—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#387—22no—
190#387—22no—
191#384C2312yes—
192#384C2312yes—
193#383—21no—
194#382—21no—
195#382C2312yes—
196#382C2312yes—
197#381—21no—
198#380—21no—
199#380C2311yescharacteristic
200#380C2312yes—
201#380C2311yescharacteristic
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—21no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D411yescharacteristic
215C4×C22⋊C4D411yescharacteristic
216#376—21no—
217#376C2311yescharacteristic
218#376C2311yescharacteristic
219C43C2312yes—
220#336C2314yes—
221#336C2314yes—
222C25—22no—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—44no—
226C2×#41—44no—
227C22×Q8—22no—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—24no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4—44no—
246C22×D4—44no—
247C22×D4—82no—
248C22×D4—82no—
249C22×D4—84no—
250C22×D4C2414yes—
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263#127—24no—
264#127—24no—
265#127—24no—
266#127—24no—
267#127—22no—
268#127—22no—
269#127—28no—
270#127—28no—
271#127—44no—
272#127—44no—
273#127—44no—
274#127—44no—
275#127—84no—
276#127—84no—
277#127—84no—
278#127—84no—
279#125—24no—
280#125—24no—
281#124—24no—
282#124—24no—
283#124—28no—
284#124—28no—
285#124—44no—
286#124—44no—
287#121—416no—
288#121—416no—
289#121—44no—
290#121—44no—
291#120—416no—
292#120—416no—
293C4×Q8—84no—
294C4×D4—44no—
295C4×D4—44no—
296#117—24no—
297#117—24no—
298#117—82no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
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⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42#4114yes—
339C2×C42#4114yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C4⋊C8—22no—
345C4⋊C8—22no—
346C4⋊C8—24no—
347#103—44no—
348#103—44no—
349#103—44no—
350#103—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—
359#102—44no—
360#102—44no—
361#102—44no—
362#102—44no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365#97—24no—
366#97—28no—
367C4×C8—24no—
368C4×C8—28no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95C2×D411yescharacteristic
384#95C2×D412yes—
385#95C2×D411yescharacteristic
386#95C2×D411yescharacteristic
387#95C2×D412yes—
388#95C2×D411yescharacteristic
389C24—24no—
390C24—22no—
391C24—48no—
392C24—44no—
393C24—84no—
394C24—82no—
395C24—82no—
396C24C22×D411yescharacteristic
397#41—84no—
398#41—84no—
399#41—88no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—48no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—44no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—88no—
441C2×D4—84no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—816no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4—84no—
448C2×D4—84no—
449C2×D4—84no—
450C2×D4—84no—
451C2×D4—84no—
452C2×D4—84no—
453C2×D4—816no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4#12414yes—
467C22×C4#12414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2511yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—82no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42#12414yes—
531C42#12414yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C42H(5,𝔽2)14yes—
537C42H(5,𝔽2)14yes—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—88no—
555C23—88no—
556C23—88no—
557C23—168no—
558C23—168no—
559C23C2×#12412yes—
560C23C23×D411yescharacteristic
561Q8—42no—
562Q8—44no—
563Q8—42no—
564Q8—88no—
565D4—44no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—42no—
570D4—44no—
571D4—42no—
572D4—84no—
573D4—88no—
574D4—88no—
575D4—88no—
576D4—88no—
577D4—88no—
578D4—84no—
579D4—88no—
580D4—168no—
581D4—168no—
582D4—168no—
583D4—168no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—88no—
619C2×C4C2×#12412yes—
620C2×C4C2×#12412yes—
621C2×C4C2×#12412yes—
622C2×C4C2×#12412yes—
623C2×C4C2×#12412yes—
624C2×C4#53514yes—
625C2×C4#53514yes—
626C2×C4#57612yes—
627C2×C4#57712yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C22×#4111yescharacteristic
630C2×C4C2×H(5,𝔽2)12yes—
631C2×C4C2×H(5,𝔽2)12yes—
632C8—42no—
633C8—42no—
634C8—44no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—42no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—82no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—1616no—
653C22C22×#12411yescharacteristic
654C22C2×#53512yes—
655C22C2×#57611yescharacteristic
656C22C2×#57711yescharacteristic
657C22#358512yes—
658C4—24no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4#313614yes—
672C4#350811yescharacteristic
673C4#350811yescharacteristic
674C4#350812yes—
675C2—42no—
676C2—162no—
677C2—164no—
678C2#3528511yescharacteristic
679C2C2×#313611yescharacteristic
680C2#6061911yescharacteristic
681C2#6083212yes—
682C2#6083312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35274C2
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
2. #34625C2semidirect
3. #33228C2semidirect
4. #30222C2semidirect
5. #28756C2semidirect
6. #28754C2semidirect
7. #28752C2semidirect
8. #28749C2semidirect
9. #25867C2semidirect
10. #25865C2semidirect
11. #21810C2semidirect
12. #21809C2semidirect
13. #21807C2semidirect
14. #21803C2semidirect
15. #20470C2semidirect
19. C2×#490C22semidirect
20. C2×#489C22semidirect
21. C2×#488C22semidirect
22. C2×#486C22semidirect
30. #2660C22semidirect
33. #2647C22semidirect
45. C4×C4⋊Q8C22semidirect
46. C4×#125C22semidirect
50. #2042C22semidirect
51. #2042C22semidirect
52. #2042C22semidirect
53. #2042C22semidirect
54. #2041C22semidirect
55. #2041C22semidirect
56. #2041C22semidirect
57. #2041C22semidirect
62. #1982C22semidirect
63. #1982C22semidirect
64. #1982C22semidirect
65. #1981C22semidirect
66. #1981C22semidirect
67. #1981C22semidirect
69. #1961C22semidirect
70. #1959C22semidirect
71. #1959C22semidirect
72. #1958C22semidirect
73. #1958C22semidirect
75. #1957C22semidirect
82. #1767C22semidirect
83. C4×C4⋊C8C22semidirect
85. #1746C22semidirect
86. #1726C22semidirect

Maximal subgroups

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

subgroupindex
#352742
#346252
#332282
#302222
#287562
#287542
#287522
#287492
#258672
#258652
#218102
#218092
#218072
#218032
#204702