Finite Group Databasebeta

← #6355050

Subgroups — #6355050

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic, complemented
2#60803C211yesmaximal, characteristic, complemented
3#60803C211yesmaximal, characteristic, complemented
4#60237C211yesmaximal, characteristic, complemented
5#19506C214yesmaximal, complemented
6#19505C214yesmaximal, complemented
7#19461C218yesmaximal, complemented
8#19237C214yesmaximal, complemented
9#19230C214yesmaximal, complemented
10#17808C211yesmaximal, characteristic, complemented
11#17731C211yesmaximal, characteristic, complemented
12C2×#1553C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×#532C2211yescharacteristic
15C2×#532C2211yescharacteristic
16#3245C2214yes—
17#3226—21no—
18#3226C2212yescomplemented
19#3214—22no—
20#3214C2214yes—
21#3213—22no—
22#3213—22no—
23#3207—22no—
24#3207C2214yes—
25#3206C2212yescomplemented
26#3206C2212yescomplemented
27#3205—21no—
28#3205C2212yescomplemented
29#3205C2212yescomplemented
30#3205C2212yescomplemented
31#3186—22no—
32#3169—22no—
33#3151—21no—
34#3151C2212yescomplemented
35#3141—21no—
36#3141C2212yescomplemented
37#3100—22no—
38#3100C2214yescomplemented
39#3036—22no—
40#3036C2214yescomplemented
41#3006—22no—
42#3006—22no—
43#2981C2214yescomplemented
44#2980—22no—
45#2974C2214yescomplemented
46#2973—22no—
47#2935C2214yescomplemented
48#2932—22no—
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51C2×#432C2211yescharacteristic
52C2×#431C2211yescharacteristic
53#2880—22no—
54#1701C2218yescomplemented
55#1699C2218yescomplemented
56#1687C2218yescomplemented
57#1685C2218yescomplemented
58#1681C2218yescomplemented
59#1679C2218yescomplemented
60#1667C2218yescomplemented
61#1665C2218yescomplemented
62#1574C2214yescomplemented
63#1572C2214yescomplemented
64#1565C2214yescomplemented
65#1557C2214yescomplemented
66#1556C2212yescomplemented
67#1555C2212yescomplemented
68#1553C2214yescomplemented
69#1553C2214yescomplemented
70#1540C2214yescomplemented
71C2×#332C2212yescomplemented
72C2×#331C2212yescomplemented
73C22×#41—21no—
74C22×#41—21no—
75C22×#41—21no—
76C2×AGL(1,ℤ/8)—24no—
77C2×AGL(1,ℤ/8)—24no—
78C2×#131—24no—
79#532—21no—
80#532—21no—
81#532—21no—
82#532—21no—
83#532—21no—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532—21no—
89#532—21no—
90#532—21no—
91#532—21no—
92#532—21no—
93#532—21no—
94#532—21no—
95#532D412yes—
96#532D412yes—
97#532C2312yes—
98#532C2312yes—
99#532C2312yes—
100#532C2312yes—
101#532C2312yes—
102#532C2312yes—
103C2×C4⋊Q8D411yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127D411yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#124—21no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#121—21no—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121—21no—
116C2×#121—21no—
117C2×#121D412yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C22×C42C2311yescharacteristic
125#500—22no—
126#496—24no—
127#496—24no—
128#493—22no—
129#493—22no—
130#493C2314yes—
131#492—22no—
132#492C2314yes—
133#489—24no—
134#489—24no—
135#487—22no—
136#487C2314yes—
137#486—22no—
138#486—22no—
139#486—22no—
140#486C2314yes—
141#480—24no—
142#480—24no—
143#474—24no—
144#474C2318yes—
145#466—28no—
146#463—28no—
147#463—28no—
148#461—24no—
149#461C2318yes—
150#459—24no—
151#459—24no—
152#459C2318yes—
153#459C2318yes—
154#449—28no—
155#449—28no—
156#448—24no—
157#448—24no—
158#447—24no—
159#447—24no—
160#447—24no—
161#447—24no—
162C4×D8—28no—
163#436C2318yes—
164#435—28no—
165#435—28no—
166#435—28no—
167C8×D4C2318yes—
168#432—21no—
169#432—21no—
170#432—21no—
171#432D412yes—
172#432D412yes—
173#432C2312yes—
174#432C2312yes—
175#432C2312yes—
176#431—21no—
177#431—21no—
178#431—21no—
179#431C2312yes—
180#431C2312yes—
181#431C2312yes—
182#431C2312yes—
183#431C2312yes—
184#423D412yes—
185#423C2312yes—
186C2×C4⋊C8C2312yes—
187C2×C4⋊C8C2312yes—
188#417—24no—
189#417—24no—
190#408—24no—
191#408—28no—
192#407—21no—
193#407—24no—
194#407D412yes—
195C2×C22⋊C8—21no—
196C2×C22⋊C8—21no—
197C2×C22⋊C8—24no—
198C4×M16—21no—
199C4×M16C2312yes—
200C2×#97—21no—
201C2×C4×C8—21no—
202#335C2318yes—
203#334C2318yes—
204#332C2318yes—
205#332C2314yes—
206#332C2314yes—
207#331C2318yes—
208#331C2314yes—
209#331C2314yes—
210C2×#41—22no—
211C2×#41—22no—
212C2×#41—22no—
213C2×#41—22no—
214C2×#41—22no—
215C2×#41—22no—
216C2×#41—22no—
217C2×#41—22no—
218C2×#41—41no—
219C2×#41—41no—
220C2×#41—41no—
221C2×#41—41no—
222C2×#41—84no—
223C2×#41—84no—
224C2×#41—84no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4—41no—
238C22×D4—41no—
239C22×D4C2×D412yes—
240C23×C4—22no—
241C23×C4—41no—
242C23×C4—41no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4#4111yescharacteristic
246AGL(1,ℤ/8)—88no—
247AGL(1,ℤ/8)—88no—
248C2×SD16—28no—
249C2×SD16—28no—
250C2×D8—28no—
251C2×D8—28no—
252C2×D8—28no—
253#131—88no—
254C2×M16—22no—
255C2×M16—28no—
256C2×M16C2×D414yes—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—28no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8C2×D412yes—
269C4⋊Q8C2×D412yes—
270C4⋊Q8C2412yes—
271C4⋊Q8C2412yes—
272#127—21no—
273#127—21no—
274#127—21no—
275#127—21no—
276#127—21no—
277#127—21no—
278#127—21no—
279#127—21no—
280#127C2×D412yes—
281#127C2×D412yes—
282#127C2412yes—
283#127C2412yes—
284#124—22no—
285#124—22no—
286#124—22no—
287#124—22no—
288#124—22no—
289#124—22no—
290#124—22no—
291#124—22no—
292#124—41no—
293#124—41no—
294#124—41no—
295#124—41no—
296#121—22no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—22no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121—22no—
305#121—22no—
306#121—22no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—41no—
311#121—41no—
312#121—41no—
313#121—41no—
314#121—41no—
315#121—41no—
316#121—41no—
317#121—41no—
318#121C2×D414yes—
319#121C2×D414yes—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—41no—
335C4×D4—41no—
336C4×D4—41no—
337C4×D4—41no—
338C4×D4—41no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4—41no—
342C4×D4C2×D414yes—
343C4×D4C2414yes—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4C2×D412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—41no—
353C2×C22⋊C4—41no—
354C2×C22⋊C4—41no—
355C2×C22⋊C4—41no—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42#4111yescharacteristic
363C2×C42#4111yescharacteristic
364C2×C42#4111yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—28no—
369C4⋊C8—22no—
370C4⋊C8—22no—
371C4⋊C8—22no—
372C4⋊C8C2×D414yes—
373C4⋊C8C2×D414yes—
374C4⋊C8C2414yes—
375C4⋊C8C2414yes—
376C4⋊C8C2414yes—
377#103—28no—
378#103—28no—
379#102—28no—
380#102—28no—
381#102—28no—
382#102—28no—
383C22⋊C8—22no—
384C22⋊C8—22no—
385C22⋊C8—22no—
386C22⋊C8—22no—
387C22⋊C8—22no—
388C22⋊C8—22no—
389C22⋊C8—28no—
390C22⋊C8—28no—
391C22⋊C8C2×D414yes—
392C22⋊C8C2×D414yes—
393#97—22no—
394#97—22no—
395#97—22no—
396#97C2×D414yes—
397C4×C8—22no—
398C4×C8—22no—
399C4×C8—22no—
400C4×C8C2414yes—
401C24—22no—
402C24—41no—
403C24—41no—
404C24#12111yescharacteristic
405#41—48no—
406#41—48no—
407#41—48no—
408#41—84no—
409#41—84no—
410#41—84no—
411#41—84no—
412#41—84no—
413#41—84no—
414#41—84no—
415#41—84no—
416#41—84no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×D4—24no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—44no—
451C2×D4—82no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—82no—
457C2×D4—82no—
458C2×D4—84no—
459C2×D4—84no—
460C2×D4—84no—
461C2×D4C22×D414yes—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—42no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—81no—
476C22×C4—84no—
477C22×C4#12111yescharacteristic
478C22×C4#12111yescharacteristic
479C22×C4#12111yescharacteristic
480C22×C4#12111yescharacteristic
481C22×C4#12111yescharacteristic
482C22×C4#12111yescharacteristic
483C22×C4#12111yescharacteristic
484C22×C4#12111yescharacteristic
485C22×C4#12111yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4111yescharacteristic
493C22×C4C2×#4111yescharacteristic
494C22×C4C2×#4111yescharacteristic
495SD16—88no—
496SD16—88no—
497D8—88no—
498D8—88no—
499D8—88no—
500M16—28no—
501M16—44no—
502M16—88no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—44no—
507C2×C8—88no—
508C2×C8—82no—
509C2×C8C22×D418yes—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—41no—
521C4⋊C4C22×D414yes—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C42#12111yescharacteristic
535C42#12111yescharacteristic
536C42#12111yescharacteristic
537C42#12111yescharacteristic
538C42#12111yescharacteristic
539C42#12111yescharacteristic
540C42C22×D411yescharacteristic
541C42C22×D411yescharacteristic
542C42C22×D411yescharacteristic
543C42C22×D411yescharacteristic
544C42C22×D411yescharacteristic
545C42C22×D411yescharacteristic
546C42C2×#4111yescharacteristic
547C42C2×#4111yescharacteristic
548C42C2×#4111yescharacteristic
549C42C2511yescharacteristic
550C23—22no—
551C23—24no—
552C23—41no—
553C23—41no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23—82no—
561C23—82no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23C2×#12111yescharacteristic
566C23C2×#12111yescharacteristic
567C23C2×#12111yescharacteristic
568Q8—44no—
569Q8—44no—
570Q8—44no—
571Q8—82no—
572Q8—82no—
573Q8—82no—
574D4—28no—
575D4—48no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—44no—
581D4—82no—
582D4—82no—
583D4—82no—
584D4—82no—
585D4—82no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—82no—
591D4—82no—
592D4—84no—
593D4—84no—
594D4—84no—
595D4—84no—
596D4—84no—
597D4—82no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—48no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—84no—
627C2×C4C2×#12111yescharacteristic
628C2×C4C2×#12111yescharacteristic
629C2×C4C2×#12111yescharacteristic
630C2×C4C2×#12111yescharacteristic
631C2×C4C2×#12111yescharacteristic
632C2×C4C2×#12111yescharacteristic
633C2×C4C2×#12111yescharacteristic
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×#12111yescharacteristic
636C2×C4#57614yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C8—28no—
641C8—44no—
642C8—44no—
643C8—44no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—164no—
663C22—164no—
664C22—164no—
665C22#304112yes—
666C22C22×#12111yescharacteristic
667C4—21no—
668C4—21no—
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4#304112yes—
680C4C2×#57612yes—
681C4#358512yes—
682C2—22no—
683C2—41no—
684C2—41no—
685C2—84no—
686C2—162no—
687C2—162no—
688C2C2×#304111yescharacteristic
689C2#6045411yescharacteristic
690C2#6046911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60940C2semidirect
2. #60803C2semidirect
3. #60803C2semidirect
4. #60237C2semidirect
5. #19506C2semidirect
6. #19505C2semidirect
7. #19461C2semidirect
8. #19237C2semidirect
9. #19230C2semidirect
10. #17808C2semidirect
11. #17731C2semidirect
12. C2×#1553C2semidirect
18. #3226C22semidirect
25. #3206C22semidirect
26. #3206C22semidirect
28. #3205C22semidirect
29. #3205C22semidirect
30. #3205C22semidirect
34. #3151C22semidirect
36. #3141C22semidirect
38. #3100C22semidirect
40. #3036C22semidirect
43. #2981C22semidirect
45. #2974C22semidirect
47. #2935C22semidirect
54. #1701C22semidirect
55. #1699C22semidirect
56. #1687C22semidirect
57. #1685C22semidirect
58. #1681C22semidirect
59. #1679C22semidirect
60. #1667C22semidirect
61. #1665C22semidirect
62. #1574C22semidirect
63. #1572C22semidirect
64. #1565C22semidirect
65. #1557C22semidirect
66. #1556C22semidirect
67. #1555C22semidirect
68. #1553C22semidirect
69. #1553C22semidirect
70. #1540C22semidirect
71. C2×#332C22semidirect
72. C2×#331C22semidirect

Maximal subgroups

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

subgroupindex
#609402
#608032
#602372
#195062
#195052
#194612
#192372
#192302
#178082
#177312
C2×#15532