Finite Group Databasebeta

← #6354835

Subgroups — #6354835

Order 512.

1879 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60864C211yesmaximal, characteristic, complemented
2C2×#3208C211yesmaximal, characteristic, complemented
3C2×#3205C211yesmaximal, characteristic
4#60236C211yesmaximal, characteristic, complemented
5#19504C218yesmaximal, complemented
6#19420C214yesmaximal, complemented
7#19384C214yesmaximal, complemented
8#19196C214yesmaximal, complemented
9#19186C214yesmaximal, complemented
10#17803C211yesmaximal, characteristic, complemented
11#17741C211yesmaximal, characteristic, complemented
12#17718C211yesmaximal, characteristic, complemented
13C2×#533C2211yescharacteristic
14C2×#532C2211yescharacteristic
15C22×#127C2211yescharacteristic
16#3235C2214yescomplemented
17#3234—22no—
18#3228—22no—
19#3227C2212yescomplemented
20#3226C2212yescomplemented
21#3208—22no—
22#3208—22no—
23#3208C2212yes—
24#3208C2212yes—
25#3208C2214yescomplemented
26#3205—22no—
27#3205—22no—
28#3205C2214yes—
29#3205C2214yes—
30#3165—21no—
31C2×#493—21no—
32C2×#492C2212yescomplemented
33#3138—21no—
34C2×#486—21no—
35C2×#486C2212yescomplemented
36#3094—22no—
37#3077C2214yescomplemented
38C2×#460—22no—
39C2×#459C2214yes—
40C2×#450—22no—
41C2×#447—22no—
42#2981C2214yescomplemented
43#2979—22no—
44#2975C2212yescomplemented
45#2974—22no—
46#2972C2212yescomplemented
47#2935—22no—
48#2931C2214yescomplemented
49#2920C2211yescharacteristic
50#2920C2211yescharacteristic
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53#2880—22no—
54#1697C22116yescomplemented
55#1685C22116yescomplemented
56#1677C22116yescomplemented
57#1665C22116yescomplemented
58#1572C2214yescomplemented
59#1571C2212yescomplemented
60#1570C2212yescomplemented
61#1564C2214yescomplemented
62#1555C2214yescomplemented
63#1554C2214yescomplemented
64#1548C2212yescomplemented
65#1546C2212yescomplemented
66#1545C2214yes—
67#1545C2214yescomplemented
68#1542C2214yescomplemented
69C22×#41—21no—
70C23×D4—21no—
71C23×D4—21no—
72C23×D4—21no—
73C22×SD16—24no—
74C22×D8—24no—
75C2×#131—24no—
76#533—21no—
77#533—21no—
78#533—21no—
79#533—21no—
80#533—21no—
81#533—21no—
82#533C2312yes—
83#533C2312yes—
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532D412yes—
89#532D412yes—
90#532C2312yes—
91#532C2312yes—
92C2×C4⋊Q8D411yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127—21no—
96C2×#127—21no—
97C2×#127—21no—
98C2×#127—21no—
99C2×#127—21no—
100C2×#127—21no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127—21no—
104C2×#127D411yescharacteristic
105C2×#127C2312yes—
106C2×#127C2312yes—
107C2×#127C2312yes—
108C2×#125—21no—
109C2×#124—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×#121—21no—
113C2×#121D412yes—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C22×C42C2311yescharacteristic
120#501—24no—
121#500—22no—
122#498—22no—
123#493—24no—
124#493—24no—
125#492—24no—
126#492C2314yes—
127#492C2314yes—
128#490—24no—
129#486—24no—
130#486—24no—
131#486—22no—
132#486—22no—
133#486—24no—
134#486C2314yes—
135#486C2314yes—
136#482—28no—
137#481—24no—
138#480—24no—
139#478—28no—
140#476C2318yescomplemented
141#474C2318yescomplemented
142#460—24no—
143#460—24no—
144#460—28no—
145#459—28no—
146#459C23116yes—
147#450—24no—
148#450—24no—
149#450—28no—
150#447—28no—
151#447—28no—
152#436—24no—
153#436—24no—
154#436C2318yescomplemented
155#435—28no—
156C8×D4—24no—
157C8×D4—24no—
158C8×D4C2318yescomplemented
159#432—21no—
160#432—21no—
161#432C2312yes—
162#432C2312yes—
163#431—21no—
164#431—21no—
165#431—21no—
166#431—21no—
167#431D412yes—
168#431D412yes—
169#431C2312yes—
170#431C2312yes—
171#431C2312yes—
172#431C2312yes—
173#431C2311yescharacteristic
174#431C2311yescharacteristic
175#431C2312yes—
176#428—24no—
177#423C2312yes—
178#423C2312yes—
179C2×C4⋊C8D412yes—
180C2×C4⋊C8C2312yes—
181#418—24no—
182C2×#102—24no—
183C2×#102—24no—
184#408—24no—
185#408—28no—
186#407—21no—
187#407—21no—
188#407—24no—
189C2×C22⋊C8—21no—
190C2×C22⋊C8—24no—
191C2×C22⋊C8D412yes—
192C4×M16—21no—
193C4×M16—21no—
194C2×C4×C8—21no—
195C2×C4×C8C2312yes—
196#335C2318yescomplemented
197#334C2318yescomplemented
198#333C2318yescomplemented
199#332C2318yescomplemented
200#331C23116yes—
201C25—41no—
202C2×#41—22no—
203C2×#41—22no—
204C2×#41—22no—
205C2×#41—22no—
206C2×#41—84no—
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—41no—
223C22×D4—41no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—41no—
227C22×D4—41no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4—41no—
231C22×D4—84no—
232C22×D4—84no—
233C22×D4—84no—
234C22×D4C2×D412yes—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4#4111yescharacteristic
239C2×SD16—28no—
240C2×SD16—28no—
241C2×SD16—88no—
242C2×D8—216no—
243C2×D8—88no—
244#131—88no—
245C2×M16—22no—
246C2×M16—22no—
247C2×M16—28no—
248C22×C8—22no—
249C22×C8—28no—
250C22×C8C2×D414yes—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8C2×D412yes—
260C4⋊Q8C2×D412yes—
261C4⋊Q8C2412yes—
262C4⋊Q8C2412yes—
263#127—21no—
264#127—21no—
265#127—22no—
266#127—22no—
267#127—22no—
268#127—22no—
269#127—22no—
270#127—22no—
271#127—22no—
272#127—22no—
273#127—22no—
274#127—41no—
275#127—41no—
276#127—41no—
277#127—41no—
278#127—41no—
279#127—41no—
280#127C2×D412yes—
281#127C2×D412yes—
282#127C2414yes—
283#125—22no—
284#125—22no—
285#125—41no—
286#125—41no—
287#124—22no—
288#124—22no—
289#124—22no—
290#124—22no—
291C22⋊Q8—22no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—41no—
296C22⋊Q8—41no—
297C22⋊Q8—41no—
298C22⋊Q8—41no—
299#121—22no—
300#121—22no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121—22no—
305#121C2×D414yes—
306#121C2×D414yes—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—41no—
312C4×Q8—41no—
313C4×Q8—41no—
314C4×Q8—41no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4C2×D414yes—
322C4×D4C2414yes—
323#117—22no—
324#117—22no—
325#117—41no—
326#117—41no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—41no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4C2×D412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—41no—
338C2×C42—41no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42#4111yescharacteristic
346C2×C42#4111yescharacteristic
347C2×C42#4111yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—28no—
352#106—28no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8C2×D412yes—
357C4⋊C8C2×D414yes—
358C4⋊C8C2×D412yes—
359C4⋊C8C2414yes—
360C4⋊C8C2414yes—
361C4⋊C8C2414yes—
362#102—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—216no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—22no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—28no—
374C22⋊C8—28no—
375C22⋊C8C2×D412yes—
376C22⋊C8C2×D412yes—
377C22⋊C8C2×D414yes—
378#97—22no—
379#97—22no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8—22no—
384C4×C8C2×D414yes—
385C4×C8C2412yes—
386C4×C8C2412yes—
387C24—22no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—84no—
392C24—84no—
393C24—84no—
394C24#12111yescharacteristic
395#41—48no—
396#41—84no—
397#41—84no—
398#41—84no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—41no—
404C2×Q8—41no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—28no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—82no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—82no—
436C2×D4—84no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—84no—
446C2×D4—84no—
447C2×D4—84no—
448C2×D4—84no—
449C2×D4C22×D414yes—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—84no—
459C22×C4#12111yescharacteristic
460C22×C4#12111yescharacteristic
461C22×C4#12111yescharacteristic
462C22×C4#12111yescharacteristic
463C22×C4#12111yescharacteristic
464C22×C4#12111yescharacteristic
465C22×C4#12111yescharacteristic
466C22×C4#12111yescharacteristic
467C22×C4#12111yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4111yescharacteristic
475C22×C4C2×#4111yescharacteristic
476C22×C4C2×#4111yescharacteristic
477SD16—88no—
478SD16—88no—
479D8—816no—
480M16—44no—
481M16—44no—
482M16—88no—
483C2×C8—28no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—44no—
488C2×C8—88no—
489C2×C8C22×D414yes—
490C2×C8C22×D414yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4C22×D414yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C42—42no—
514C42—42no—
515C42#12111yescharacteristic
516C42#12111yescharacteristic
517C42#12111yescharacteristic
518C42#12111yescharacteristic
519C42#12111yescharacteristic
520C42#12111yescharacteristic
521C42C22×D411yescharacteristic
522C42C22×D411yescharacteristic
523C42C22×D411yescharacteristic
524C42C22×D411yescharacteristic
525C42C22×D411yescharacteristic
526C42C22×D411yescharacteristic
527C42C2×#4111yescharacteristic
528C42C2×#4111yescharacteristic
529C42C2×#4111yescharacteristic
530C42C2511yescharacteristic
531C23—22no—
532C23—22no—
533C23—24no—
534C23—41no—
535C23—44no—
536C23—44no—
537C23—82no—
538C23—82no—
539C23—84no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23—84no—
547C23—84no—
548C23—1616no—
549C23C2×#12111yescharacteristic
550C23C2×#12111yescharacteristic
551C23C2×#12111yescharacteristic
552Q8—44no—
553Q8—44no—
554Q8—82no—
555Q8—82no—
556Q8—82no—
557Q8—82no—
558D4—28no—
559D4—48no—
560D4—44no—
561D4—44no—
562D4—44no—
563D4—48no—
564D4—48no—
565D4—82no—
566D4—82no—
567D4—84no—
568D4—84no—
569D4—84no—
570D4—84no—
571D4—84no—
572D4—84no—
573D4—84no—
574D4—84no—
575D4—84no—
576D4—84no—
577D4—84no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—84no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×#12111yescharacteristic
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×#12111yescharacteristic
609C2×C4C2×#12111yescharacteristic
610C2×C4C2×#12111yescharacteristic
611C2×C4#57614yes—
612C2×C4#57714yes—
613C2×C4C23×D411yescharacteristic
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C8—24no—
617C8—24no—
618C8—44no—
619C8—44no—
620C8—44no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—164no—
636C22—168no—
637C22—168no—
638C22—168no—
639C22#304112yes—
640C22#305712yes—
641C22C22×#12111yescharacteristic
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4#304112yes—
653C4#305712yes—
654C4#306812yes—
655C4C2×#57612yes—
656C4C2×#57712yes—
657C2—22no—
658C2—22no—
659C2—41no—
660C2—84no—
661C2—164no—
662C2C2×#304111yescharacteristic
663C2C2×#305711yescharacteristic
664C2C2×#306811yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#3205C2
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. #60864C2semidirect
2. C2×#3208C2semidirect
4. #60236C2semidirect
5. #19504C2semidirect
6. #19420C2semidirect
7. #19384C2semidirect
8. #19196C2semidirect
9. #19186C2semidirect
10. #17803C2semidirect
11. #17741C2semidirect
12. #17718C2semidirect
16. #3235C22semidirect
19. #3227C22semidirect
20. #3226C22semidirect
25. #3208C22semidirect
32. C2×#492C22semidirect
35. C2×#486C22semidirect
37. #3077C22semidirect
42. #2981C22semidirect
44. #2975C22semidirect
46. #2972C22semidirect
48. #2931C22semidirect
54. #1697C22semidirect
55. #1685C22semidirect
56. #1677C22semidirect
57. #1665C22semidirect
58. #1572C22semidirect
59. #1571C22semidirect
60. #1570C22semidirect
61. #1564C22semidirect
62. #1555C22semidirect
63. #1554C22semidirect
64. #1548C22semidirect
65. #1546C22semidirect
67. #1545C22semidirect
68. #1542C22semidirect
140. #476C23semidirect
141. #474C23semidirect
154. #436C23semidirect
158. C8×D4C23semidirect
196. #335C23semidirect
197. #334C23semidirect
198. #333C23semidirect
199. #332C23semidirect

Maximal subgroups

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

subgroupindex
#608642
C2×#32082
C2×#32052
#602362
#195042
#194202
#193842
#191962
#191862
#178032
#177412
#177182