Finite Group Databasebeta

← #6500748

Subgroups — #6500748

Order 512.

2366 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35149C211yesmaximal, characteristic
2#34361C211yesmaximal, characteristic, complemented
3#27594C212yesmaximal, complemented
4#27592C212yesmaximal, complemented
5#24876C214yesmaximal, complemented
6#24872C214yesmaximal, complemented
7#24866C214yesmaximal, complemented
8#24862C214yesmaximal, complemented
9#24016C212yesmaximal, complemented
10#24013C212yesmaximal, complemented
11#21939C212yesmaximal, complemented
12#21739C211yesmaximal, characteristic, complemented
13#20597C212yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#481C2212yescomplemented
19C2×#485C2214yescomplemented
20C2×#480C2212yescomplemented
21C2×#416C2212yes—
22C2×#416C2212yes—
23C2×#416C2212yes—
24C2×#416C2212yes—
25C22×C22⋊C8C2212yes—
26#2849—22no—
27#2613—22no—
28#2432—22no—
29#2431—22no—
30#2407C2214yes—
31#2407C2214yes—
32#2375—22no—
33#2374—22no—
34#2354C2214yescomplemented
35#2352C2214yescomplemented
36#2333—22no—
37#2332—22no—
38#2314C2212yes—
39#2311C2212yes—
40#2310C2211yescharacteristic
41#2309C2211yescharacteristic
42#2280C2212yes—
43#2280C2211yescharacteristic
44#2097C2214yescomplemented
45#2094C2212yescomplemented
46#2093C2212yescomplemented
47#2073C2212yescomplemented
48#2073C2212yescomplemented
49#2072C2214yescomplemented
50#2036C2214yescomplemented
51#2035C2214yescomplemented
52#2034C2214yescomplemented
53#2033C2214yescomplemented
54#1928C2214yescomplemented
55#1927C2214yescomplemented
56#1926C2214yescomplemented
57#1925C2214yescomplemented
58#1875C2212yescomplemented
59#1875C2212yescomplemented
60#1872C2214yescomplemented
61#1869C2214yescomplemented
62#1869C2214yescomplemented
63#1854C2214yes—
64#1854C2214yes—
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1818C2212yes—
68#1808C2214yescomplemented
69#1807C2214yescomplemented
70#1788C2214yes—
71#1788C2214yes—
72#1787C2214yes—
73#1786C2214yes—
74#1780C2212yes—
75C22×#41—22no—
76C22×#41—28no—
77C22×#41—42no—
78C22×#41—42no—
79C22×#41—44no—
80C22×#41C2312yes—
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4—21no—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87C24×C4C2311yescharacteristic
88C23×C8C2312yes—
89C23×C8C2312yes—
90#536—44no—
91#535—42no—
92#535—42no—
93C2×C4⋊Q8—21no—
94C2×#127—21no—
95C2×#124—21no—
96C2×#124—21no—
97#525—28no—
98C2×#123—22no—
99C2×#123D412yes—
100C2×C22⋊Q8D412yes—
101C2×C22⋊Q8C2314yes—
102C2×#121D412yes—
103C2×#121C2312yes—
104C2×#121C2312yes—
105C2×#120—22no—
106C2×#120D412yes—
107C2×C4×D4C2314yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115#485—48no—
116#481—44no—
117#480—44no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122#416—44no—
123#416—44no—
124#416—44no—
125#416—44no—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8—28no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2312yes—
140#394—24no—
141#394—24no—
142#392—24no—
143#392—24no—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#390—22no—
148#390—24no—
149#390—22no—
150#390C2312yes—
151#390C2312yes—
152#388—24no—
153#388—24no—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#386—22no—
159#386—22no—
160#386—24no—
161#386—24no—
162#386—24no—
163#385—24no—
164#385—24no—
165#384C2312yes—
166#382C2312yes—
167#381—22no—
168#381—21no—
169#381—22no—
170#380C2314yes—
171#380C2314yes—
172#380C2314yes—
173#380C2314yes—
174#379—21no—
175#379—22no—
176#379—22no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—24no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C25—21no—
191C25C2×D411yescharacteristic
192C2×#41—24no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—41no—
202C2×#41—41no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—48no—
209C2×#41—48no—
210C2×#41—432no—
211C2×#41—432no—
212C2×#41—48no—
213C22×Q8—24no—
214C22×Q8—21no—
215C22×Q8—24no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8C2×D411yescharacteristic
219C22×Q8C2412yes—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—24no—
228C22×D4—42no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4C2×D414yes—
233C22×D4C2×D412yes—
234C22×D4C2×D411yescharacteristic
235C22×D4C2411yescharacteristic
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—41no—
242C23×C4—41no—
243C23×C4—42no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4#4118yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2412yes—
253C22×C8—24no—
254C22×C8—24no—
255C22×C8—42no—
256C22×C8—42no—
257C22×C8—42no—
258C22×C8—42no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—44no—
262#127—44no—
263#124—44no—
264#124—44no—
265#123—28no—
266#123—48no—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C22⋊Q8—44no—
270#121—28no—
271#121—24no—
272#121—24no—
273#121—42no—
274#121—42no—
275#120—28no—
276#120—48no—
277C4×D4—28no—
278C4×D4—28no—
279#117—24no—
280#117—41no—
281#117—41no—
282#117—42no—
283#117—42no—
284#117—42no—
285#117—42no—
286#117—42no—
287#117—44no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42C2412yes—
323#107—44no—
324#107—44no—
325#106—44no—
326#106—44no—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331#102—28no—
332#102—28no—
333#102—44no—
334#102—44no—
335C22⋊C8—28no—
336C22⋊C8—28no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339C22⋊C8—44no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348C24—24no—
349C24—21no—
350C24—21no—
351C24—41no—
352C24—41no—
353C24—44no—
354C24—42no—
355C24—42no—
356C24—84no—
357C24#12314yes—
358C24C22×D412yes—
359C24C22×D411yescharacteristic
360C24C22×D412yes—
361#41—216no—
362#41—216no—
363#41—48no—
364#41—48no—
365#41—48no—
366#41—48no—
367#41—48no—
368#41—48no—
369#41—464no—
370#41—84no—
371#41—84no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8H(5,𝔽2)14yes—
386C2×D4—24no—
387C2×D4—28no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—432no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4H(5,𝔽2)12yes—
418C2×D4H(5,𝔽2)12yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—432no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—84no—
453C22×C4#12318yes—
454C22×C4#12314yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2511yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—84no—
470C2×C8—84no—
471C4⋊C4—24no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C22⋊C4—24no—
484C22⋊C4—28no—
485C22⋊C4—28no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C42—24no—
501C42—42no—
502C42—42no—
503C42—84no—
504C42—82no—
505C42—82no—
506C42—82no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—44no—
522C23—48no—
523C23—48no—
524C23—44no—
525C23—44no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—88no—
531C23—88no—
532C23—84no—
533C23—84no—
534C23—88no—
535C23C2×#12312yes—
536C23C2×#12314yes—
537C23C23×D411yescharacteristic
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—82no—
544D4—24no—
545D4—24no—
546D4—48no—
547D4—416no—
548D4—416no—
549D4—42no—
550D4—42no—
551D4—82no—
552D4—88no—
553D4—84no—
554D4—164no—
555D4—164no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—164no—
593C2×C4C2×#12314yes—
594C2×C4C2×#12312yes—
595C2×C4C22×#4112yes—
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4C2×H(5,𝔽2)11yescharacteristic
598C2×C4C2×H(5,𝔽2)11yescharacteristic
599C8—44no—
600C8—44no—
601C22—22no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—44no—
607C22—42no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—88no—
620C22—816no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C22×#12311yescharacteristic
626C22#348411yescharacteristic
627C22#348411yescharacteristic
628C22#358414yes—
629C4—24no—
630C4—22no—
631C4—42no—
632C4—44no—
633C4—44no—
634C4—84no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4#349012yes—
639C4#349012yes—
640C2—44no—
641C2—42no—
642C2—82no—
643C2—82no—
644C2—82no—
645C2#3535011yescharacteristic
646C2#6051612yes—
647C2#6163612yes—
648C2#6163712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35149C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34361C2semidirect
3. #27594C2semidirect
4. #27592C2semidirect
5. #24876C2semidirect
6. #24872C2semidirect
7. #24866C2semidirect
8. #24862C2semidirect
9. #24016C2semidirect
10. #24013C2semidirect
11. #21939C2semidirect
12. #21739C2semidirect
13. #20597C2semidirect
18. C2×#481C22semidirect
19. C2×#485C22semidirect
20. C2×#480C22semidirect
34. #2354C22semidirect
35. #2352C22semidirect
44. #2097C22semidirect
45. #2094C22semidirect
46. #2093C22semidirect
47. #2073C22semidirect
48. #2073C22semidirect
49. #2072C22semidirect
50. #2036C22semidirect
51. #2035C22semidirect
52. #2034C22semidirect
53. #2033C22semidirect
54. #1928C22semidirect
55. #1927C22semidirect
56. #1926C22semidirect
57. #1925C22semidirect
58. #1875C22semidirect
59. #1875C22semidirect
60. #1872C22semidirect
61. #1869C22semidirect
62. #1869C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
68. #1808C22semidirect
69. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#351492
#343612
#275942
#275922
#248762
#248722
#248662
#248622
#240162
#240132
#219392
#217392
#205972