Finite Group Databasebeta

← #7535019

Subgroups — #7535019

Order 512.

2337 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59816C211yesmaximal, characteristic
2#36534C211yesmaximal, characteristic, complemented
3#31270C212yesmaximal, complemented
4#31260C212yesmaximal, complemented
5#29145C212yesmaximal, complemented
6#28154C214yesmaximal, complemented
7#24981C214yesmaximal, complemented
8#24252C214yesmaximal, complemented
9#23979C212yesmaximal, complemented
10#23969C212yesmaximal, complemented
11#23175C211yesmaximal, characteristic, complemented
12#22423C214yesmaximal, complemented
13#21505C212yesmaximal, complemented
14C2×#556C2211yescharacteristic, complemented
15C2×#554C2211yescharacteristic, complemented
16C2×D42—21no—
17C2×D42C2214yes—
18C2×#460C2212yes—
19C2×#450C2212yescomplemented
20C2×#450C2212yescomplemented
21C2×#440C2214yescomplemented
22C2×#435C2214yescomplemented
23C2×#418C2212yescomplemented
24C22×#102C2212yescomplemented
25#2862C2211yescharacteristic, complemented
26#2861—21no—
27#2679—21no—
28#2679—21no—
29#2565—21no—
30#2563C2211yescharacteristic, complemented
31#2560—24no—
32#2547C2211yescharacteristic, complemented
33#2544—21no—
34#2537—21no—
35#2537C2214yes—
36#2536C2214yescomplemented
37#2408—24no—
38#2408C2212yes—
39#2408C2212yes—
40#2352C2214yescomplemented
41C4×#127C2212yes—
42C4×#127C2211yescharacteristic
43C4×#124—21no—
44C42×Q8C2211yescharacteristic, complemented
45#2076C2214yescomplemented
46#2075C2214yescomplemented
47#2033C2214yescomplemented
48#2033C2214yescomplemented
49#2023C2214yescomplemented
50#2023C2214yescomplemented
51#2000C2214yescomplemented
52#2000C2214yescomplemented
53#1982C2212yes—
54#1961C2214yes—
55#1955C2214yescomplemented
56#1947C2212yescomplemented
57#1925C2214yescomplemented
58#1895C2214yescomplemented
59#1894C2212yescomplemented
60#1893C2214yescomplemented
61#1892C2212yescomplemented
62#1880C2214yescomplemented
63#1875C2214yescomplemented
64#1874C2214yescomplemented
65#1870C2212yescomplemented
66#1869C2212yescomplemented
67#1836C2214yescomplemented
68#1825C2212yescomplemented
69#1824C2212yescomplemented
70#1804C2214yescomplemented
71#1804C2214yescomplemented
72#1803C2212yescomplemented
73#1786C2214yescomplemented
74#1786C2214yescomplemented
75#1767C2212yescomplemented
76#1762C2214yescomplemented
77C23×D4—24no—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C22×SD16C2314yes—
81C22×M16C2312yes—
82C23×C8C2312yes—
83#556—42no—
84#556—42no—
85#554—42no—
86#554—42no—
87D42—216no—
88D42—48no—
89D42—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#127D411yescharacteristic
93C2×#127C2312yes—
94C2×#126—22no—
95C2×#125—21no—
96C2×#124—21no—
97C2×#124—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×#121—24no—
102C2×#121—24no—
103C2×#121C2314yes—
104C2×#121C2314yes—
105C2×#120—22no—
106C2×#120—22no—
107C2×#120—24no—
108C2×#120—24no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4C2314yes—
116C2×C4×D4C2314yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117C2312yes—
120C22×C4⋊C4C2312yes—
121#460—28no—
122#450—44no—
123#450—44no—
124#440—48no—
125#435—48no—
126C2×#107C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129#418—44no—
130C2×#103C2314yes—
131C2×#102—44no—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2314yes—
144#398—24no—
145#397—22no—
146#394—24no—
147#394—22no—
148#394—22no—
149#394—24no—
150#394—24no—
151#393—22no—
152#392—22no—
153#392—22no—
154#391—21no—
155#391—22no—
156#391—21no—
157#391—21no—
158#390—21no—
159#390—22no—
160#390—22no—
161#390D414yes—
162#390D411yescharacteristic
163#390D411yescharacteristic
164#390C2314yes—
165#388—24no—
166#388—24no—
167#388—24no—
168#387—24no—
169#386—24no—
170#384D411yescharacteristic
171#384C2312yes—
172#382—21no—
173#382C2314yes—
174#380—24no—
175#380C2314yes—
176#380C2312yes—
177#380C2314yes—
178#380C2312yes—
179#379—22no—
180#379—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2311yescharacteristic
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—24no—
186C43—21no—
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195C25—22no—
196C25—22no—
197C22×Q8—21no—
198C22×Q8—21no—
199C22×Q8C2411yescharacteristic
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—416no—
215C22×D4—416no—
216C22×D4—88no—
217C22×D4C2×D414yes—
218C22×D4C2×D411yescharacteristic
219C22×D4C2×D411yescharacteristic
220C22×D4C2414yes—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C2×SD16—28no—
226C2×SD16—28no—
227C2×SD16—44no—
228C2×M16—42no—
229C2×M16—44no—
230C22×C8—42no—
231C22×C8—44no—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—41no—
235C4⋊Q8—41no—
236C4⋊Q8—41no—
237C4⋊Q8—41no—
238#127—24no—
239#127—24no—
240#127#4118yes—
241#126—48no—
242#125—44no—
243#124—44no—
244#124—44no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—42no—
248C22⋊Q8—42no—
249C22⋊Q8—48no—
250#121—28no—
251#121—28no—
252#121—44no—
253#121—44no—
254#121—416no—
255#121—416no—
256#120—416no—
257#120—416no—
258#120—48no—
259#120—48no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—44no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—28no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4—84no—
276#117—24no—
277#117—42no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2412yes—
323#107—44no—
324C4⋊C8—24no—
325C4⋊C8—24no—
326#103—44no—
327#102—28no—
328#102—28no—
329#102—44no—
330#102—44no—
331#102—44no—
332#102—44no—
333#102—44no—
334C22⋊C8—44no—
335C22⋊C8—44no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338#97—28no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—21no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347C24—24no—
348C24—48no—
349C24—48no—
350C24—44no—
351C24—44no—
352C24—82no—
353C24—82no—
354C24—84no—
355C24—84no—
356C24—82no—
357C24—82no—
358C24C22×D412yes—
359C24C22×D412yes—
360C2×Q8—21no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—82no—
368C2×Q8H(5,𝔽2)12yes—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—41no—
380C2×D4—44no—
381C2×D4—41no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—84no—
403C2×D4—84no—
404C2×D4—864no—
405C2×D4AGL(1,ℤ/8)14yes—
406C2×D4C2×#4118yes—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—84no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C2511yescharacteristic
433SD16—416no—
434M16—84no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—84no—
442C4⋊C4—24no—
443C4⋊C4—28no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—82no—
463C4⋊C4H(5,𝔽2)12yes—
464C22⋊C4—28no—
465C22⋊C4—28no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C42—24no—
481C42—22no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—42no—
486C42—44no—
487C42—44no—
488C42—42no—
489C42—82no—
490C42C2×#4114yes—
491C42H(5,𝔽2)14yes—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—816no—
509C23—88no—
510C23—88no—
511C23—816no—
512C23—164no—
513C23—168no—
514C23—168no—
515C23—164no—
516C23—164no—
517C23—164no—
518C23C2×AGL(1,ℤ/8)14yes—
519C23C23×D411yescharacteristic
520Q8—24no—
521Q8—41no—
522Q8—41no—
523Q8—84no—
524Q8—162no—
525D4—28no—
526D4—28no—
527D4—44no—
528D4—416no—
529D4—416no—
530D4—44no—
531D4—44no—
532D4—82no—
533D4—88no—
534D4—88no—
535D4—88no—
536D4—816no—
537D4—816no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—816no—
572C2×C4—164no—
573C2×C4#54614yes—
574C2×C4#54614yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C22×#4112yes—
577C2×C4C2×H(5,𝔽2)11yescharacteristic
578C2×C4C2×H(5,𝔽2)11yescharacteristic
579C2×C4C2×H(5,𝔽2)12yes—
580C8—44no—
581C8—44no—
582C22—24no—
583C22—28no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—48no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—48no—
592C22—44no—
593C22—48no—
594C22—42no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—1616no—
600C22—1616no—
601C22C2×#54612yes—
602C22#348411yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22#358512yes—
605C22#358611yescharacteristic
606C4—24no—
607C4—22no—
608C4—24no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—82no—
615C4—82no—
616C4—84no—
617C4—82no—
618C4#348111yescharacteristic
619C4#351912yes—
620C4#354911yescharacteristic
621C2—44no—
622C2—82no—
623C2—88no—
624C2#5904111yescharacteristic
625C2#6115512yes—
626C2#6121912yes—
627C2#6157211yescharacteristic
628C2#6167311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59816C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36534C2semidirect
3. #31270C2semidirect
4. #31260C2semidirect
5. #29145C2semidirect
6. #28154C2semidirect
7. #24981C2semidirect
8. #24252C2semidirect
9. #23979C2semidirect
10. #23969C2semidirect
11. #23175C2semidirect
12. #22423C2semidirect
13. #21505C2semidirect
14. C2×#556C22semidirect
15. C2×#554C22semidirect
19. C2×#450C22semidirect
20. C2×#450C22semidirect
21. C2×#440C22semidirect
22. C2×#435C22semidirect
23. C2×#418C22semidirect
24. C22×#102C22semidirect
25. #2862C22semidirect
30. #2563C22semidirect
32. #2547C22semidirect
36. #2536C22semidirect
40. #2352C22semidirect
44. C42×Q8C22semidirect
45. #2076C22semidirect
46. #2075C22semidirect
47. #2033C22semidirect
48. #2033C22semidirect
49. #2023C22semidirect
50. #2023C22semidirect
51. #2000C22semidirect
52. #2000C22semidirect
55. #1955C22semidirect
56. #1947C22semidirect
57. #1925C22semidirect
58. #1895C22semidirect
59. #1894C22semidirect
60. #1893C22semidirect
61. #1892C22semidirect
62. #1880C22semidirect
63. #1875C22semidirect
64. #1874C22semidirect
65. #1870C22semidirect
66. #1869C22semidirect
67. #1836C22semidirect
68. #1825C22semidirect
69. #1824C22semidirect
70. #1804C22semidirect
71. #1804C22semidirect
72. #1803C22semidirect
73. #1786C22semidirect
74. #1786C22semidirect
75. #1767C22semidirect
76. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#598162
#365342
#312702
#312602
#291452
#281542
#249812
#242522
#239792
#239692
#231752
#224232
#215052