Finite Group Databasebeta

← #7569883

Subgroups — #7569883

Order 512.

2161 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#36260C211yesmaximal, characteristic
3#32038C212yesmaximal, complemented
4#32027C212yesmaximal, complemented
5#28753C212yesmaximal, complemented
6#28610C214yesmaximal, complemented
7#26244C214yesmaximal, complemented
8#25898C214yesmaximal, complemented
9#24054C212yesmaximal, complemented
10#24032C212yesmaximal, complemented
11#23175C211yesmaximal, characteristic, complemented
12#22419C214yesmaximal, complemented
13#21614C212yesmaximal, complemented
14C2×#559—21no—
15C2×#547C2211yescharacteristic
16C2×#546C2214yes—
17C2×D42C2211yescharacteristic
18C2×#489C2214yescomplemented
19C2×#484C2212yescomplemented
20C2×#480C2212yes—
21C2×#476C2212yes—
22C2×#435C2214yescomplemented
23C2×#417C2212yes—
24C22×#102C2212yes—
25#2858C2211yescharacteristic
26#2856—21no—
27#2724—21no—
28#2724—21no—
29#2582—21no—
30#2547C2214yescomplemented
31#2542C2211yescharacteristic
32#2541C2211yescharacteristic
33#2540C2214yes—
34#2539—24no—
35#2501—21no—
36#2486C2212yes—
37#2486C2212yescomplemented
38#2447—24no—
39#2433—21no—
40D4×C4⋊C4C2214yescomplemented
41C4×C4⋊Q8C2212yescomplemented
42C4×#127C2211yescharacteristic
43C2×#390—21no—
44C42×D4C2211yescharacteristic
45#2043C2214yescomplemented
46#2041C2214yes—
47#2036C2214yescomplemented
48#2035C2214yescomplemented
49#2035C2214yescomplemented
50#2034C2214yescomplemented
51#2034C2214yescomplemented
52#2033C2214yes—
53#1987C2212yescomplemented
54#1983C2214yescomplemented
55#1983C2214yescomplemented
56#1959C2214yescomplemented
57#1955C2214yescomplemented
58#1947C2212yescomplemented
59#1919C2214yescomplemented
60#1895C2214yescomplemented
61#1894C2212yescomplemented
62#1893C2212yes—
63#1892C2214yescomplemented
64#1874C2214yescomplemented
65#1872C2214yescomplemented
66#1870C2212yescomplemented
67#1869C2212yes—
68#1862C2214yes—
69#1802C2212yes—
70#1788C2212yes—
71#1788C2214yes—
72#1786C2212yes—
73#1786C2214yes—
74#1776C2214yescomplemented
75#1767C2212yescomplemented
76#1762C2214yes—
77C22×#41C2312yes—
78C23×D4—21no—
79C23×D4C2312yes—
80C22×M16C2312yes—
81C23×C8C2312yes—
82#559—48no—
83#547—42no—
84#547—42no—
85#546—216no—
86#546—48no—
87D42—42no—
88D42—42no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127—21no—
92C2×#127D411yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#124—24no—
95C2×#124C2314yes—
96C2×#123—22no—
97C2×#123—22no—
98C2×#123—24no—
99C2×#123—22no—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2314yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121C2312yes—
106C2×#121C2314yes—
107C2×#120—24no—
108C2×#120—22no—
109C2×C4×Q8—22no—
110C2×C4×D4—24no—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—24no—
120C22×C42—21no—
121#489—48no—
122#484—44no—
123#480—44no—
124#476—28no—
125#435—48no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130#417—44no—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102—44no—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147#399—24no—
148#399—22no—
149#398—22no—
150#398—22no—
151#397—22no—
152#397—22no—
153#397—24no—
154#397—24no—
155#392—24no—
156#392—22no—
157#391—24no—
158#391—22no—
159#390—21no—
160#390—21no—
161#390—24no—
162#390—24no—
163#390—21no—
164#390—21no—
165#390—22no—
166#390D411yescharacteristic
167#390D414yes—
168#390D411yescharacteristic
169#389—24no—
170#389—22no—
171#389C2314yes—
172#385—24no—
173#385—24no—
174#384—21no—
175#384D411yescharacteristic
176#384C2312yes—
177#381—22no—
178#381—22no—
179#380C2314yes—
180#380C2314yes—
181#380C2312yes—
182#380C2312yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2311yescharacteristic
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—22no—
189C43C2311yescharacteristic
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2314yes—
195C25—22no—
196C2×#41—28no—
197C2×#41—24no—
198C2×#41—44no—
199C2×#41—42no—
200C22×Q8—22no—
201C22×Q8C2412yes—
202C22×D4—28no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—42no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—82no—
215C22×D4C2×D414yes—
216C22×D4C2×D411yescharacteristic
217C22×D4C2×D411yescharacteristic
218C22×D4C2412yes—
219C22×D4C2411yescharacteristic
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C2×M16—42no—
228C2×M16—44no—
229C22×C8—42no—
230C22×C8—44no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—41no—
236C4⋊Q8—41no—
237#127—24no—
238#127—41no—
239#127—41no—
240#127—82no—
241#124—28no—
242#124—28no—
243#124—416no—
244#123—416no—
245#123—48no—
246#123—48no—
247#123—48no—
248C22⋊Q8—28no—
249C22⋊Q8—24no—
250C22⋊Q8—44no—
251C22⋊Q8—42no—
252#121—28no—
253#121—24no—
254#121—416no—
255#121—42no—
256#121—44no—
257#121—44no—
258#121—44no—
259#120—416no—
260#120—48no—
261C4×Q8—48no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—28no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—416no—
272C4×D4#4118yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—416no—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C42—24no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—82no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2412yes—
321#107—28no—
322#106—28no—
323C4⋊C8—24no—
324C4⋊C8—24no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#102—28no—
330#102—28no—
331#102—44no—
332#102—44no—
333#102—44no—
334#102—44no—
335C22⋊C8—44no—
336C22⋊C8—44no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339#97—28no—
340#95—21no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348C24—21no—
349C24—22no—
350C24—48no—
351C24—44no—
352C24—44no—
353C24—82no—
354C24—84no—
355C24C22×D412yes—
356C24C22×D412yes—
357#41—48no—
358#41—88no—
359#41—84no—
360C2×Q8—24no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—22no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—44no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—48no—
387C2×D4—42no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—82no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4—816no—
397C2×D4—82no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4AGL(1,ℤ/8)14yes—
401C2×D4C2×#4114yes—
402C2×D4H(5,𝔽2)12yes—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C2511yescharacteristic
437M16—84no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—84no—
445C4⋊C4—28no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4C2×#4114yes—
465C4⋊C4H(5,𝔽2)12yes—
466C22⋊C4—24no—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—41no—
486C42—42no—
487C42—42no—
488C42—48no—
489C42—82no—
490C42C2×#4114yes—
491C42H(5,𝔽2)14yes—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—41no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—88no—
512C23—88no—
513C23—816no—
514C23—168no—
515C23—164no—
516C23C2×AGL(1,ℤ/8)14yes—
517C23C23×D411yescharacteristic
518Q8—42no—
519Q8—44no—
520Q8—42no—
521Q8—88no—
522D4—24no—
523D4—22no—
524D4—44no—
525D4—416no—
526D4—42no—
527D4—44no—
528D4—42no—
529D4—82no—
530D4—84no—
531D4—88no—
532D4—88no—
533D4—88no—
534D4—168no—
535D4—162no—
536D4#55914yescomplemented
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—88no—
571C2×C4—88no—
572C2×C4—164no—
573C2×C4—164no—
574C2×C4#54614yes—
575C2×C4#54614yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C22×#4112yes—
578C2×C4C2×H(5,𝔽2)11yescharacteristic
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C2×C4C2×H(5,𝔽2)12yes—
581C8—44no—
582C8—44no—
583C22—24no—
584C22—24no—
585C22—28no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—44no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—42no—
598C22—82no—
599C22—84no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—1616no—
604C22C2×#54612yes—
605C22#348411yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22#358511yescharacteristic
608C22#358612yes—
609C4—24no—
610C4—22no—
611C4—24no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—82no—
618C4—84no—
619C4—84no—
620C4C2×#55911yescharacteristic
621C4#350812yes—
622C4#354311yescharacteristic
623C2—44no—
624C2—44no—
625C2—82no—
626C2—162no—
627C2#5914311yescharacteristic
628C2#6116112yes—
629C2#6121912yes—
630C2#6156911yescharacteristic
631C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36260C2
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
1. #59827C2semidirect
3. #32038C2semidirect
4. #32027C2semidirect
5. #28753C2semidirect
6. #28610C2semidirect
7. #26244C2semidirect
8. #25898C2semidirect
9. #24054C2semidirect
10. #24032C2semidirect
11. #23175C2semidirect
12. #22419C2semidirect
13. #21614C2semidirect
18. C2×#489C22semidirect
19. C2×#484C22semidirect
22. C2×#435C22semidirect
30. #2547C22semidirect
37. #2486C22semidirect
40. D4×C4⋊C4C22semidirect
41. C4×C4⋊Q8C22semidirect
45. #2043C22semidirect
47. #2036C22semidirect
48. #2035C22semidirect
49. #2035C22semidirect
50. #2034C22semidirect
51. #2034C22semidirect
53. #1987C22semidirect
54. #1983C22semidirect
55. #1983C22semidirect
56. #1959C22semidirect
57. #1955C22semidirect
58. #1947C22semidirect
59. #1919C22semidirect
60. #1895C22semidirect
61. #1894C22semidirect
63. #1892C22semidirect
64. #1874C22semidirect
65. #1872C22semidirect
66. #1870C22semidirect
74. #1776C22semidirect
75. #1767C22semidirect
536. D4#559semidirect

Maximal subgroups

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

subgroupindex
#598272
#362602
#320382
#320272
#287532
#286102
#262442
#258982
#240542
#240322
#231752
#224192
#216142