Finite Group Databasebeta

← #7487774

Subgroups — #7487774

Order 512.

2429 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59816C211yesmaximal, characteristic
2#39167C211yesmaximal, characteristic, complemented
3#31393C214yesmaximal, complemented
4#31260C212yesmaximal, complemented
5#30510C212yesmaximal, complemented
6#30415C214yesmaximal, complemented
7#30039C212yesmaximal, complemented
8#29367C212yesmaximal, complemented
9#24448C214yesmaximal, complemented
10#23979C212yesmaximal, complemented
11#23855C212yesmaximal, complemented
12#23849C214yesmaximal, complemented
13#23356C211yesmaximal, characteristic, complemented
14C23×SD16C2212yescomplemented
15C2×#556C2211yescharacteristic, complemented
16C2×D4×Q8C2211yescharacteristic, complemented
17C2×D42—21no—
18C2×D42C2214yes—
19C2×#468C2214yescomplemented
20C2×#448C2214yescomplemented
21C2×#450C2212yescomplemented
22C2×#450C2214yescomplemented
23C2×#450C2212yescomplemented
24C2×#418C2212yescomplemented
25#2862C2211yescharacteristic, complemented
26#2723—24no—
27#2679—21no—
28#2679—21no—
29#2640—21no—
30#2567—21no—
31#2547C2211yescharacteristic, complemented
32#2544—21no—
33#2542C2211yescharacteristic, complemented
34#2537—21no—
35#2537C2214yes—
36#2415C2214yescomplemented
37#2409C2214yescomplemented
38#2408—24no—
39#2408C2212yes—
40#2408C2212yes—
41#2326—21no—
42C4×#127C2212yes—
43C4×#127C2211yescharacteristic
44#2295C2211yescharacteristic, complemented
45#2100C2212yescomplemented
46#2095C2212yescomplemented
47#2075C2214yescomplemented
48#2072C2214yescomplemented
49#2041C2212yes—
50#2036C2214yescomplemented
51#2033C2214yescomplemented
52#2023C2214yescomplemented
53#2014C2212yes—
54#2014C2214yes—
55#2013C2214yescomplemented
56#2000C2214yescomplemented
57#2000C2214yescomplemented
58#2000C2214yescomplemented
59#1925C2214yescomplemented
60#1894C2212yescomplemented
61#1893C2214yescomplemented
62#1892C2212yescomplemented
63#1880C2214yescomplemented
64#1877C2214yescomplemented
65#1877C2214yescomplemented
66#1877C2214yescomplemented
67#1875C2214yescomplemented
68#1870C2212yescomplemented
69#1870C2214yescomplemented
70#1869C2212yescomplemented
71#1869C2214yescomplemented
72#1862C2214yescomplemented
73#1839C2212yescomplemented
74#1825C2212yescomplemented
75#1787C2212yescomplemented
76#1786C2214yescomplemented
77C23×Q8C2312yes—
78C23×D4—24no—
79C23×D4C2312yes—
80C23×D4C2312yes—
81C22×SD16—48no—
82C22×SD16—44no—
83C22×SD16C2314yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×M16C2312yes—
87C23×C8C2312yes—
88#556—42no—
89#556—42no—
90D4×Q8—42no—
91D4×Q8—42no—
92D42—48no—
93D42—48no—
94D42—48no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127D411yescharacteristic
99C2×#127C2312yes—
100C2×#126—22no—
101C2×#124—21no—
102C2×#124—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121C2314yes—
110C2×#121C2314yes—
111C2×#120—22no—
112C2×#120—22no—
113C2×#120—24no—
114C2×#120—24no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4C2314yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2314yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117C2312yes—
125#468—48no—
126#450—44no—
127#450—48no—
128#450—44no—
129#448—48no—
130C2×#107C2314yes—
131#418—44no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144#399—22no—
145#397—24no—
146#397—22no—
147#395—24no—
148#394—24no—
149#394—22no—
150#394—22no—
151#394—24no—
152#394—24no—
153#393—24no—
154#392—22no—
155#392—22no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391C2314yes—
160#391C2312yes—
161#390—21no—
162#390—22no—
163#390—22no—
164#390D411yescharacteristic
165#390D411yescharacteristic
166#390D414yes—
167#390C2314yes—
168#389—24no—
169#389C2312yes—
170#388—24no—
171#387—22no—
172#386—24no—
173#385—24no—
174#384—21no—
175#384D411yescharacteristic
176#384C2312yes—
177#382—22no—
178#380—24no—
179#380C2312yes—
180#380C2314yes—
181#380C2312yes—
182#379—22no—
183#379—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2311yescharacteristic
186#376—21no—
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#336C2314yes—
193#336C2312yes—
194#336C2312yes—
195#336C2314yes—
196C25—22no—
197C25—22no—
198C22×Q8—21no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8C2412yes—
205C22×Q8C2411yescharacteristic
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—416no—
221C22×D4—416no—
222C22×D4—88no—
223C22×D4C2×D414yes—
224C22×D4C2×D411yescharacteristic
225C22×D4C2×D411yescharacteristic
226C22×D4C2414yes—
227C23×C4—22no—
228C23×C4—24no—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C2×SD16—28no—
232C2×SD16—28no—
233C2×SD16—28no—
234C2×SD16—28no—
235C2×SD16—44no—
236C2×SD16—44no—
237C2×SD16—44no—
238C2×SD16—44no—
239C2×SD16—44no—
240C2×SD16—432no—
241C2×M16—42no—
242C2×M16—44no—
243C22×C8—42no—
244C22×C8—44no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—41no—
248C4⋊Q8—41no—
249C4⋊Q8—41no—
250C4⋊Q8—41no—
251C4⋊Q8—44no—
252#127—24no—
253#127—28no—
254#126—48no—
255#124—44no—
256#124—44no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—42no—
260C22⋊Q8—42no—
261C22⋊Q8—48no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264#121—28no—
265#121—28no—
266#121—44no—
267#121—44no—
268#121—416no—
269#121—416no—
270#120—416no—
271#120—416no—
272#120—48no—
273#120—48no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—44no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—41no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—84no—
288#117—24no—
289#117—42no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331#107—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335#102—28no—
336#102—28no—
337#102—44no—
338#102—44no—
339#102—44no—
340C22⋊C8—44no—
341C22⋊C8—42no—
342C22⋊C8—44no—
343C22⋊C8—42no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—24no—
353C24—24no—
354C24—48no—
355C24—48no—
356C24—44no—
357C24—44no—
358C24—82no—
359C24—82no—
360C24—84no—
361C24—84no—
362C24—82no—
363C24—82no—
364C24C22×D412yes—
365C24C22×D412yes—
366C2×Q8—21no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—48no—
377C2×Q8—84no—
378C2×Q8—84no—
379C2×Q8—88no—
380C2×Q8H(5,𝔽2)12yes—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—22no—
385C2×D4—28no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—41no—
392C2×D4—44no—
393C2×D4—41no—
394C2×D4—44no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—82no—
407C2×D4—82no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—84no—
413C2×D4—84no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—864no—
417C2×D4AGL(1,ℤ/8)14yes—
418C2×D4H(5,𝔽2)18yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—44no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C2511yescharacteristic
445SD16—416no—
446SD16—416no—
447SD16—88no—
448M16—84no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—48no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—41no—
464C4⋊C4—44no—
465C4⋊C4—41no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4H(5,𝔽2)12yes—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C42—24no—
493C42—22no—
494C42—24no—
495C42—24no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—44no—
500C42—42no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—816no—
518C23—88no—
519C23—88no—
520C23—816no—
521C23—164no—
522C23—168no—
523C23—168no—
524C23—164no—
525C23—164no—
526C23—164no—
527C23C2×AGL(1,ℤ/8)14yes—
528C23C23×D411yescharacteristic
529Q8—24no—
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—44no—
534Q8—84no—
535Q8—88no—
536D4—28no—
537D4—28no—
538D4—44no—
539D4—416no—
540D4—416no—
541D4—44no—
542D4—44no—
543D4—82no—
544D4—88no—
545D4—88no—
546D4—88no—
547D4—816no—
548D4—816no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—816no—
584C2×C4—88no—
585C2×C4C2×AGL(1,ℤ/8)12yes—
586C2×C4#57714yes—
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×H(5,𝔽2)14yes—
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C8—44no—
591C8—44no—
592C22—24no—
593C22—28no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—42no—
598C22—48no—
599C22—42no—
600C22—44no—
601C22—48no—
602C22—42no—
603C22—48no—
604C22—44no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—1616no—
610C22—1616no—
611C22#348411yescharacteristic
612C22#348412yes—
613C22C22×AGL(1,ℤ/8)11yescharacteristic
614C22C2×#57712yes—
615C22#358611yescharacteristic
616C4—22no—
617C4—24no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—42no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4#353212yes—
629C4#353712yes—
630C2—44no—
631C2—82no—
632C2—88no—
633C2#5200411yescharacteristic
634C2#6157211yescharacteristic
635C2#6160012yes—
636C2#6166411yescharacteristic
637C2#6168012yes—
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. #39167C2semidirect
3. #31393C2semidirect
4. #31260C2semidirect
5. #30510C2semidirect
6. #30415C2semidirect
7. #30039C2semidirect
8. #29367C2semidirect
9. #24448C2semidirect
10. #23979C2semidirect
11. #23855C2semidirect
12. #23849C2semidirect
13. #23356C2semidirect
14. C23×SD16C22semidirect
15. C2×#556C22semidirect
16. C2×D4×Q8C22semidirect
19. C2×#468C22semidirect
20. C2×#448C22semidirect
21. C2×#450C22semidirect
22. C2×#450C22semidirect
23. C2×#450C22semidirect
24. C2×#418C22semidirect
25. #2862C22semidirect
31. #2547C22semidirect
33. #2542C22semidirect
36. #2415C22semidirect
37. #2409C22semidirect
44. #2295C22semidirect
45. #2100C22semidirect
46. #2095C22semidirect
47. #2075C22semidirect
48. #2072C22semidirect
50. #2036C22semidirect
51. #2033C22semidirect
52. #2023C22semidirect
55. #2013C22semidirect
56. #2000C22semidirect
57. #2000C22semidirect
58. #2000C22semidirect
59. #1925C22semidirect
60. #1894C22semidirect
61. #1893C22semidirect
62. #1892C22semidirect
63. #1880C22semidirect
64. #1877C22semidirect
65. #1877C22semidirect
66. #1877C22semidirect
67. #1875C22semidirect
68. #1870C22semidirect
69. #1870C22semidirect
70. #1869C22semidirect
71. #1869C22semidirect
72. #1862C22semidirect
73. #1839C22semidirect
74. #1825C22semidirect
75. #1787C22semidirect
76. #1786C22semidirect

Maximal subgroups

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

subgroupindex
#598162
#391672
#313932
#312602
#305102
#304152
#300392
#293672
#244482
#239792
#238552
#238492
#233562