Finite Group Databasebeta

← #6500761

Subgroups — #6500761

Order 512.

2914 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35222C211yesmaximal, characteristic
2#34361C211yesmaximal, characteristic, complemented
3#27594C212yesmaximal, complemented
4#27591C212yesmaximal, complemented
5#24890C214yesmaximal, complemented
6#24886C214yesmaximal, complemented
7#24882C214yesmaximal, complemented
8#24878C214yesmaximal, complemented
9#24015C212yesmaximal, complemented
10#24013C212yesmaximal, complemented
11#21943C212yesmaximal, complemented
12#21743C211yesmaximal, characteristic, complemented
13#20598C212yesmaximal, complemented
14C24×D4C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#481C2212yescomplemented
19C2×#482C2214yescomplemented
20C2×#480C2212yescomplemented
21C2×#418C2212yes—
22C2×#418C2212yes—
23C2×#418C2212yes—
24C2×#418C2212yes—
25C2×#407C2212yes—
26#2846—22no—
27#2613—22no—
28#2435—22no—
29#2433—22no—
30#2408C2214yes—
31#2408C2214yes—
32#2375—22no—
33#2374—22no—
34#2354C2214yescomplemented
35#2352C2214yescomplemented
36#2333—22no—
37#2332—22no—
38#2314C2212yescomplemented
39#2311C2212yescomplemented
40#2309C2211yescharacteristic
41#2309C2211yescharacteristic
42#2280C2212yes—
43#2280C2211yescharacteristic
44#2094C2212yescomplemented
45#2093C2212yescomplemented
46#2092C2214yescomplemented
47#2073C2212yescomplemented
48#2073C2212yescomplemented
49#2071C2214yescomplemented
50#2034C2214yescomplemented
51#2034C2214yescomplemented
52#2033C2214yescomplemented
53#2033C2214yescomplemented
54#1926C2214yescomplemented
55#1926C2214yescomplemented
56#1925C2214yescomplemented
57#1925C2214yescomplemented
58#1875C2212yescomplemented
59#1875C2212yescomplemented
60#1874C2214yescomplemented
61#1870C2214yescomplemented
62#1870C2214yescomplemented
63#1854C2214yescomplemented
64#1854C2214yescomplemented
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1818C2212yescomplemented
68#1809C2214yescomplemented
69#1809C2214yescomplemented
70#1786C2214yes—
71#1786C2214yes—
72#1786C2214yes—
73#1786C2214yes—
74#1780C2212yescomplemented
75C26—21no—
76C22×#41C2311yescharacteristic
77C23×D4—21no—
78C23×D4—22no—
79C23×D4—28no—
80C23×D4—42no—
81C23×D4—42no—
82C23×D4—44no—
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90#535—42no—
91#535—42no—
92#535—44no—
93C2×#127—21no—
94C2×#127—21no—
95C2×#124—21no—
96C2×#124—21no—
97#525—28no—
98C2×#123D412yes—
99C2×C22⋊Q8D412yes—
100C2×#121D412yes—
101C2×#121C2312yes—
102C2×#121C2314yes—
103C2×#121C2312yes—
104C2×#120—22no—
105C2×#120—22no—
106C2×#120—24no—
107C2×#120—24no—
108C2×#120D412yes—
109C2×C4×D4C2314yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2312yes—
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C2×#117C2312yescomplemented
116C22×C4⋊C4C2312yescomplemented
117#482—48no—
118#481—44no—
119#480—44no—
120C2×#107C2314yescomplemented
121C2×#107C2314yescomplemented
122C2×#106C2314yescomplemented
123C2×#106C2314yescomplemented
124#418—44no—
125#418—44no—
126#418—44no—
127#418—44no—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136#407—28no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2312yescomplemented
140C2×C22⋊C8C2314yescomplemented
141C2×C22⋊C8C2312yescomplemented
142#394—24no—
143#394—24no—
144#392—24no—
145#392—24no—
146#391—22no—
147#390—22no—
148#390—24no—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#390C2312yes—
153#390C2312yes—
154#388—24no—
155#388—24no—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#386—22no—
161#386—22no—
162#386—24no—
163#384C2312yescomplemented
164#382C2312yescomplemented
165#381—22no—
166#381—21no—
167#380C2314yes—
168#380C2314yes—
169#380C2314yes—
170#380C2314yescomplemented
171#379—21no—
172#379—22no—
173#379—22no—
174#379—22no—
175#379—24no—
176#379—24no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4C2314yescomplemented
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—24no—
183#340C2314yescomplemented
184#340C2314yescomplemented
185#340C2314yescomplemented
186#340C2314yescomplemented
187#336C2314yescomplemented
188#336C2312yescomplemented
189#336C2312yescomplemented
190C25—21no—
191C25—21no—
192C25—22no—
193C25—22no—
194C25—44no—
195C25—44no—
196C25—44no—
197C25—88no—
198C25C2×D411yescharacteristic
199C2×#41—24no—
200C2×#41—41no—
201C2×#41—41no—
202C2×#41—42no—
203C22×Q8C2×D411yescharacteristic
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—24no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4—42no—
231C22×D4—432no—
232C22×D4—432no—
233C22×D4—48no—
234C22×D4—48no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—48no—
238C22×D4—44no—
239C22×D4C2×D414yes—
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C22×D4C2411yescharacteristic
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—42no—
247C23×C4—41no—
248C23×C4—41no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4#4118yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C2×M16—24no—
256C2×M16—24no—
257C2×M16—42no—
258C2×M16—42no—
259C2×M16—42no—
260C2×M16—42no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263#127—44no—
264#127—44no—
265#124—44no—
266#124—44no—
267#123—28no—
268C22⋊Q8—28no—
269#121—28no—
270#121—24no—
271#121—24no—
272#121—28no—
273#121—42no—
274#121—44no—
275#121—42no—
276#120—28no—
277#120—48no—
278#120—48no—
279#120—88no—
280#120—88no—
281C4×D4—28no—
282C4×D4—28no—
283#117—24no—
284#117—41no—
285#117—42no—
286#117—41no—
287#117—42no—
288#117—42no—
289#117—42no—
290#117—42no—
291#117—44no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
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⋊C4C2×D412yes—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42C2412yes—
328#107—44no—
329#107—44no—
330#106—44no—
331#106—44no—
332#102—28no—
333#102—28no—
334#102—28no—
335#102—28no—
336#102—44no—
337#102—44no—
338#102—44no—
339#102—44no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—44no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—21no—
355C24—21no—
356C24—41no—
357C24—41no—
358C24—44no—
359C24—44no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—42no—
364C24—42no—
365C24—48no—
366C24—48no—
367C24—432no—
368C24—44no—
369C24—44no—
370C24—42no—
371C24—42no—
372C24—84no—
373C24—84no—
374C24—84no—
375C24—84no—
376C24—832no—
377C24—832no—
378C24—832no—
379C24#12314yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C22×D411yescharacteristic
383#41—84no—
384#41—84no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×D4—216no—
389C2×D4—216no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—464no—
429C2×D4—432no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4H(5,𝔽2)12yes—
442C2×D4H(5,𝔽2)12yes—
443C2×D4H(5,𝔽2)14yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4#12318yes—
468C22×C4#12314yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2511yescharacteristic
477M16—84no—
478M16—84no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—24no—
516C42—42no—
517C42—42no—
518C42—84no—
519C42—82no—
520C42—82no—
521C42—82no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—44no—
538C23—48no—
539C23—416no—
540C23—416no—
541C23—48no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—88no—
556C23—88no—
557C23—88no—
558C23—88no—
559C23—88no—
560C23—88no—
561C23—88no—
562C23—88no—
563C23—816no—
564C23—816no—
565C23—816no—
566C23—816no—
567C23—8128no—
568C23—88no—
569C23—88no—
570C23C2×#12312yes—
571C23C2×#12314yes—
572C23C23×D411yescharacteristic
573Q8—82no—
574D4—24no—
575D4—24no—
576D4—24no—
577D4—24no—
578D4—416no—
579D4—416no—
580D4—48no—
581D4—48no—
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—82no—
587D4—88no—
588D4—84no—
589D4—168no—
590D4—164no—
591D4—164no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—164no—
623C2×C4C2×#12314yes—
624C2×C4C2×#12312yes—
625C2×C4C22×#4112yes—
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C8—44no—
630C8—44no—
631C22—22no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—416no—
637C22—44no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—832no—
653C22—832no—
654C22—816no—
655C22—816no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—84no—
663C22C22×#12311yescharacteristic
664C22#348411yescharacteristic
665C22#348411yescharacteristic
666C22#358514yes—
667C4—24no—
668C4—22no—
669C4—42no—
670C4—44no—
671C4—84no—
672C4—82no—
673C4—84no—
674C4—82no—
675C4#349012yes—
676C4#349012yes—
677C2—42no—
678C2—44no—
679C2—82no—
680C2—84no—
681C2—82no—
682C2—82no—
683C2#3535011yescharacteristic
684C2#6051812yes—
685C2#6164312yes—
686C2#6164412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35222C2
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. #27591C2semidirect
5. #24890C2semidirect
6. #24886C2semidirect
7. #24882C2semidirect
8. #24878C2semidirect
9. #24015C2semidirect
10. #24013C2semidirect
11. #21943C2semidirect
12. #21743C2semidirect
13. #20598C2semidirect
18. C2×#481C22semidirect
19. C2×#482C22semidirect
20. C2×#480C22semidirect
34. #2354C22semidirect
35. #2352C22semidirect
38. #2314C22semidirect
39. #2311C22semidirect
44. #2094C22semidirect
45. #2093C22semidirect
46. #2092C22semidirect
47. #2073C22semidirect
48. #2073C22semidirect
49. #2071C22semidirect
50. #2034C22semidirect
51. #2034C22semidirect
52. #2033C22semidirect
53. #2033C22semidirect
54. #1926C22semidirect
55. #1926C22semidirect
56. #1925C22semidirect
57. #1925C22semidirect
58. #1875C22semidirect
59. #1875C22semidirect
60. #1874C22semidirect
61. #1870C22semidirect
62. #1870C22semidirect
63. #1854C22semidirect
64. #1854C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
67. #1818C22semidirect
68. #1809C22semidirect
69. #1809C22semidirect
74. #1780C22semidirect
115. C2×#117C23semidirect
116. C22×C4⋊C4C23semidirect
120. C2×#107C23semidirect
121. C2×#107C23semidirect
122. C2×#106C23semidirect
123. C2×#106C23semidirect
139. C2×C22⋊C8C23semidirect
140. C2×C22⋊C8C23semidirect
141. C2×C22⋊C8C23semidirect
163. #384C23semidirect
164. #382C23semidirect
170. #380C23semidirect
178. C4×C4⋊C4C23semidirect
183. #340C23semidirect
184. #340C23semidirect
185. #340C23semidirect
186. #340C23semidirect
187. #336C23semidirect
188. #336C23semidirect
189. #336C23semidirect

Maximal subgroups

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

subgroupindex
#352222
#343612
#275942
#275912
#248902
#248862
#248822
#248782
#240152
#240132
#219432
#217432
#205982