Finite Group Databasebeta

← #6644830

Subgroups — #6644830

Order 512.

1777 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35031C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic, complemented
3#33098C211yesmaximal, characteristic, complemented
4#33094C211yesmaximal, characteristic, complemented
5#32266C214yesmaximal, complemented
6#31329C214yesmaximal, complemented
7#31327C214yesmaximal, complemented
8#31268C214yesmaximal, complemented
9#29642C211yesmaximal, characteristic, complemented
10#29639C211yesmaximal, characteristic, complemented
11#27604C212yesmaximal, complemented
12#24553C212yesmaximal, complemented
13#24552C212yesmaximal, complemented
14#24007C212yesmaximal, complemented
15#21738C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#481C2212yes—
21C22×C22⋊C8C2212yes—
22#2855C2211yescharacteristic
23#2854C2211yescharacteristic
24#2848C2211yescharacteristic
25#2847C2211yescharacteristic
26#2622—21no—
27#2613—21no—
28#2602—22no—
29#2600—22no—
30#2563—22no—
31#2563—22no—
32#2563C2214yes—
33#2563C2214yes—
34#2485C2212yes—
35#2415C2212yes—
36#2414C2212yes—
37#2407C2212yes—
38C2×#393—21no—
39C2×#393—21no—
40#2322—21no—
41#2321—21no—
42C2×#390—21no—
43C2×#382—21no—
44#2280C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48#2077C2214yescomplemented
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2075C2214yescomplemented
52#2058C2214yescomplemented
53#2057C2214yescomplemented
54#2036C2214yescomplemented
55#2035C2214yescomplemented
56#2034C2214yescomplemented
57#2033C2214yescomplemented
58#2024C2214yescomplemented
59#2023C2214yescomplemented
60#1927C2214yescomplemented
61#1925C2214yescomplemented
62#1920C2214yescomplemented
63#1918C2214yescomplemented
64#1893C2214yescomplemented
65#1892C2214yescomplemented
66#1874C2212yes—
67#1872C2212yes—
68#1869C2214yes—
69#1869C2214yes—
70#1854C2214yescomplemented
71#1854C2214yescomplemented
72#1853C2212yes—
73#1852C2212yes—
74#1825C2214yescomplemented
75#1824C2212yes—
76#1824C2212yes—
77#1817C2211yescharacteristic
78#1817C2211yescharacteristic
79#1808C2214yescomplemented
80#1807C2214yescomplemented
81#1788C2214yescomplemented
82#1779C2212yes—
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×D4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C23×C8C2314yes—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8C2314yes—
92#525—44no—
93C2×#123—21no—
94C2×#123—21no—
95C2×C22⋊Q8C2311yescharacteristic
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2311yescharacteristic
98C2×#121C2311yescharacteristic
99C2×#121C2312yes—
100C2×#121C2311yescharacteristic
101C2×#120—21no—
102C2×#120—21no—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—44no—
106C2×C4×D4D412yes—
107C2×C4×D4D412yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—22no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122#484—44no—
123#481—44no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8—44no—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138#397—22no—
139#397—22no—
140#394—22no—
141#394—22no—
142#394—22no—
143#394—22no—
144#393—24no—
145#393—24no—
146#393—24no—
147#393—24no—
148#392—22no—
149#392—22no—
150#391C2312yes—
151#390—24no—
152#390—24no—
153#390—22no—
154#390C2312yes—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#386—21no—
165#386—21no—
166#386—22no—
167#386—22no—
168#385—21no—
169#385—21no—
170#385—22no—
171#385—22no—
172#382—24no—
173#382—24no—
174#381—21no—
175#381—21no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380C2314yes—
181#380C2314yes—
182#380C2314yes—
183#380C2314yes—
184#380C2312yes—
185#380C2312yes—
186#380C2312yes—
187#380C2311yescharacteristic
188#380C2311yescharacteristic
189#379—21no—
190#379—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4C2314yes—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C2×#95—21no—
196C2×#95—22no—
197C2×#95—22no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25C2×D411yescharacteristic
209C25C2×D411yescharacteristic
210C2×#41—42no—
211C2×#41—44no—
212C22×Q8—24no—
213C22×Q8—44no—
214C22×Q8—84no—
215C22×Q8C2411yescharacteristic
216C22×D4—24no—
217C22×D4—21no—
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—24no—
222C22×D4—42no—
223C22×D4C2×D412yes—
224C22×D4C2×D412yes—
225C22×D4C2411yescharacteristic
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C22×C8—44no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—48no—
252#123—44no—
253#123—44no—
254C22⋊Q8—24no—
255C22⋊Q8—42no—
256C22⋊Q8—44no—
257#121—24no—
258#121—42no—
259#121—44no—
260#120—44no—
261#120—44no—
262C4×D4—44no—
263C4×D4—44no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—44no—
269C4×D4—44no—
270#117—42no—
271#117—44no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—84no—
315C2×C22⋊C4—84no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42C2×D412yes—
326C2×C42C2412yes—
327#107—44no—
328#107—44no—
329#106—44no—
330#106—44no—
331#103—44no—
332#103—44no—
333#102—44no—
334#102—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—44no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—21no—
346#95—22no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24#13714yes—
365C24C22×D411yescharacteristic
366C24C22×D411yescharacteristic
367C24C22×D412yes—
368C24C22×D411yescharacteristic
369#41—88no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—48no—
373C2×Q8—84no—
374C2×Q8—84no—
375C2×Q8—816no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—416no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—44no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—82no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4—82no—
436C22×C4#13514yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C4C2511yescharacteristic
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—88no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—42no—
497C42—48no—
498C42—82no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—48no—
519C23—48no—
520C23—44no—
521C23—42no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—82no—
527C23—82no—
528C23—84no—
529C23—84no—
530C23—82no—
531C23—82no—
532C23#53412yes—
533C23C2×#13512yes—
534C23C2×#13714yes—
535C23C2×#13712yes—
536C23C23×D411yescharacteristic
537Q8—44no—
538Q8—816no—
539D4—48no—
540D4—48no—
541D4—44no—
542D4—84no—
543D4—84no—
544D4—84no—
545D4—164no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4#53412yes—
594C2×C4C2×#13514yes—
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4#58514yes—
597C8—44no—
598C8—44no—
599C22—24no—
600C22—24no—
601C22—22no—
602C22—22no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—88no—
621C22—88no—
622C22—82no—
623C22—82no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22#319014yes—
628C22C2×#53411yescharacteristic
629C22#348512yes—
630C22C22×#13511yescharacteristic
631C22C22×#13711yescharacteristic
632C22#358512yes—
633C4—24no—
634C4—22no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4—81no—
643C4—84no—
644C4—81no—
645C4—81no—
646C4#354514yes—
647C2—24no—
648C2—44no—
649C2—81no—
650C2—81no—
651C2—161no—
652C2#3711911yescharacteristic
653C2C2×#319011yescharacteristic
654C2#6075311yescharacteristic
655C2#6170012yes—
656C2#6170912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35031C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2563C2semidirect
3. #33098C2semidirect
4. #33094C2semidirect
5. #32266C2semidirect
6. #31329C2semidirect
7. #31327C2semidirect
8. #31268C2semidirect
9. #29642C2semidirect
10. #29639C2semidirect
11. #27604C2semidirect
12. #24553C2semidirect
13. #24552C2semidirect
14. #24007C2semidirect
15. #21738C2semidirect
48. #2077C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2075C22semidirect
52. #2058C22semidirect
53. #2057C22semidirect
54. #2036C22semidirect
55. #2035C22semidirect
56. #2034C22semidirect
57. #2033C22semidirect
58. #2024C22semidirect
59. #2023C22semidirect
60. #1927C22semidirect
61. #1925C22semidirect
62. #1920C22semidirect
63. #1918C22semidirect
64. #1893C22semidirect
65. #1892C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
74. #1825C22semidirect
79. #1808C22semidirect
80. #1807C22semidirect
81. #1788C22semidirect

Maximal subgroups

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

subgroupindex
#350312
C2×#25632
#330982
#330942
#322662
#313292
#313272
#312682
#296422
#296392
#276042
#245532
#245522
#240072
#217382