Finite Group Databasebeta

← #6529856

Subgroups — #6529856

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2C2×#2985C211yesmaximal, characteristic, complemented
3#37640C211yesmaximal, characteristic, complemented
4#35172C211yesmaximal, characteristic, complemented
5#28276C214yesmaximal, complemented
6#28197C214yesmaximal, complemented
7#25477C212yesmaximal, complemented
8#25470C212yesmaximal, complemented
9#25300C212yesmaximal, complemented
10#25292C212yesmaximal, complemented
11#24618C214yesmaximal, complemented
12#24512C214yesmaximal, complemented
13#21988C211yesmaximal, characteristic, complemented
14#21863C211yesmaximal, characteristic, complemented
15#20732C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3109—28no—
20C2×#481C2212yes—
21C2×#476C2212yes—
22C2×#459C2212yes—
23C2×#447C2212yes—
24#2985—28no—
25C2×#446C2214yescomplemented
26C2×#446C2212yescomplemented
27C2×C8×Q8C2212yescomplemented
28C2×#442C2214yes—
29C2×#433C2214yes—
30C2×#432C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2848—22no—
33#2820—22no—
34#2626—22no—
35#2624C2214yes—
36#2620—22no—
37#2616—22no—
38#2580C2214yes—
39#2567C2212yescomplemented
40#2566—22no—
41#2563—21no—
42#2548C2212yescomplemented
43#2547—21no—
44#2542—21no—
45#2542—21no—
46#2538C2212yes—
47#2538C2212yes—
48#2538C2212yescomplemented
49#2538C2212yescomplemented
50#2317C2212yes—
51#2309C2212yes—
52#2285C2211yescharacteristic
53#2285C2211yescharacteristic
54#2088C2214yescomplemented
55#2088C2214yescomplemented
56#2067C2214yes—
57#2067C2214yescomplemented
58#2065C2214yes—
59#2065C2214yescomplemented
60#2054C2214yes—
61#2054C2214yes—
62#1990C2211yescharacteristic, complemented
63#1990C2211yescharacteristic, complemented
64#1972C2214yes—
65#1970C2214yes—
66#1957C2211yescharacteristic, complemented
67#1957C2211yescharacteristic, complemented
68#1948C2212yescomplemented
69#1947C2212yescomplemented
70#1943C2214yescomplemented
71#1941C2214yescomplemented
72#1936C2214yes—
73#1936C2214yescomplemented
74#1934C2214yes—
75#1934C2214yescomplemented
76#1875C2214yes—
77#1854C2214yes—
78#1847C2212yescomplemented
79#1836C2212yescomplemented
80#1823C2212yes—
81#1804C2214yescomplemented
82#1802C2212yes—
83#1802C2212yes—
84#1786C2212yes—
85C4×#102C2212yes—
86C4×#102C2212yes—
87C22×#41—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×D8C2314yes—
91#533—24no—
92#533—24no—
93#532—24no—
94#532—24no—
95C2×C4⋊Q8D411yescharacteristic
96C2×C4⋊Q8D411yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127C2311yescharacteristic
99C2×#125D412yes—
100C2×#124—22no—
101C2×#124—21no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8D412yes—
107C2×C22⋊Q8D412yes—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—24no—
116C2×C4×D4—24no—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C42C2311yescharacteristic
126#481—28no—
127#476—28no—
128#459—28no—
129#447—28no—
130#446—28no—
131#446—216no—
132C8×Q8—28no—
133#442—216no—
134#433—216no—
135#432D418yes—
136#431—24no—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155C2×#97C2312yes—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2314yes—
159#399—24no—
160#399—24no—
161#397—24no—
162#396—24no—
163#396—24no—
164#396—24no—
165#396—24no—
166#395—24no—
167#394—24no—
168#394—24no—
169#394—24no—
170#393—24no—
171#391—22no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390C2314yes—
176#390C2314yes—
177#390C2314yes—
178#390C2314yes—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#385—24no—
184#385—24no—
185#385—24no—
186#384—22no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#382—22no—
193#382—21no—
194#382C2312yes—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#379—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4C2312yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2314yes—
215#340C2314yes—
216C25—21no—
217C2×#41—44no—
218C2×#41—82no—
219C2×#41—82no—
220C22×Q8—21no—
221C22×Q8C2×D412yes—
222C22×Q8C2×D412yes—
223C22×D4—24no—
224C22×D4—22no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—42no—
231C22×D4C2412yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—44no—
238C23×C4—82no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×D8—28no—
244C2×D8—28no—
245C2×D8—44no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253#127—22no—
254#127—22no—
255#125—28no—
256#124—44no—
257#124—84no—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260C22⋊Q8—84no—
261C22⋊Q8—84no—
262C22⋊Q8—84no—
263C22⋊Q8—84no—
264#121—24no—
265#121—24no—
266#121—44no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280#117—24no—
281#117—24no—
282#117—44no—
283#117—82no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4C2×D414yes—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D414yes—
333C2×C42#4114yes—
334C2×C42C2414yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—44no—
338#106—28no—
339#106—44no—
340C4⋊C8—28no—
341C4⋊C8—28no—
342C4⋊C8—24no—
343C4⋊C8C2×D418yes—
344#102—24no—
345#102—24no—
346#102—28no—
347#102—28no—
348#102—42no—
349#102—44no—
350#102—42no—
351#102—44no—
352C22⋊C8—24no—
353C22⋊C8C2×D418yes—
354#97—28no—
355#97C2×D418yes—
356C4×C8—28no—
357C4×C8—22no—
358C4×C8—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—81no—
373C24—81no—
374C24—81no—
375C24—81no—
376C24C22×D411yescharacteristic
377#41—816no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×Q8—24no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—42no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—81no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D42−1+414yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—816no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4#12114yes—
437C22×C4#12114yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C2×#4112yes—
447C22×C4C2511yescharacteristic
448D8—416no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8C22×D418yes—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C22⋊C4—28no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—82no—
503C42—24no—
504C42—24no—
505C42—44no—
506C42—44no—
507C42#12114yes—
508C42#12114yes—
509C42C22×D412yes—
510C42C22×D412yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—24no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23C2×#12112yes—
541C23C23×D411yescharacteristic
542Q8—48no—
543Q8—84no—
544Q8—84no—
545D4—24no—
546D4—24no—
547D4—48no—
548D4—42no—
549D4—44no—
550D4—42no—
551D4—44no—
552D4—84no—
553D4—88no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—88no—
595C2×C4C2×#12112yes—
596C2×C4C2×#12112yes—
597C2×C4C2×#12112yes—
598C2×C4C2×#12112yes—
599C2×C4C2×#12112yes—
600C2×C4#57612yes—
601C2×C4#57712yes—
602C2×C4C23×D411yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C2×2−1+412yes—
605C2×C4#58512yes—
606C8—28no—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22C22×#12111yescharacteristic
632C22#348611yescharacteristic
633C22#348711yescharacteristic
634C22C2×#57611yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358511yescharacteristic
637C22#358511yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—44no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4#306214yes—
651C4#348912yes—
652C4#349512yes—
653C2—42no—
654C2—84no—
655C2—162no—
656C2#3566111yescharacteristic
657C2C2×#306211yescharacteristic
658C2#6045411yescharacteristic
659C2#6184711yescharacteristic
660C2#6185011yescharacteristic
661C2#6191011yescharacteristic
662C2#6191311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2985C2semidirect
3. #37640C2semidirect
4. #35172C2semidirect
5. #28276C2semidirect
6. #28197C2semidirect
7. #25477C2semidirect
8. #25470C2semidirect
9. #25300C2semidirect
10. #25292C2semidirect
11. #24618C2semidirect
12. #24512C2semidirect
13. #21988C2semidirect
14. #21863C2semidirect
15. #20732C2semidirect
25. C2×#446C22semidirect
26. C2×#446C22semidirect
27. C2×C8×Q8C22semidirect
39. #2567C22semidirect
42. #2548C22semidirect
48. #2538C22semidirect
49. #2538C22semidirect
54. #2088C22semidirect
55. #2088C22semidirect
57. #2067C22semidirect
59. #2065C22semidirect
62. #1990C22semidirect
63. #1990C22semidirect
66. #1957C22semidirect
67. #1957C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1943C22semidirect
71. #1941C22semidirect
73. #1936C22semidirect
75. #1934C22semidirect
78. #1847C22semidirect
79. #1836C22semidirect
81. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
C2×#29852
#376402
#351722
#282762
#281972
#254772
#254702
#253002
#252922
#246182
#245122
#219882
#218632
#207322