Finite Group Databasebeta

← #6470738

Subgroups — #6470738

Order 512.

2450 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3188C212yesmaximal, complemented
2#37450C211yesmaximal, characteristic, complemented
3#34200C211yesmaximal, characteristic, complemented
4#33116C212yesmaximal, complemented
5#32913C212yesmaximal, complemented
6#29765C212yesmaximal, complemented
7#29725C212yesmaximal, complemented
8#25214C214yesmaximal, complemented
9#25210C214yesmaximal, complemented
10#25080C214yesmaximal, complemented
11#25076C214yesmaximal, complemented
12#21931C211yesmaximal, characteristic, complemented
13C2×#1874C212yesmaximal, complemented
14C2×#525C2211yescharacteristic
15C22×#121C2212yes—
16C2×#520C2211yescharacteristic, complemented
17C23×C4⋊C4C2211yescharacteristic
18#3188—216no—
19C2×#484C2212yescomplemented
20C2×#481C2212yescomplemented
21C2×#482C2214yescomplemented
22C2×#466C2214yes—
23C2×#465C2214yes—
24C2×#460C2212yescomplemented
25C2×#459C2212yescomplemented
26C2×#449C2214yescomplemented
27C2×#450C2212yescomplemented
28C2×#447C2212yescomplemented
29C22×#102C2214yescomplemented
30C22×C22⋊C8C2212yes—
31#2815—22no—
32#2619C2212yes—
33#2619C2212yes—
34#2617—22no—
35#2617—22no—
36#2504C2214yes—
37#2433C2214yes—
38#2403—22no—
39#2361C2214yescomplemented
40#2360C2214yescomplemented
41#2352—22no—
42D4×C4⋊C4—22no—
43#2322—22no—
44#2317C2211yescharacteristic, complemented
45#2311C2211yescharacteristic, complemented
46C4×#120—22no—
47#2280C2211yescharacteristic
48#2274C2211yescharacteristic, complemented
49#2085C2214yescomplemented
50#2074C2212yescomplemented
51#2073C2212yescomplemented
52#2067C2214yescomplemented
53#2065C2214yescomplemented
54#2055C2214yescomplemented
55#2054C2214yescomplemented
56#2041C2214yescomplemented
57#2014C2212yescomplemented
58#2011C2212yescomplemented
59#1982C2214yescomplemented
60#1949C2214yescomplemented
61#1926C2214yescomplemented
62#1925C2214yescomplemented
63#1880C2214yescomplemented
64#1879C2214yescomplemented
65#1874C2214yes—
66#1874C2214yes—
67#1874C2214yescomplemented
68#1874C2214yescomplemented
69#1850C2214yescomplemented
70#1850C2214yescomplemented
71#1805C2212yescomplemented
72#1804C2212yescomplemented
73#1798C2212yescomplemented
74#1798C2212yescomplemented
75C2×#340C2214yescomplemented
76C22×#41C2311yescharacteristic
77C23×D4—22no—
78C23×D4C2312yes—
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C22×SD16C2314yes—
82C22×D8C2314yes—
83C23×C8C2314yes—
84C2×#127C2312yes—
85C2×#127C2312yes—
86#525—24no—
87#525D418yescomplemented
88C2×#123—21no—
89C2×C22⋊Q8C2312yes—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—28no—
93C2×#121—28no—
94C2×#121C2314yes—
95C2×#121C2314yes—
96C2×#121C2312yes—
97C2×#120—21no—
98C2×#120—24no—
99C2×#120—24no—
100#520—44no—
101C2×C4×Q8C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—24no—
105C2×C4×D4C2312yes—
106C2×#117—21no—
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—44no—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—22no—
119#484—28no—
120#482—216no—
121#481—28no—
122#466—216no—
123#465—216no—
124#460—44no—
125#459—44no—
126#450—28no—
127#449—216no—
128#447—28no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#102—48no—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8—28no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#397—24no—
148#397—24no—
149#394—22no—
150#394—22no—
151#394—24no—
152#392—22no—
153#392—22no—
154#392—24no—
155#392—24no—
156#392—24no—
157#391C2312yes—
158#390C2312yes—
159#390C2314yes—
160#390C2314yes—
161#389—24no—
162#389C2314yes—
163#388—24no—
164#387—24no—
165#386—22no—
166#386—24no—
167#386—24no—
168#386—24no—
169#385—22no—
170#385—24no—
171#385—24no—
172#384—21no—
173#382—21no—
174#382C2312yes—
175#382C2312yes—
176#381D412yes—
177#380—22no—
178#380D412yes—
179#380D412yes—
180#379D412yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2312yes—
191#336C2312yes—
192#336C2314yes—
193C25—22no—
194C25C2×D411yescharacteristic
195C2×#41—22no—
196C2×#41—42no—
197C2×#41—42no—
198C2×#41C2×D414yes—
199C22×Q8C2411yescharacteristic
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—48no—
213C22×D4—84no—
214C22×D4—84no—
215C22×D4C2414yes—
216C22×D4C2411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—84no—
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C2×SD16—28no—
234C2×SD16—44no—
235C2×SD16—44no—
236C2×D8—28no—
237C2×D8—44no—
238C2×D8—44no—
239C22×C8—28no—
240C22×C8—44no—
241C22×C8—44no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244#127—44no—
245#127—42no—
246#127—42no—
247#123—28no—
248C22⋊Q8—24no—
249C22⋊Q8C2×D418yes—
250#121—24no—
251#121—28no—
252#121—28no—
253#121—28no—
254#121—28no—
255#121—48no—
256#121—48no—
257#121C2×D418yes—
258#120—28no—
259#120—88no—
260#120—88no—
261C4×Q8—24no—
262C4×Q8—42no—
263C4×D4—216no—
264C4×D4—24no—
265C4×D4—42no—
266C4×D4—48no—
267C4×D4—88no—
268#117—22no—
269#117—42no—
270#117—42no—
271#117—44no—
272#117C2×D414yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42C2412yes—
316C2×C42C2412yes—
317#107—28no—
318#106—28no—
319C4⋊C8—44no—
320C4⋊C8—42no—
321C4⋊C8—42no—
322#103—28no—
323#102—28no—
324#102—28no—
325#102—28no—
326#102—28no—
327#102—28no—
328#102—44no—
329#102—44no—
330C22⋊C8—24no—
331C22⋊C8—28no—
332C22⋊C8—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—24no—
338#95—22no—
339#95C2×D414yes—
340#95C2×D414yes—
341C24—24no—
342C24—24no—
343C24—21no—
344C24—24no—
345C24—41no—
346C24—41no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—82no—
351C24—82no—
352C24—82no—
353C24—82no—
354C24#12114yes—
355C24C22×D411yescharacteristic
356C24C22×D412yes—
357#41—48no—
358#41—48no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8C22×D412yes—
363C2×D4—28no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—24no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—416no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—88no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—832no—
397C2×D4C22×D412yes—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—84no—
423C22×C4—82no—
424C22×C4—84no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—816no—
429C22×C4#12118yes—
430C22×C4#12114yes—
431C22×C4C22×D412yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2511yescharacteristic
441SD16—88no—
442D8—88no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—88no—
450C4⋊C4—24no—
451C4⋊C4—28no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—84no—
465C4⋊C4C22×D414yes—
466C4⋊C4H(5,𝔽2)14yes—
467C22⋊C4—24no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—44no—
487C42—84no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—21no—
493C23—21no—
494C23—216no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—44no—
499C23—44no—
500C23—48no—
501C23—44no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—88no—
514C23—88no—
515C23—88no—
516C23—164no—
517C23—164no—
518C23—164no—
519C23—164no—
520C23C2×#12112yes—
521C23C2×#12114yes—
522C23#57614yes—
523C23#57714yes—
524C23C23×D411yescharacteristic
525Q8—24no—
526Q8—42no—
527D4—24no—
528D4—48no—
529D4—416no—
530D4—42no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—88no—
536D4—816no—
537D4—816no—
538D4—168no—
539D4—164no—
540D4—164no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—84no—
573C2×C4—816no—
574C2×C4—88no—
575C2×C4—84no—
576C2×C4—164no—
577C2×C4C2×#12112yes—
578C2×C4C2×#12114yes—
579C2×C4C23×D411yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)12yes—
582C2×C4#58512yes—
583C8—44no—
584C8—44no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—22no—
593C22—416no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—44no—
598C22—42no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—1616no—
610C22#304518yes—
611C22C22×#12111yescharacteristic
612C22#348411yescharacteristic
613C22#348511yescharacteristic
614C22C2×#57612yes—
615C22C2×#57712yes—
616C4—22no—
617C4—24no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—84no—
625C4—82no—
626C4#345912yes—
627C4#349612yes—
628C2—28no—
629C2—42no—
630C2—82no—
631C2—164no—
632C2#3498411yescharacteristic
633C2C2×#304512yes—
634C2#6163011yescharacteristic
635C2#6163311yescharacteristic
636C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3188C2semidirect
2. #37450C2semidirect
3. #34200C2semidirect
4. #33116C2semidirect
5. #32913C2semidirect
6. #29765C2semidirect
7. #29725C2semidirect
8. #25214C2semidirect
9. #25210C2semidirect
10. #25080C2semidirect
11. #25076C2semidirect
12. #21931C2semidirect
13. C2×#1874C2semidirect
16. C2×#520C22semidirect
19. C2×#484C22semidirect
20. C2×#481C22semidirect
21. C2×#482C22semidirect
24. C2×#460C22semidirect
25. C2×#459C22semidirect
26. C2×#449C22semidirect
27. C2×#450C22semidirect
28. C2×#447C22semidirect
29. C22×#102C22semidirect
39. #2361C22semidirect
40. #2360C22semidirect
44. #2317C22semidirect
45. #2311C22semidirect
48. #2274C22semidirect
49. #2085C22semidirect
50. #2074C22semidirect
51. #2073C22semidirect
52. #2067C22semidirect
53. #2065C22semidirect
54. #2055C22semidirect
55. #2054C22semidirect
56. #2041C22semidirect
57. #2014C22semidirect
58. #2011C22semidirect
59. #1982C22semidirect
60. #1949C22semidirect
61. #1926C22semidirect
62. #1925C22semidirect
63. #1880C22semidirect
64. #1879C22semidirect
67. #1874C22semidirect
68. #1874C22semidirect
69. #1850C22semidirect
70. #1850C22semidirect
71. #1805C22semidirect
72. #1804C22semidirect
73. #1798C22semidirect
74. #1798C22semidirect
75. C2×#340C22semidirect
87. #525D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31882
#374502
#342002
#331162
#329132
#297652
#297252
#252142
#252102
#250802
#250762
#219312
C2×#18742