Finite Group Databasebeta

← #6480286

Subgroups — #6480286

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35285C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic, complemented
3#33038C214yesmaximal, complemented
4#30066C212yesmaximal, complemented
5#30038C212yesmaximal, complemented
6#29558C212yesmaximal, complemented
7#29552C212yesmaximal, complemented
8#29360C212yesmaximal, complemented
9#28387C212yesmaximal, complemented
10#27591C212yesmaximal, complemented
11#24304C212yesmaximal, complemented
12#24208C212yesmaximal, complemented
13#23939C212yesmaximal, complemented
14#22056C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C2×#478C2214yescomplemented
20C2×#459C2214yes—
21C2×#447C2214yes—
22#2858C2212yes—
23#2858C2212yes—
24#2641C2212yes—
25#2641C2212yes—
26#2548—22no—
27#2548—22no—
28#2435—22no—
29#2435—22no—
30#2374—22no—
31#2364—22no—
32#2363—22no—
33#2359—22no—
34#2330C2211yescharacteristic, complemented
35#2328C2211yescharacteristic, complemented
36#2324C2212yes—
37#2314C2211yescharacteristic
38#2309C2212yes—
39#2309C2212yes—
40C4×#127C2211yescharacteristic, complemented
41C4×#125C2211yescharacteristic, complemented
42C4×#121C2212yes—
43#2293C2212yescomplemented
44#2275C2212yescomplemented
45C4×#117C2211yescharacteristic
46#2092C2214yescomplemented
47#2071C2214yescomplemented
48#2058C2212yescomplemented
49#2057C2212yescomplemented
50#2041C2212yes—
51#2041C2212yes—
52#2025C2214yescomplemented
53#2023C2212yescomplemented
54#2023C2212yescomplemented
55#2011C2212yes—
56#2011C2212yes—
57#2011C2214yescomplemented
58#1989C2214yescomplemented
59#1982C2214yes—
60#1972C2214yescomplemented
61#1970C2214yescomplemented
62#1936C2214yescomplemented
63#1934C2214yescomplemented
64#1926C2214yescomplemented
65#1926C2214yescomplemented
66#1925C2214yescomplemented
67#1925C2214yescomplemented
68#1879C2214yescomplemented
69#1879C2214yescomplemented
70#1879C2214yescomplemented
71#1879C2214yescomplemented
72#1875C2214yes—
73#1854C2214yes—
74#1839C2212yescomplemented
75#1832C2212yescomplemented
76#1818C2212yes—
77#1797C2212yes—
78#1786C2214yes—
79#1768C2212yescomplemented
80#1750C2212yes—
81#1745C2211yescharacteristic, complemented
82#1744C2211yescharacteristic, complemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×D8C2314yes—
86C22×D8C2314yes—
87#540—28no—
88#533—44no—
89C2×C4⋊Q8C2312yes—
90C2×#127—22no—
91C2×#127—44no—
92C2×#127C2311yescharacteristic
93C2×#127C2312yes—
94C2×#127C2312yes—
95C2×#125C2312yes—
96C2×#125C2311yescharacteristic
97C2×#124—22no—
98C2×#124—22no—
99C2×#124—22no—
100C2×#124—22no—
101C2×#123—22no—
102C2×C22⋊Q8—22no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×C4×Q8—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#478—48no—
117#459—48no—
118#447—216no—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133#394—24no—
134#394—24no—
135#394—24no—
136#394—24no—
137#390—24no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2314yes—
148#390C2314yes—
149#390C2314yes—
150#390C2314yes—
151#390C2312yes—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#385—22no—
168#384C2311yescharacteristic
169#382C2311yescharacteristic
170#380C2312yes—
171#379—24no—
172#379—22no—
173#379—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C2×#95—22no—
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C25—22no—
203C22×Q8—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—28no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—44no—
214C22×D4—48no—
215C22×D4—84no—
216C22×D4—84no—
217C22×D4C2414yes—
218C22×D4C2412yes—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4C2×D412yes—
223C23×C4C2411yescharacteristic
224C2×D8—28no—
225C2×D8—28no—
226C2×D8—44no—
227C2×D8—44no—
228C2×D8—44no—
229C2×D8—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—44no—
233#127—24no—
234#127—28no—
235#127—44no—
236#127—48no—
237#125—24no—
238#125—28no—
239#124—84no—
240#124—84no—
241#124—84no—
242#124—84no—
243#123—216no—
244C22⋊Q8—48no—
245#121—216no—
246#121—24no—
247#121#4118yes—
248C4×Q8—48no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—28no—
252C4×D4—44no—
253C4×D4—84no—
254C4×D4—84no—
255#117—24no—
256#117—42no—
257#117—42no—
258#117—42no—
259#117—84no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D411yescharacteristic
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42C2×D412yes—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310#107—44no—
311#106—44no—
312C4⋊C8—28no—
313#102—28no—
314#102—28no—
315#102—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321C22⋊C8—28no—
322#97—42no—
323C4×C8—42no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95C2×D412yes—
332#95C2×D412yes—
333#95C2×D412yes—
334#95C2×D412yes—
335C24—22no—
336C24—24no—
337C24—44no—
338C24—44no—
339C24—44no—
340C24—84no—
341C24—84no—
342C24C22×D411yescharacteristic
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—44no—
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—88no—
368C2×D4—88no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—832no—
376C2×D4C2×#4114yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C2×#4114yes—
405C22×C4C2511yescharacteristic
406D8—88no—
407D8—88no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4C2×#4114yes—
429C22⋊C4—28no—
430C22⋊C4—24no—
431C22⋊C4—28no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—82no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—42no—
459C42—42no—
460C42—44no—
461C42—82no—
462C42—82no—
463C42—82no—
464C42H(5,𝔽2)14yes—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—88no—
484C23—816no—
485C23—168no—
486C23—164no—
487C23—164no—
488C23#52512yes—
489C23C23×D411yescharacteristic
490Q8—88no—
491D4—28no—
492D4—44no—
493D4—44no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—88no—
498D4—88no—
499D4—88no—
500D4—816no—
501D4—88no—
502D4—88no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—88no—
538C2×C4—164no—
539C2×C4—164no—
540C2×C4#52512yes—
541C2×C4#53514yes—
542C2×C4#57614yes—
543C2×C4C22×#4112yes—
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—22no—
549C22—24no—
550C22—41no—
551C22—41no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—1616no—
568C22C2×#52511yescharacteristic
569C22C2×#53511yescharacteristic
570C22#344911yescharacteristic
571C22C2×#57612yes—
572C22#358512yes—
573C4—22no—
574C4—24no—
575C4—24no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4#351414yes—
587C2—42no—
588C2—84no—
589C2—164no—
590C2#3512911yescharacteristic
591C2#6059012yes—
592C2#6081112yes—
593C2#6101111yescharacteristic
594C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35285C2semidirect
2. #34280C2semidirect
3. #33038C2semidirect
4. #30066C2semidirect
5. #30038C2semidirect
6. #29558C2semidirect
7. #29552C2semidirect
8. #29360C2semidirect
9. #28387C2semidirect
10. #27591C2semidirect
11. #24304C2semidirect
12. #24208C2semidirect
13. #23939C2semidirect
14. #22056C2semidirect
15. #20785C2semidirect
19. C2×#478C22semidirect
34. #2330C22semidirect
35. #2328C22semidirect
40. C4×#127C22semidirect
41. C4×#125C22semidirect
43. #2293C22semidirect
44. #2275C22semidirect
46. #2092C22semidirect
47. #2071C22semidirect
48. #2058C22semidirect
49. #2057C22semidirect
52. #2025C22semidirect
53. #2023C22semidirect
54. #2023C22semidirect
57. #2011C22semidirect
58. #1989C22semidirect
60. #1972C22semidirect
61. #1970C22semidirect
62. #1936C22semidirect
63. #1934C22semidirect
64. #1926C22semidirect
65. #1926C22semidirect
66. #1925C22semidirect
67. #1925C22semidirect
68. #1879C22semidirect
69. #1879C22semidirect
70. #1879C22semidirect
71. #1879C22semidirect
74. #1839C22semidirect
75. #1832C22semidirect
79. #1768C22semidirect
81. #1745C22semidirect
82. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352852
#342802
#330382
#300662
#300382
#295582
#295522
#293602
#283872
#275912
#243042
#242082
#239392
#220562
#207852