Finite Group Databasebeta

← #6434500

Subgroups — #6434500

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35296C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic, complemented
3#30137C214yesmaximal, complemented
4#30077C212yesmaximal, complemented
5#30038C212yesmaximal, complemented
6#29959C212yesmaximal, complemented
7#29910C212yesmaximal, complemented
8#29721C212yesmaximal, complemented
9#29622C212yesmaximal, complemented
10#23828C212yesmaximal, complemented
11#23810C212yesmaximal, complemented
12#23682C212yesmaximal, complemented
13#23676C212yesmaximal, complemented
14#21458C214yesmaximal, complemented
15#20781C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#518C2212yes—
18C2×#447C2214yes—
19C2×#442C2214yescomplemented
20C2×C4×D8C2214yescomplemented
21#2864C2211yescharacteristic, complemented
22#2858C2211yescharacteristic, complemented
23#2854C2212yes—
24#2644C2212yescomplemented
25#2628C2211yescharacteristic
26#2619C2211yescharacteristic
27#2595—22no—
28#2557—22no—
29#2474—22no—
30#2471—22no—
31#2366—22no—
32#2366—22no—
33D4×C22⋊C4—22no—
34D4×C22⋊C4—22no—
35#2331C2211yescharacteristic, complemented
36#2328C2212yes—
37#2328C2212yes—
38#2328C2211yescharacteristic, complemented
39#2324C2212yes—
40#2309C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43#2293C2212yescomplemented
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yescomplemented
47#2092C2214yescomplemented
48#2092C2214yescomplemented
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2073C2214yescomplemented
52#2071C2212yescomplemented
53#2071C2212yescomplemented
54#2067C2214yescomplemented
55#2065C2214yescomplemented
56#2054C2214yescomplemented
57#2054C2214yescomplemented
58#2041C2214yes—
59#2041C2212yes—
60#2041C2212yes—
61#2023C2214yescomplemented
62#2011C2214yes—
63#2011C2212yes—
64#2011C2212yes—
65#2011C2214yescomplemented
66#1879C2214yescomplemented
67#1879C2214yescomplemented
68#1875C2214yes—
69#1864C2214yescomplemented
70#1862C2214yescomplemented
71#1854C2214yes—
72#1839C2212yescomplemented
73#1832C2212yescomplemented
74#1817C2212yes—
75#1798C2212yes—
76#1786C2214yes—
77#1767C2211yescharacteristic, complemented
78C4×C4⋊C8C2211yescharacteristic, complemented
79#1762C2214yescomplemented
80C4×#102C2214yescomplemented
81#1750C2212yes—
82#1746C2212yescomplemented
83C23×D4—22no—
84C23×D4C2312yes—
85C22×D8C2314yes—
86C22×D8C2314yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127—28no—
89C2×#127—22no—
90C2×#127C2312yes—
91C2×#127C2312yes—
92C2×#127C2311yescharacteristic
93C2×#124—22no—
94C2×#123—22no—
95C2×#121C2312yes—
96#518—44no—
97#518—44no—
98C2×C4×D4—24no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2314yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C2×#117—22no—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C22⋊C4—24no—
111C22×C42C2311yescharacteristic
112#447—216no—
113#442—48no—
114C4×D8—48no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2314yes—
128C2×#97C2312yes—
129C2×C4×C8C2312yes—
130#400—21no—
131#399—21no—
132#397—22no—
133#397—22no—
134#397—22no—
135#396—22no—
136#395—21no—
137#394—21no—
138#394—21no—
139#393—21no—
140#392—21no—
141#392—21no—
142#391—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#390C2314yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—24no—
172#386—24no—
173#386—24no—
174#386—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#384C2312yes—
179#382C2312yes—
180#382C2311yescharacteristic
181#382C2311yescharacteristic
182#380—22no—
183#380C2312yes—
184#379—22no—
185#379—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4D412yes—
193#376C2311yescharacteristic
194#376C2311yescharacteristic
195C43C2311yescharacteristic
196C43C2311yescharacteristic
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205#336C2314yes—
206C25—22no—
207C22×Q8—21no—
208C22×D4—216no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4—24no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—42no—
221C22×D4—84no—
222C22×D4—84no—
223C22×D4#4118yes—
224C22×D4C2412yes—
225C22×D4C2414yes—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4C2×D412yes—
230C23×C4C2411yescharacteristic
231C2×D8—28no—
232C2×D8—28no—
233C2×D8—44no—
234C2×D8—44no—
235C2×D8—44no—
236C2×D8—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—42no—
240#127—216no—
241#127—28no—
242#127—28no—
243#127—42no—
244#124—84no—
245#123—84no—
246#121—28no—
247C4×D4—28no—
248C4×D4—28no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—44no—
252C4×D4—44no—
253C4×D4—416no—
254C4×D4—84no—
255C4×D4—84no—
256C4×D4—84no—
257C4×D4—84no—
258#117—22no—
259#117—22no—
260#117—42no—
261#117—42no—
262#117—42no—
263#117—84no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—416no—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D411yescharacteristic
299C2×C22⋊C4C2×D411yescharacteristic
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—44no—
315#106—44no—
316C4⋊C8—42no—
317C4⋊C8—42no—
318#102—28no—
319#102—28no—
320#102—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325#102—44no—
326C22⋊C8—28no—
327#97—24no—
328C4×C8—24no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—22no—
339#95—22no—
340#95C2×D412yes—
341#95C2×D412yes—
342#95C2×D412yes—
343#95C2×D412yes—
344C24—24no—
345C24—28no—
346C24—22no—
347C24—44no—
348C24—44no—
349C24—84no—
350C24—84no—
351C24C22×D411yescharacteristic
352C2×Q8—42no—
353C2×Q8—82no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—216no—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—832no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4C2×#4118yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—48no—
399C22×C4—44no—
400C22×C4—48no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C22×D412yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2511yescharacteristic
418D8—416no—
419D8—416no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—44no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C22⋊C4—24no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—22no—
471C42—22no—
472C42—42no—
473C42—42no—
474C42—44no—
475C42—82no—
476C42H(5,𝔽2)12yes—
477C42H(5,𝔽2)12yes—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—28no—
483C23—416no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—816no—
498C23—164no—
499C23—164no—
500C23—168no—
501C23#52512yes—
502C23C23×D411yescharacteristic
503Q8—162no—
504D4—28no—
505D4—28no—
506D4—416no—
507D4—416no—
508D4—416no—
509D4—44no—
510D4—44no—
511D4—44no—
512D4—44no—
513D4—816no—
514D4—162no—
515D4—162no—
516D4—164no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—88no—
553C2×C4—168no—
554C2×C4—168no—
555C2×C4#52512yes—
556C2×C4#53514yes—
557C2×C4#57614yes—
558C2×C4C22×#4112yes—
559C2×C4C2×H(5,𝔽2)11yescharacteristic
560C2×C4C2×H(5,𝔽2)11yescharacteristic
561C2×C4#58512yes—
562C8—44no—
563C8—44no—
564C22—22no—
565C22—24no—
566C22—41no—
567C22—41no—
568C22—42no—
569C22—48no—
570C22—48no—
571C22—48no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—1616no—
584C22C2×#52511yescharacteristic
585C22C2×#53511yescharacteristic
586C22#344911yescharacteristic
587C22C2×#57612yes—
588C22#358512yes—
589C4—22no—
590C4—24no—
591C4—24no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—82no—
600C4—84no—
601C4—82no—
602C4#345912yes—
603C4#345912yes—
604C2—42no—
605C2—88no—
606C2—162no—
607C2#3459311yescharacteristic
608C2#6057812yes—
609C2#6083212yes—
610C2#6101111yescharacteristic
611C2#6101711yescharacteristic
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. #35296C2semidirect
2. #34210C2semidirect
3. #30137C2semidirect
4. #30077C2semidirect
5. #30038C2semidirect
6. #29959C2semidirect
7. #29910C2semidirect
8. #29721C2semidirect
9. #29622C2semidirect
10. #23828C2semidirect
11. #23810C2semidirect
12. #23682C2semidirect
13. #23676C2semidirect
14. #21458C2semidirect
15. #20781C2semidirect
19. C2×#442C22semidirect
20. C2×C4×D8C22semidirect
21. #2864C22semidirect
22. #2858C22semidirect
24. #2644C22semidirect
35. #2331C22semidirect
38. #2328C22semidirect
43. #2293C22semidirect
46. #2101C22semidirect
47. #2092C22semidirect
48. #2092C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
51. #2073C22semidirect
52. #2071C22semidirect
53. #2071C22semidirect
54. #2067C22semidirect
55. #2065C22semidirect
56. #2054C22semidirect
57. #2054C22semidirect
61. #2023C22semidirect
65. #2011C22semidirect
66. #1879C22semidirect
67. #1879C22semidirect
69. #1864C22semidirect
70. #1862C22semidirect
72. #1839C22semidirect
73. #1832C22semidirect
77. #1767C22semidirect
78. C4×C4⋊C8C22semidirect
79. #1762C22semidirect
80. C4×#102C22semidirect
82. #1746C22semidirect

Maximal subgroups

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

subgroupindex
#352962
#342102
#301372
#300772
#300382
#299592
#299102
#297212
#296222
#238282
#238102
#236822
#236762
#214582
#207812