Finite Group Databasebeta

← #6472851

Subgroups — #6472851

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2#35002C211yesmaximal, characteristic, complemented
3#33051C214yesmaximal, complemented
4#30188C214yesmaximal, complemented
5#27713C214yesmaximal, complemented
6#27711C214yesmaximal, complemented
7#27266C214yesmaximal, complemented
8#27262C214yesmaximal, complemented
9#21759C211yesmaximal, characteristic, complemented
10#21758C211yesmaximal, characteristic
11C2×#1893C211yesmaximal, characteristic
12C2×#1892C211yesmaximal, characteristic, complemented
13C2×#1752C211yesmaximal, characteristic, complemented
14C24×C8C2211yescharacteristic
15C2×#543—21no—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C2×#489C2214yescomplemented
19C22×#103C2212yescomplemented
20C22×#102C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2212yescomplemented
23#2857C2214yescomplemented
24#2850C2214yes—
25#2659—22no—
26#2639—22no—
27#2591—22no—
28#2561—22no—
29#2549—22no—
30#2549—22no—
31#2504C2212yes—
32#2503C2212yes—
33#2440C2214yes—
34#2440C2214yescomplemented
35#2436C2212yes—
36#2435C2212yes—
37#2315—21no—
38#2308—21no—
39#2308—21no—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43#2101C2214yescomplemented
44#2101C2214yescomplemented
45#2072C2214yescomplemented
46#2071C2214yescomplemented
47#2042C2214yescomplemented
48#2042C2214yescomplemented
49#2024C2214yescomplemented
50#2023C2214yescomplemented
51#1993C2214yescomplemented
52#1959C2214yescomplemented
53#1936C2214yescomplemented
54#1936C2214yescomplemented
55#1935C2214yescomplemented
56#1935C2214yescomplemented
57#1893C2212yes—
58#1893C2214yes—
59#1893C2214yes—
60#1893C2212yes—
61#1892C2214yes—
62#1892C2212yescomplemented
63#1892C2214yescomplemented
64#1892C2212yescomplemented
65#1874C2214yes—
66#1872C2214yes—
67#1836C2214yescomplemented
68#1824C2212yescomplemented
69#1823C2212yes—
70#1787C2212yescomplemented
71#1786C2212yes—
72#1752C2214yes—
73#1752C2214yescomplemented
74C2×#340C2212yes—
75C2×#340C2212yescomplemented
76C2×#336C2212yescomplemented
77C23×Q8C2311yescharacteristic
78C23×D4—21no—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—44no—
83C23×C8C2311yescharacteristic
84C23×C8C2312yes—
85C23×C8C2312yes—
86C23×C8C2312yes—
87C23×C8C2311yescharacteristic
88#543—48no—
89C2×C4⋊Q8C2314yes—
90C2×#124C2314yes—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—24no—
103C2×#121—24no—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42C2312yes—
118#489—48no—
119C2×#103—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102—44no—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8—24no—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×#97—44no—
133C2×#97C2314yes—
134C2×#97C2314yes—
135#400—24no—
136#398—24no—
137#398—24no—
138#397—24no—
139#395—24no—
140#395—24no—
141#394—24no—
142#393—24no—
143#391—24no—
144#391C2314yes—
145#390C2314yes—
146#389—24no—
147#389C2314yes—
148#389C2314yes—
149#388—24no—
150#388—24no—
151#386—24no—
152#386—24no—
153#386—22no—
154#386—22no—
155#386—22no—
156#386—22no—
157#385—24no—
158#385—24no—
159#384C2314yes—
160#382C2314yes—
161#381—22no—
162#381—22no—
163#380—24no—
164#380—21no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380C2312yes—
169#380C2312yes—
170#380C2312yes—
171#380C2312yes—
172#379—22no—
173#379—22no—
174C4×C22⋊C4D412yes—
175C4×C22⋊C4D412yes—
176C4×C22⋊C4C2312yes—
177C4×C22⋊C4C2312yes—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95D412yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191C25—21no—
192C25C2×D411yescharacteristic
193C22×Q8—24no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8C2412yes—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—21no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4C2412yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C2×D412yes—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4#4114yes—
224C23×C4#4114yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C22×C8—22no—
229C22×C8—24no—
230C22×C8—24no—
231C22×C8—24no—
232C22×C8—22no—
233C22×C8—24no—
234C22×C8—24no—
235C22×C8—24no—
236C22×C8—42no—
237C22×C8—42no—
238C22×C8—42no—
239C22×C8—42no—
240C22×C8—42no—
241C22×C8—416no—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247#124—44no—
248#124—44no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—416no—
254C22⋊Q8—416no—
255C22⋊Q8—416no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258#121—24no—
259#121—24no—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—44no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C2×D414yes—
313C2×C42C2414yes—
314#103—28no—
315#103—44no—
316#103—44no—
317#103—44no—
318#102—28no—
319#102—44no—
320#102—44no—
321#102—44no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324#97—28no—
325#97—44no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95C2×D414yes—
337#95C2×D414yes—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24—82no—
346C24#12414yes—
347C24AGL(1,ℤ/8)14yes—
348C24#13714yes—
349C24C22×D411yescharacteristic
350C24C22×D411yescharacteristic
351C24C22×D411yescharacteristic
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—42no—
359C2×Q8—82no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—816no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4#12414yes—
417C22×C4#12414yes—
418C22×C4#12414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2511yescharacteristic
431C2×C8—28no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—48no—
441C2×C8—48no—
442C2×C8—48no—
443C2×C8—44no—
444C2×C8H(5,𝔽2)14yes—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—84no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C42—24no—
481C42—24no—
482C42—44no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—164no—
502C23—164no—
503C23#489116yescomplemented
504C23C2×#12412yes—
505C23C2×#12412yes—
506C23C2×#12412yes—
507C23C2×AGL(1,ℤ/8)12yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23C2×AGL(1,ℤ/8)12yes—
510C23C2×#13712yes—
511C23C2×#13712yes—
512C23C2×#13712yes—
513C23C23×D411yescharacteristic
514Q8—44no—
515Q8—42no—
516Q8—42no—
517Q8—84no—
518Q8—168no—
519D4—44no—
520D4—42no—
521D4—42no—
522D4—84no—
523D4—88no—
524D4—88no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—164no—
565C2×C4—164no—
566C2×C4C2×#12412yes—
567C2×C4C2×#12412yes—
568C2×C4C2×#12412yes—
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C2×H(5,𝔽2)12yes—
572C2×C4#58512yes—
573C8—24no—
574C8—44no—
575C8—42no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—416no—
589C22—84no—
590C22—84no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—168no—
595C22C2×#48914yes—
596C22C2×#48914yes—
597C22C2×#48914yes—
598C22C22×#12411yescharacteristic
599C22#349612yes—
600C22C22×AGL(1,ℤ/8)11yescharacteristic
601C22C22×#13711yescharacteristic
602C22#358412yes—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—44no—
607C4—44no—
608C4—84no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4#345912yes—
614C4#349011yescharacteristic
615C4#349111yescharacteristic
616C2—24no—
617C2—24no—
618C2—24no—
619C2—162no—
620C2#3499311yescharacteristic
621C2C22×#48911yescharacteristic
622C2#6061211yescharacteristic
623C2#6225312yes—
624C2#6225412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35002C2semidirect
3. #33051C2semidirect
4. #30188C2semidirect
5. #27713C2semidirect
6. #27711C2semidirect
7. #27266C2semidirect
8. #27262C2semidirect
9. #21759C2semidirect
12. C2×#1892C2semidirect
13. C2×#1752C2semidirect
18. C2×#489C22semidirect
19. C22×#103C22semidirect
22. C22×#97C22semidirect
23. #2857C22semidirect
34. #2440C22semidirect
43. #2101C22semidirect
44. #2101C22semidirect
45. #2072C22semidirect
46. #2071C22semidirect
47. #2042C22semidirect
48. #2042C22semidirect
49. #2024C22semidirect
50. #2023C22semidirect
51. #1993C22semidirect
52. #1959C22semidirect
53. #1936C22semidirect
54. #1936C22semidirect
55. #1935C22semidirect
56. #1935C22semidirect
62. #1892C22semidirect
63. #1892C22semidirect
64. #1892C22semidirect
67. #1836C22semidirect
68. #1824C22semidirect
70. #1787C22semidirect
73. #1752C22semidirect
75. C2×#340C22semidirect
76. C2×#336C22semidirect
503. C23#489semidirect

Maximal subgroups

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

subgroupindex
#350192
#350022
#330512
#301882
#277132
#277112
#272662
#272622
#217592
#217582
C2×#18932
C2×#18922
C2×#17522