Finite Group Databasebeta

← #6434501

Subgroups — #6434501

Order 512.

1904 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35324C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30138C214yesmaximal, complemented
4#30077C212yesmaximal, complemented
5#30039C212yesmaximal, complemented
6#29959C212yesmaximal, complemented
7#29911C212yesmaximal, complemented
8#29723C212yesmaximal, complemented
9#29625C212yesmaximal
10#23828C212yesmaximal, complemented
11#23811C212yesmaximal, complemented
12#23684C212yesmaximal, complemented
13#23678C212yesmaximal, complemented
14#21459C214yesmaximal, complemented
15#20781C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C2×#518C2212yes—
18C2×#450C2214yes—
19C2×#440C2214yescomplemented
20C2×C4×SD16C2214yescomplemented
21#2864C2211yescharacteristic, complemented
22#2862C2211yescharacteristic, complemented
23#2855C2212yes—
24#2645C2212yescomplemented
25#2628C2211yescharacteristic
26#2621C2211yescharacteristic
27#2588—22no—
28#2575—22no—
29#2528—22no—
30#2527—22no—
31#2366—22no—
32#2366—22no—
33D4×C22⋊C4—22no—
34D4×C22⋊C4—22no—
35#2331C2211yescharacteristic, complemented
36#2328C2212yes—
37#2328C2212yes—
38#2327C2211yescharacteristic, complemented
39#2323C2212yes—
40#2309C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43#2295C2212yescomplemented
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2101C2214yescomplemented
47#2097C2214yescomplemented
48#2092C2214yescomplemented
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2073C2214yescomplemented
52#2072C2212yescomplemented
53#2072C2212yescomplemented
54#2067C2214yescomplemented
55#2065C2214yescomplemented
56#2055C2214yescomplemented
57#2055C2214yescomplemented
58#2041C2214yes—
59#2041C2212yes—
60#2041C2212yes—
61#2023C2214yescomplemented
62#2014C2214yes—
63#2014C2212yes—
64#2014C2212yes—
65#2013C2214yescomplemented
66#1877C2214yescomplemented
67#1877C2214yescomplemented
68#1873C2214yes—
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×SD16C2314yes—
86C22×SD16C2314yes—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127—28no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×C22⋊Q8C2312yes—
94#518—44no—
95#518—44no—
96C2×C4×Q8C2312yes—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4C2314yes—
103C2×C4×D4C2314yes—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C22⋊C4—24no—
108C22×C42C2311yescharacteristic
109#450—216no—
110#440—48no—
111C4×SD16—48no—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8C2314yes—
125C2×#97C2312yes—
126C2×C4×C8C2312yes—
127#401—22no—
128#400—21no—
129#400—22no—
130#400—22no—
131#400—22no—
132#399—21no—
133#398—21no—
134#398—21no—
135#397—21no—
136#397—21no—
137#395—21no—
138#393—21no—
139#391—22no—
140#391—22no—
141#391C2312yes—
142#391C2312yes—
143#391C2312yes—
144#391C2312yes—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#390C2314yes—
153#389—22no—
154#389C2312yes—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#386—24no—
171#386—24no—
172#386—24no—
173#386—24no—
174#385—22no—
175#384C2312yes—
176#382—22no—
177#382C2311yescharacteristic
178#382C2312yes—
179#382C2311yescharacteristic
180#381—22no—
181#380—22no—
182#380C2312yes—
183#379—22no—
184#379—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4D412yes—
191C4×C22⋊C4D412yes—
192#376—22no—
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×Q8C2412yes—
209C22×D4—216no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—84no—
220C22×D4—84no—
221C22×D4#4118yes—
222C22×D4C2414yes—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4C2×D412yes—
226C23×C4C2411yescharacteristic
227C2×SD16—28no—
228C2×SD16—28no—
229C2×SD16—44no—
230C2×SD16—44no—
231C2×SD16—44no—
232C2×SD16—44no—
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—42no—
236C4⋊Q8—42no—
237#127—216no—
238#127—28no—
239#127—28no—
240C22⋊Q8—28no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—44no—
246C4×D4—44no—
247C4×D4—416no—
248C4×D4—84no—
249C4×D4—84no—
250C4×D4—84no—
251C4×D4—84no—
252#117—22no—
253#117—22no—
254#117—42no—
255#117—42no—
256#117—42no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C4⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2414yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—416no—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2411yescharacteristic
298C2×C42C2411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—44no—
304#106—44no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307#103—44no—
308#103—44no—
309#102—28no—
310#102—28no—
311#102—44no—
312#102—44no—
313#102—44no—
314#102—44no—
315C22⋊C8—28no—
316#97—24no—
317C4×C8—24no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—22no—
336#95—22no—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24—24no—
342C24—28no—
343C24—44no—
344C24—44no—
345C24—84no—
346C24—84no—
347C24C22×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—24no—
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—24no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—832no—
376C2×D4C2×#4118yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—42no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C2×#4114yes—
404C22×C4C2511yescharacteristic
405SD16—416no—
406SD16—416no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C22⋊C4—24no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—82no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—22no—
449C42—22no—
450C42—21no—
451C42—21no—
452C42—22no—
453C42—22no—
454C42—42no—
455C42—42no—
456C42—44no—
457C42—44no—
458C42—82no—
459C42H(5,𝔽2)12yes—
460C42H(5,𝔽2)12yes—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—28no—
466C23—416no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—816no—
474C23—164no—
475C23—164no—
476C23—168no—
477C23#52512yes—
478C23C23×D411yescharacteristic
479Q8—44no—
480Q8—44no—
481Q8—162no—
482D4—28no—
483D4—28no—
484D4—416no—
485D4—416no—
486D4—416no—
487D4—44no—
488D4—44no—
489D4—816no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—168no—
529C2×C4—168no—
530C2×C4#52512yes—
531C2×C4#53514yes—
532C2×C4#57714yes—
533C2×C4C22×#4112yes—
534C2×C4C2×H(5,𝔽2)11yescharacteristic
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4#58512yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—24no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—48no—
545C22—48no—
546C22—48no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—1616no—
552C22C2×#52511yescharacteristic
553C22C2×#53511yescharacteristic
554C22#344911yescharacteristic
555C22C2×#57712yes—
556C22#358512yes—
557C4—22no—
558C4—24no—
559C4—24no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4—82no—
571C4#345912yes—
572C4#345912yes—
573C2—42no—
574C2—88no—
575C2#3459311yescharacteristic
576C2#6057912yes—
577C2#6083312yes—
578C2#6101211yescharacteristic
579C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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. #35324C2semidirect
3. #30138C2semidirect
4. #30077C2semidirect
5. #30039C2semidirect
6. #29959C2semidirect
7. #29911C2semidirect
8. #29723C2semidirect
10. #23828C2semidirect
11. #23811C2semidirect
12. #23684C2semidirect
13. #23678C2semidirect
14. #21459C2semidirect
15. #20781C2semidirect
19. C2×#440C22semidirect
20. C2×C4×SD16C22semidirect
21. #2864C22semidirect
22. #2862C22semidirect
24. #2645C22semidirect
35. #2331C22semidirect
38. #2327C22semidirect
43. #2295C22semidirect
46. #2101C22semidirect
47. #2097C22semidirect
48. #2092C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
51. #2073C22semidirect
52. #2072C22semidirect
53. #2072C22semidirect
54. #2067C22semidirect
55. #2065C22semidirect
56. #2055C22semidirect
57. #2055C22semidirect
61. #2023C22semidirect
65. #2013C22semidirect
66. #1877C22semidirect
67. #1877C22semidirect
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
#353242
#342102
#301382
#300772
#300392
#299592
#299112
#297232
#296252
#238282
#238112
#236842
#236782
#214592
#207812