Finite Group Databasebeta

← #6632142

Subgroups — #6632142

Order 512.

2396 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35086C211yesmaximal, characteristic, complemented
2C2×#2436C211yesmaximal, characteristic
3#32882C212yesmaximal, complemented
4#31308C214yesmaximal, complemented
5#30998C214yesmaximal, complemented
6#30534C214yesmaximal, complemented
7#30473C214yesmaximal, complemented
8#29598C212yesmaximal
9#29112C211yesmaximal, characteristic, complemented
10#28304C211yesmaximal, characteristic, complemented
11#28121C212yesmaximal, complemented
12#25859C211yesmaximal, characteristic, complemented
13#25730C211yesmaximal, characteristic, complemented
14#24217C212yesmaximal, complemented
15#21904C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C2×#497C2212yescomplemented
21C2×#488C2212yescomplemented
22C2×#484C2212yes—
23C2×#464C2214yescomplemented
24C2×#451C2212yes—
25C2×#451C2212yes—
26C2×#451C2214yescomplemented
27C2×#451C2214yescomplemented
28C2×#450C2212yes—
29#2857C2211yescharacteristic
30#2851C2211yescharacteristic
31#2848C2212yes—
32#2652—22no—
33#2505—22no—
34#2496C2212yescomplemented
35#2489C2212yescomplemented
36#2480—22no—
37#2436—22no—
38#2436—22no—
39#2436C2214yes—
40#2436C2214yes—
41#2429C2212yescomplemented
42#2421C2212yescomplemented
43C2×#393—22no—
44#2334—22no—
45C4×C22⋊Q8C2212yes—
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#386—22no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2095C2212yescomplemented
51#2074C2212yescomplemented
52#2068C2214yescomplemented
53#2056C2214yescomplemented
54#2042C2214yescomplemented
55#2042C2214yescomplemented
56#2033C2214yes—
57#2033C2214yes—
58#2026C2212yescomplemented
59#2016C2212yescomplemented
60#2015C2214yescomplemented
61#2015C2214yescomplemented
62#2000C2214yes—
63#2000C2214yes—
64#1983C2214yescomplemented
65#1969C2214yescomplemented
66#1959C2212yescomplemented
67#1958C2212yescomplemented
68#1935C2214yescomplemented
69#1928C2214yescomplemented
70#1924C2214yescomplemented
71#1923C2214yescomplemented
72#1878C2214yescomplemented
73#1824C2212yes—
74#1797C2212yes—
75#1787C2212yes—
76#1787C2212yes—
77#1787C2214yescomplemented
78#1787C2214yescomplemented
79#1786C2212yes—
80#1779C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C23×Q8—21no—
84C23×Q8—22no—
85C23×Q8—44no—
86C23×Q8—44no—
87C23×Q8—44no—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×Q8C2314yes—
91C23×D4—21no—
92C23×D4C2311yescharacteristic
93C24×C4—21no—
94C24×C4C2311yescharacteristic
95C22×Q16C2314yes—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#126—22no—
100C2×#125C2311yescharacteristic
101C2×#124—21no—
102C2×#124—21no—
103C2×#124—22no—
104C2×#124—44no—
105C2×#124—44no—
106C2×#124C2312yes—
107C2×#124C2312yes—
108C2×#124C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×#121—22no—
113C2×#121—44no—
114C2×#121C2314yes—
115C2×C4×Q8C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4D412yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#497—44no—
124#488—44no—
125#484—44no—
126#464—48no—
127#451—44no—
128#451—44no—
129#451—48no—
130#451—48no—
131#450—44no—
132C2×#107C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#400—22no—
153#398—22no—
154#397—22no—
155#397—22no—
156#397—22no—
157#396—22no—
158#393—24no—
159#393—24no—
160#393—24no—
161#393—24no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391C2312yes—
166#391C2314yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—24no—
182#386—24no—
183#386—24no—
184#386—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384C2311yescharacteristic
191#382—22no—
192#382C2311yescharacteristic
193#380—22no—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#379—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—22no—
206C2×#95—22no—
207C2×#95D412yes—
208#340C2314yes—
209#340C2314yes—
210#336C2314yes—
211#336C2312yes—
212#336C2312yes—
213C25—21no—
214C25C2×D411yescharacteristic
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×Q8—24no—
220C22×Q8—28no—
221C22×Q8—28no—
222C22×Q8—24no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8—44no—
232C22×Q8—44no—
233C22×Q8—48no—
234C22×Q8—48no—
235C22×Q8—48no—
236C22×Q8—464no—
237C22×Q8C2412yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4—24no—
241C22×D4—21no—
242C22×D4—24no—
243C22×D4—42no—
244C22×D4—42no—
245C22×D4—44no—
246C22×D4—48no—
247C22×D4C2412yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×Q16—28no—
265C2×Q16—28no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×SD16—44no—
271C2×SD16—44no—
272C2×SD16—44no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—42no—
276#126—48no—
277#125—42no—
278#124—28no—
279#124—24no—
280#124—24no—
281#124—42no—
282#124—42no—
283#124—44no—
284#124—44no—
285#124—44no—
286#124—48no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—44no—
290C22⋊Q8—44no—
291#121—44no—
292#121—44no—
293#121—48no—
294C4×Q8—44no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—44no—
340C4⋊C8—44no—
341#103—28no—
342#103—28no—
343#103—44no—
344#103—44no—
345#103—44no—
346#103—44no—
347#103—44no—
348#102—44no—
349#102—44no—
350#102—44no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—44no—
354C22⋊C8—44no—
355#97—42no—
356C4×C8—42no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24—22no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—41no—
381C24—41no—
382C24—42no—
383C24—42no—
384C24—44no—
385C24—42no—
386C24—44no—
387C24C22×D412yes—
388C24C22×D411yescharacteristic
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—416no—
404C2×Q8—416no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—48no—
408C2×Q8—416no—
409C2×Q8—416no—
410C2×Q8—416no—
411C2×Q8—464no—
412C2×Q8—432no—
413C2×Q8—416no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8H(5,𝔽2)14yes—
417C2×Q8H(5,𝔽2)14yes—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—42no—
428C2×D4—88no—
429C2×D4—816no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—432no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4H(5,𝔽2)14yes—
475C22×C4C2511yescharacteristic
476Q16—88no—
477Q16—88no—
478SD16—88no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—41no—
519C42—41no—
520C42—42no—
521C42—44no—
522C42—82no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—28no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—84no—
543C23—84no—
544C23—88no—
545C23—88no—
546C23—84no—
547C23#53512yes—
548C23#57514yes—
549C23#57814yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—28no—
553Q8—48no—
554Q8—48no—
555Q8—432no—
556Q8—44no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560D4—44no—
561D4—44no—
562D4—84no—
563D4—816no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—432no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4#53512yes—
613C2×C4C2×H(5,𝔽2)12yes—
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4C2×H(5,𝔽2)12yes—
616C8—44no—
617C8—44no—
618C22—22no—
619C22—22no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—416no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—168no—
636C22#322314yes—
637C22C2×#53511yescharacteristic
638C22#348411yescharacteristic
639C22#348411yescharacteristic
640C22C2×#57512yes—
641C22C2×#57812yes—
642C4—22no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4#353214yes—
656C2—24no—
657C2—44no—
658C2—162no—
659C2#3670311yescharacteristic
660C2C2×#322311yescharacteristic
661C2#6079911yescharacteristic
662C2#6160312yes—
663C2#6160312yes—
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
1. #35086C2semidirect
3. #32882C2semidirect
4. #31308C2semidirect
5. #30998C2semidirect
6. #30534C2semidirect
7. #30473C2semidirect
9. #29112C2semidirect
10. #28304C2semidirect
11. #28121C2semidirect
12. #25859C2semidirect
13. #25730C2semidirect
14. #24217C2semidirect
15. #21904C2semidirect
20. C2×#497C22semidirect
21. C2×#488C22semidirect
23. C2×#464C22semidirect
26. C2×#451C22semidirect
27. C2×#451C22semidirect
34. #2496C22semidirect
35. #2489C22semidirect
41. #2429C22semidirect
42. #2421C22semidirect
50. #2095C22semidirect
51. #2074C22semidirect
52. #2068C22semidirect
53. #2056C22semidirect
54. #2042C22semidirect
55. #2042C22semidirect
58. #2026C22semidirect
59. #2016C22semidirect
60. #2015C22semidirect
61. #2015C22semidirect
64. #1983C22semidirect
65. #1969C22semidirect
66. #1959C22semidirect
67. #1958C22semidirect
68. #1935C22semidirect
69. #1928C22semidirect
70. #1924C22semidirect
71. #1923C22semidirect
72. #1878C22semidirect
77. #1787C22semidirect
78. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350862
C2×#24362
#328822
#313082
#309982
#305342
#304732
#295982
#291122
#283042
#281212
#258592
#257302
#242172
#219042