Finite Group Databasebeta

← #6506496

Subgroups — #6506496

Order 512.

2342 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37306C211yesmaximal, characteristic, complemented
2#35616C211yesmaximal, characteristic
3#29116C212yesmaximal, complemented
4#29113C212yesmaximal, complemented
5#27281C214yesmaximal, complemented
6#27274C214yesmaximal, complemented
7#27265C214yesmaximal, complemented
8#27262C214yesmaximal, complemented
9#25866C212yesmaximal, complemented
10#25864C212yesmaximal, complemented
11#21771C212yesmaximal, complemented
12#20602C212yesmaximal
13#20532C211yesmaximal, characteristic, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C2×#489C2214yescomplemented
19C2×#489C2212yescomplemented
20C2×#487C2212yescomplemented
21C2×#417C2212yes—
22C2×#418C2212yes—
23C2×#416C2212yes—
24C22×#102C2212yes—
25C2×#408C2212yes—
26#2857C2212yescomplemented
27#2811—21no—
28#2809—21no—
29#2804—21no—
30#2692—24no—
31#2683—21no—
32#2628C2212yescomplemented
33#2440C2214yescomplemented
34#2439C2214yescomplemented
35#2435—21no—
36#2435—21no—
37#2435C2214yes—
38#2432—21no—
39#2432—21no—
40#2432C2214yes—
41#2405—24no—
42#2311C2212yes—
43#2309C2211yescharacteristic
44#2297C2211yescharacteristic
45#2297C2211yescharacteristic
46#2073C2214yescomplemented
47#2072C2212yescomplemented
48#2071C2212yescomplemented
49#2043C2214yescomplemented
50#2042C2214yescomplemented
51#2042C2214yescomplemented
52#2042C2214yescomplemented
53#2024C2212yescomplemented
54#2023C2214yescomplemented
55#2023C2212yescomplemented
56#1960C2212yescomplemented
57#1959C2214yescomplemented
58#1958C2212yescomplemented
59#1936C2214yescomplemented
60#1936C2214yescomplemented
61#1935C2214yescomplemented
62#1934C2214yescomplemented
63#1894C2214yescomplemented
64#1892C2214yescomplemented
65#1873C2214yescomplemented
66#1872C2214yescomplemented
67#1825C2214yescomplemented
68#1788C2214yes—
69#1788C2214yes—
70#1787C2214yescomplemented
71#1786C2214yes—
72#1786C2214yes—
73#1753C2214yescomplemented
74#1752C2214yescomplemented
75#1750C2212yescomplemented
76#1749C2212yescomplemented
77C2×H(5,𝔽2)—28no—
78C2×H(5,𝔽2)—42no—
79C2×H(5,𝔽2)—44no—
80C2×H(5,𝔽2)—42no—
81C22×#41—21no—
82C22×#41C2312yes—
83C22×#41C2311yescharacteristic
84C22×#41C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4—22no—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C22×M16C2312yes—
91C23×C8C2312yes—
92#536—28no—
93#536—42no—
94#536—42no—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×C4⋊Q8C2312yes—
99C2×#124—21no—
100C2×#124—21no—
101C2×#124—21no—
102C2×#124—44no—
103C2×#124C2314yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2314yes—
112C2×#117C2311yescharacteristic
113C2×#117C2312yes—
114C2×#117C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—22no—
117C22×C42C2311yescharacteristic
118#489—44no—
119#489—48no—
120#487—44no—
121#418—44no—
122#417—44no—
123#416—44no—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102—44no—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137#408—28no—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143#400—22no—
144#399—22no—
145#398—24no—
146#396—24no—
147#395—22no—
148#395—24no—
149#394—24no—
150#394—24no—
151#394—24no—
152#394—24no—
153#393—22no—
154#393—24no—
155#391—22no—
156#391—22no—
157#391—24no—
158#391D411yescharacteristic
159#391D411yescharacteristic
160#391C2312yes—
161#390—22no—
162#390—21no—
163#390—21no—
164#390D411yescharacteristic
165#390D411yescharacteristic
166#390C2314yes—
167#390C2312yes—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#387—24no—
172#386—22no—
173#386—22no—
174#386—22no—
175#386D414yes—
176#385—24no—
177#385—24no—
178#385—24no—
179#384C2312yes—
180#382C2312yes—
181#380—22no—
182#380C2314yes—
183#380C2314yes—
184#379—22no—
185#379—24no—
186#379—24no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#336C2312yes—
195#336C2312yes—
196#336C2314yes—
197C25—21no—
198C25—22no—
199H(5,𝔽2)—432no—
200H(5,𝔽2)—432no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—42no—
209C2×#41—41no—
210C2×#41—41no—
211C2×#41—42no—
212C2×#41—42no—
213C2×#41—42no—
214C2×#41—44no—
215C2×#41—44no—
216C2×#41—44no—
217C2×#41#4118yes—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8—41no—
224C22×Q8—41no—
225C22×Q8C2×D411yescharacteristic
226C22×Q8C2412yes—
227C22×Q8C2411yescharacteristic
228C22×D4—24no—
229C22×D4—24no—
230C22×D4—21no—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—24no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4—41no—
241C22×D4—41no—
242C22×D4—44no—
243C22×D4—48no—
244C22×D4—44no—
245C22×D4—48no—
246C22×D4—48no—
247C22×D4—44no—
248C22×D4C2×D412yes—
249C22×D4C2×D411yescharacteristic
250C22×D4C2411yescharacteristic
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C2×M16—24no—
260C2×M16—42no—
261C2×M16—42no—
262C22×C8—24no—
263C22×C8—42no—
264C22×C8—42no—
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—24no—
268C4⋊Q8—42no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272C4⋊Q8—44no—
273#124—28no—
274#124—24no—
275#124—44no—
276#124—42no—
277#124—44no—
278#124—44no—
279#124—44no—
280#123—48no—
281#123—48no—
282C22⋊Q8—28no—
283C22⋊Q8—28no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—88no—
287C22⋊Q8—84no—
288#117—24no—
289#117—41no—
290#117—41no—
291#117—42no—
292#117—42no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—48no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—42no—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2412yes—
334#103—28no—
335#103—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#102—28no—
341#102—28no—
342#102—28no—
343#102—44no—
344#102—44no—
345#102—44no—
346C22⋊C8—28no—
347C22⋊C8—28no—
348#97—44no—
349#97—42no—
350C4×C8—42no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95C2×D414yes—
360#95C2×D414yes—
361C24—24no—
362C24—21no—
363C24—22no—
364C24—44no—
365C24—44no—
366C24—42no—
367C24—44no—
368C24—42no—
369C24—82no—
370C24—82no—
371C24—84no—
372C24—84no—
373C24C22×D412yes—
374C24C22×D411yescharacteristic
375#41—216no—
376#41—48no—
377#41—48no—
378#41—48no—
379#41—48no—
380#41—48no—
381#41—48no—
382#41—48no—
383#41—432no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—41no—
392C2×Q8—41no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—84no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—84no—
402C2×Q8—84no—
403C2×Q8#12414yes—
404C2×Q8H(5,𝔽2)12yes—
405C2×D4—216no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—41no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—42no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—42no—
431C2×D4—432no—
432C2×D4—464no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4#12418yes—
438C2×D4#12414yes—
439C2×D4H(5,𝔽2)12yes—
440C2×D4H(5,𝔽2)14yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4C2511yescharacteristic
477M16—84no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—84no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—82no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—48no—
511C22⋊C4—84no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—41no—
522C42—41no—
523C42—42no—
524C42—82no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—22no—
529C23—21no—
530C23—22no—
531C23—21no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—416no—
536C23—48no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—88no—
546C23—164no—
547C23—168no—
548C23—168no—
549C23—164no—
550C23—164no—
551C23—162no—
552C23—162no—
553C23C2×#12414yes—
554C23C2×#12412yes—
555C23C23×D411yescharacteristic
556Q8—24no—
557Q8—44no—
558Q8—42no—
559Q8—44no—
560Q8—44no—
561Q8—84no—
562Q8—164no—
563Q8—164no—
564D4—24no—
565D4—24no—
566D4—24no—
567D4—48no—
568D4—44no—
569D4—416no—
570D4—48no—
571D4—48no—
572D4—416no—
573D4—42no—
574D4—42no—
575D4—42no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—48no—
598C2×C4—416no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—164no—
616C2×C4C2×#12412yes—
617C2×C4C2×#12414yes—
618C2×C4C22×#4112yes—
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)12yes—
622C8—44no—
623C8—44no—
624C22—22no—
625C22—24no—
626C22—28no—
627C22—24no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—41no—
632C22—42no—
633C22—41no—
634C22—42no—
635C22—44no—
636C22—48no—
637C22—42no—
638C22—42no—
639C22—82no—
640C22—84no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—168no—
645C22—1616no—
646C22C22×#12411yescharacteristic
647C22#348411yescharacteristic
648C22#348411yescharacteristic
649C22#358512yes—
650C22#358612yes—
651C4—24no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—82no—
659C4—84no—
660C4—82no—
661C4—84no—
662C4#349012yes—
663C4#349012yes—
664C2—42no—
665C2—44no—
666C2—84no—
667C2—82no—
668C2#3536711yescharacteristic
669C2#6062512yes—
670C2#6164712yes—
671C2#6167311yescharacteristic
672C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37306C2semidirect
3. #29116C2semidirect
4. #29113C2semidirect
5. #27281C2semidirect
6. #27274C2semidirect
7. #27265C2semidirect
8. #27262C2semidirect
9. #25866C2semidirect
10. #25864C2semidirect
11. #21771C2semidirect
13. #20532C2semidirect
18. C2×#489C22semidirect
19. C2×#489C22semidirect
20. C2×#487C22semidirect
26. #2857C22semidirect
32. #2628C22semidirect
33. #2440C22semidirect
34. #2439C22semidirect
46. #2073C22semidirect
47. #2072C22semidirect
48. #2071C22semidirect
49. #2043C22semidirect
50. #2042C22semidirect
51. #2042C22semidirect
52. #2042C22semidirect
53. #2024C22semidirect
54. #2023C22semidirect
55. #2023C22semidirect
56. #1960C22semidirect
57. #1959C22semidirect
58. #1958C22semidirect
59. #1936C22semidirect
60. #1936C22semidirect
61. #1935C22semidirect
62. #1934C22semidirect
63. #1894C22semidirect
64. #1892C22semidirect
65. #1873C22semidirect
66. #1872C22semidirect
67. #1825C22semidirect
70. #1787C22semidirect
73. #1753C22semidirect
74. #1752C22semidirect
75. #1750C22semidirect
76. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#373062
#356162
#291162
#291132
#272812
#272742
#272652
#272622
#258662
#258642
#217712
#206022
#205322