Finite Group Databasebeta

← #6471235

Subgroups — #6471235

Order 512.

1836 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2367C211yesmaximal, characteristic
3#33100C212yesmaximal, complemented
4#32954C211yesmaximal, characteristic, complemented
5#32906C211yesmaximal, characteristic, complemented
6#29950C211yesmaximal, characteristic
7#29714C211yesmaximal, characteristic
8#29626C212yesmaximal
9#27073C214yesmaximal, complemented
10#26898C214yesmaximal, complemented
11#26763C214yesmaximal, complemented
12#26595C214yesmaximal, complemented
13#22044C212yesmaximal, complemented
14#21912C211yesmaximal, characteristic, complemented
15C4×#450C212yesmaximal
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#475C2212yescomplemented
20C2×#462C2212yescomplemented
21C2×#451C2212yes—
22C2×#450C2212yes—
23C2×C4×SD16C2214yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yescomplemented
32#2507C2212yescomplemented
33#2451C2212yescomplemented
34#2446C2212yescomplemented
35#2367—22no—
36#2367—22no—
37#2367C2214yes—
38#2367C2214yes—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46C2×#385—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4—21no—
50#2093C2212yes—
51#2071C2212yes—
52#2067C2214yes—
53#2055C2214yes—
54#2042C2214yes—
55#2042C2214yes—
56#2023C2212yes—
57#2015C2214yes—
58#2015C2214yes—
59#2013C2212yes—
60#1969C2214yescomplemented
61#1933C2214yescomplemented
62#1928C2214yescomplemented
63#1928C2214yescomplemented
64#1927C2214yescomplemented
65#1919C2214yescomplemented
66#1918C2214yescomplemented
67#1916C2214yescomplemented
68#1916C2214yescomplemented
69#1878C2214yescomplemented
70#1878C2214yescomplemented
71#1877C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1865C2214yescomplemented
75#1864C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1817C2212yes—
79#1803C2212yescomplemented
80#1803C2212yescomplemented
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1786C2212yes—
85C4×#102C2214yes—
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×Q16C2314yes—
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#124—24no—
95C2×#124—24no—
96C2×#124—22no—
97C2×#124C2314yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—44no—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#475—44no—
125#462—44no—
126#451—28no—
127#450—28no—
128C4×SD16—48no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C4×C8C2314yes—
144#400—22no—
145#400—22no—
146#400—22no—
147#398—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#395—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2314yes—
159#391C2312yes—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—24no—
206C4×C22⋊C4—24no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×Q8C2412yes—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—42no—
233C22×D4—42no—
234C22×D4#4114yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×Q16—28no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260#124—216no—
261#124—28no—
262#124—28no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—44no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—42no—
269C4×Q8—42no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×D4—28no—
274C4×D4—44no—
275C4×D4—48no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—82no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42—82no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340#107—44no—
341#106—28no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344#103—28no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—44no—
351#102—44no—
352C22⋊C8—24no—
353C22⋊C8—24no—
354C4×C8—44no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
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—24no—
377C24—22no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—48no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—48no—
399C2×Q8—42no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8C2×#4114yes—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—48no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—48no—
411C2×D4C2×#4114yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—84no—
449C22×C4—816no—
450C22×C4#13514yes—
451C22×C4#13714yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2511yescharacteristic
464Q16—88no—
465SD16—416no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4H(5,𝔽2)14yes—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—41no—
517C42—41no—
518C42—44no—
519C42—44no—
520C42—82no—
521C42—84no—
522C42—82no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—24no—
528C23—48no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23—84no—
541C23#52512yes—
542C23C2×#13512yes—
543C23C2×#13712yes—
544C23C23×D411yescharacteristic
545Q8—24no—
546Q8—24no—
547Q8—48no—
548Q8—42no—
549Q8—42no—
550Q8—44no—
551Q8—44no—
552Q8—162no—
553D4—28no—
554D4—48no—
555D4—44no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
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—42no—
607C2×C4—42no—
608C2×C4—816no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—84no—
612C2×C4#52512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13714yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4#58512yes—
620C8—44no—
621C8—44no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—816no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22#310714yes—
637C22C2×#52511yescharacteristic
638C22#348411yescharacteristic
639C22#348711yescharacteristic
640C22C22×#13511yescharacteristic
641C22C22×#13711yescharacteristic
642C22#358511yescharacteristic
643C22#358611yescharacteristic
644C4—22no—
645C4—22no—
646C4—24no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—44no—
653C4—42no—
654C4—44no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4#345912yes—
661C4#350512yes—
662C2—24no—
663C2—84no—
664C2#3498711yescharacteristic
665C2C2×#310711yescharacteristic
666C2#6056511yescharacteristic
667C2#6156411yescharacteristic
668C2#6157011yescharacteristic
669C2#6188511yescharacteristic
670C2#6188911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35015C2semidirect
3. #33100C2semidirect
4. #32954C2semidirect
5. #32906C2semidirect
9. #27073C2semidirect
10. #26898C2semidirect
11. #26763C2semidirect
12. #26595C2semidirect
13. #22044C2semidirect
14. #21912C2semidirect
19. C2×#475C22semidirect
20. C2×#462C22semidirect
31. #2511C22semidirect
32. #2507C22semidirect
33. #2451C22semidirect
34. #2446C22semidirect
60. #1969C22semidirect
61. #1933C22semidirect
62. #1928C22semidirect
63. #1928C22semidirect
64. #1927C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
67. #1916C22semidirect
68. #1916C22semidirect
69. #1878C22semidirect
70. #1878C22semidirect
74. #1865C22semidirect
79. #1803C22semidirect
80. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#23672
#331002
#329542
#329062
#299502
#297142
#296262
#270732
#268982
#267632
#265952
#220442
#219122
C4×#4502