Finite Group Databasebeta

← #6471604

Subgroups — #6471604

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#33009C212yesmaximal
4#32956C211yesmaximal, characteristic
5#32905C211yesmaximal, characteristic
6#29969C211yesmaximal, characteristic, complemented
7#29913C212yesmaximal, complemented
8#29757C211yesmaximal, characteristic, complemented
9#27576C214yesmaximal, complemented
10#27320C214yesmaximal, complemented
11#26396C214yesmaximal, complemented
12#26186C214yesmaximal, complemented
13#22085C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21453C212yesmaximal
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#476C2212yes—
21C2×#461C2212yes—
22C2×#450C2212yes—
23C2×#440C2214yes—
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#2349—22no—
36#2349—22no—
37#2349C2214yes—
38#2349C2214yes—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#385—21no—
45C2×#385—21no—
46C2×#382C2211yescharacteristic
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C22×#95—21no—
50#2101C2214yescomplemented
51#2100C2212yescomplemented
52#2097C2214yescomplemented
53#2087C2214yescomplemented
54#2085C2214yescomplemented
55#2083C2212yescomplemented
56#2074C2214yescomplemented
57#2068C2214yescomplemented
58#2057C2212yescomplemented
59#2043C2212yescomplemented
60#1943C2214yescomplemented
61#1935C2214yescomplemented
62#1918C2214yescomplemented
63#1917C2214yescomplemented
64#1917C2214yescomplemented
65#1877C2214yes—
66#1877C2214yes—
67#1877C2214yes—
68#1877C2214yes—
69#1872C2212yes—
70#1872C2212yes—
71#1865C2214yescomplemented
72#1864C2214yes—
73#1862C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1824C2212yes—
79#1817C2212yes—
80#1802C2212yes—
81#1802C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—21no—
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C4×Q8C2312yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—44no—
104C2×C4×D4—44no—
105C2×C4×D4C2314yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#484—28no—
123#476—44no—
124#461—44no—
125#450—28no—
126#440—48no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141#400—22no—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#395—22no—
149#391—21no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2312yes—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#385—22no—
179#385—22no—
180#385—24no—
181#385—24no—
182#385—24no—
183#385—24no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—22no—
187#382C2314yes—
188#381—22no—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—24no—
193#380—24no—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—24no—
207C2×#95—24no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C22×Q8—21no—
223C22×Q8C2412yes—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4#4114yes—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—28no—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4—82no—
245C23×C4—82no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×SD16—28no—
254C2×SD16—28no—
255C2×SD16—28no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—42no—
259C4⋊Q8—42no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262C22⋊Q8—44no—
263C4×Q8—44no—
264C4×D4—216no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—28no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—44no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—48no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4#4114yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336#107—28no—
337#107—44no—
338#106—44no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341#103—28no—
342#103—44no—
343#103—44no—
344#102—28no—
345#102—28no—
346#102—28no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349#97—44no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
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#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24C22×D411yescharacteristic
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4C2×#4114yes—
398C2×D4H(5,𝔽2)14yes—
399C2×D42−1+414yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—816no—
448C22×C4—816no—
449C22×C4—816no—
450C22×C4#13714yes—
451C22×C4C22×D412yes—
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×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2511yescharacteristic
463SD16—416no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4C2×#4114yes—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—42no—
517C42—82no—
518C42—82no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—216no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—48no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—48no—
535C23—42no—
536C23—42no—
537C23#52512yes—
538C23C2×#13712yes—
539C23#57512yes—
540C23C23×D411yescharacteristic
541Q8—44no—
542Q8—44no—
543Q8—162no—
544D4—216no—
545D4—24no—
546D4—24no—
547D4—24no—
548D4#53918yes—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
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—88no—
602C2×C4—88no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—816no—
606C2×C4—816no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4#52512yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)12yes—
615C2×C4C2×2−1+412yes—
616C8—44no—
617C8—44no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—28no—
625C22—432no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22#311614yes—
633C22C2×#52511yescharacteristic
634C22#348411yescharacteristic
635C22#348611yescharacteristic
636C22C22×#13711yescharacteristic
637C22C2×#57511yescharacteristic
638C22#358511yescharacteristic
639C22#358511yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—44no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4C2×#53912yes—
657C4#349212yes—
658C2—24no—
659C2—48no—
660C2#3498911yescharacteristic
661C2C2×#311611yescharacteristic
662C2#6056411yescharacteristic
663C2#6157011yescharacteristic
664C2#6158111yescharacteristic
665C2#6182611yescharacteristic
666C2#6183111yescharacteristic
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
6. #29969C2semidirect
7. #29913C2semidirect
8. #29757C2semidirect
9. #27576C2semidirect
10. #27320C2semidirect
11. #26396C2semidirect
12. #26186C2semidirect
13. #22085C2semidirect
14. #21916C2semidirect
31. #2511C22semidirect
32. #2507C22semidirect
33. #2451C22semidirect
34. #2446C22semidirect
50. #2101C22semidirect
51. #2100C22semidirect
52. #2097C22semidirect
53. #2087C22semidirect
54. #2085C22semidirect
55. #2083C22semidirect
56. #2074C22semidirect
57. #2068C22semidirect
58. #2057C22semidirect
59. #2043C22semidirect
60. #1943C22semidirect
61. #1935C22semidirect
62. #1918C22semidirect
63. #1917C22semidirect
64. #1917C22semidirect
71. #1865C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#23492
#330092
#329562
#329052
#299692
#299132
#297572
#275762
#273202
#263962
#261862
#220852
#219162
#214532