Finite Group Databasebeta

← #6477844

Subgroups — #6477844

Order 512.

2108 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2361C211yesmaximal, characteristic
3#32982C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#32905C211yesmaximal, characteristic, complemented
6#29968C211yesmaximal, characteristic, complemented
7#29756C211yesmaximal, characteristic, complemented
8#29596C212yesmaximal
9#27096C214yesmaximal, complemented
10#26928C214yesmaximal, complemented
11#26786C214yesmaximal, complemented
12#26658C214yesmaximal, complemented
13#22043C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#483C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4—21no—
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#483C2212yes—
21C2×#483C2214yescomplemented
22C2×#476C2212yescomplemented
23C2×#461C2212yescomplemented
24C2×#450C2212yes—
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2848C2212yes—
28#2646—21no—
29#2626—21no—
30#2589—22no—
31#2587—22no—
32#2497C2212yescomplemented
33#2492C2212yescomplemented
34#2429C2212yescomplemented
35#2423C2212yescomplemented
36#2361—22no—
37#2361—22no—
38#2361C2214yes—
39#2361C2214yes—
40#2329—21no—
41#2312—21no—
42C4×C22⋊Q8C2212yes—
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#386—21no—
47C2×#385—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2212yescomplemented
51#2088C2214yescomplemented
52#2084C2212yescomplemented
53#2066C2214yescomplemented
54#2058C2212yescomplemented
55#2043C2212yescomplemented
56#2034C2214yes—
57#2034C2214yes—
58#2000C2214yes—
59#2000C2214yes—
60#1968C2214yescomplemented
61#1934C2214yescomplemented
62#1927C2214yescomplemented
63#1926C2214yescomplemented
64#1923C2214yescomplemented
65#1923C2214yescomplemented
66#1920C2214yescomplemented
67#1880C2214yescomplemented
68#1877C2214yescomplemented
69#1872C2212yes—
70#1872C2212yes—
71#1872C2214yescomplemented
72#1864C2214yescomplemented
73#1853C2212yes—
74#1853C2212yes—
75#1853C2214yescomplemented
76#1823C2212yes—
77#1823C2214yescomplemented
78#1802C2212yescomplemented
79#1802C2212yescomplemented
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1786C2212yes—
83#1779C2212yes—
84C4×#106C2214yescomplemented
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×SD16C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×#121—24no—
100C2×#121—22no—
101C2×#121—24no—
102C2×#121C2314yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—44no—
109C2×C4×D4—48no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—44no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#483—28no—
130#483—48no—
131#476—44no—
132#461—44no—
133#450—28no—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—22no—
152#399—22no—
153#398—22no—
154#398—22no—
155#396—22no—
156#396—22no—
157#393—22no—
158#391—21no—
159#391—22no—
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#390C2314yes—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—24no—
179#388—24no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—24no—
184#386—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#380—21no—
194#380—21no—
195#380—21no—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#379—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222C25—21no—
223C25C2×D411yescharacteristic
224C22×Q8—21no—
225C22×Q8C2412yes—
226C22×D4—28no—
227C22×D4—24no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4#4114yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C22×C8C2414yes—
264C22×C8C2414yes—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272#121—216no—
273#121—28no—
274#121—28no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—42no—
279C4×D4—42no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—416no—
283C4×D4—416no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4SD1618yescomplemented
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2412yes—
351#106—28no—
352#106—28no—
353#106—44no—
354C4⋊C8—42no—
355C4⋊C8—42no—
356#103—28no—
357#103—28no—
358#103—44no—
359#102—28no—
360#102—44no—
361#102—44no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364C22⋊C8—44no—
365C22⋊C8—44no—
366C4×C8—44no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24C2×SD1614yes—
399C24C22×D412yes—
400C24C22×D411yescharacteristic
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—42no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×D4—216no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—48no—
419C2×D4—432no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4C2×#4114yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—84no—
472C22×C4—816no—
473C22×C4—816no—
474C22×C4C2×SD1614yes—
475C22×C4C2×SD1614yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2511yescharacteristic
487SD16—88no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—48no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4C2×#4114yes—
521C4⋊C4H(5,𝔽2)14yes—
522C22⋊C4—216no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4C2×SD1618yes—
536C22⋊C4C2×SD1618yes—
537C42—22no—
538C42—22no—
539C42—24no—
540C42—42no—
541C42—41no—
542C42—41no—
543C42—48no—
544C42—82no—
545C42—82no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—22no—
552C23—28no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23#52512yes—
578C23C22×SD1614yes—
579C23C22×SD1612yes—
580C23#57512yes—
581C23C23×D411yescharacteristic
582Q8—44no—
583Q8—44no—
584Q8—162no—
585D4—24no—
586D4—24no—
587D4—48no—
588D4—432no—
589D4—42no—
590D4—42no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—48no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—816no—
647C2×C4—84no—
648C2×C4#52512yes—
649C2×C4C22×SD1614yes—
650C2×C4C22×SD1612yes—
651C2×C4C22×SD1612yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4#58512yes—
656C8—44no—
657C8—44no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—416no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—816no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22#310114yes—
681C22C2×#52511yescharacteristic
682C22#348411yescharacteristic
683C22#348511yescharacteristic
684C22C23×SD1611yescharacteristic
685C22C2×#57511yescharacteristic
686C22#358511yescharacteristic
687C22#358611yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—41no—
697C4—41no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#346012yes—
704C4#350512yes—
705C2—24no—
706C2—44no—
707C2—84no—
708C2#3509111yescharacteristic
709C2C2×#310111yescharacteristic
710C2#6056511yescharacteristic
711C2#6156011yescharacteristic
712C2#6158311yescharacteristic
713C2#6169511yescharacteristic
714C2#6171311yescharacteristic
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. #35014C2semidirect
3. #32982C2semidirect
4. #32956C2semidirect
5. #32905C2semidirect
6. #29968C2semidirect
7. #29756C2semidirect
9. #27096C2semidirect
10. #26928C2semidirect
11. #26786C2semidirect
12. #26658C2semidirect
13. #22043C2semidirect
14. #21902C2semidirect
15. C4×#483C2semidirect
21. C2×#483C22semidirect
22. C2×#476C22semidirect
23. C2×#461C22semidirect
32. #2497C22semidirect
33. #2492C22semidirect
34. #2429C22semidirect
35. #2423C22semidirect
50. #2100C22semidirect
51. #2088C22semidirect
52. #2084C22semidirect
53. #2066C22semidirect
54. #2058C22semidirect
55. #2043C22semidirect
60. #1968C22semidirect
61. #1934C22semidirect
62. #1927C22semidirect
63. #1926C22semidirect
64. #1923C22semidirect
65. #1923C22semidirect
66. #1920C22semidirect
67. #1880C22semidirect
68. #1877C22semidirect
71. #1872C22semidirect
72. #1864C22semidirect
75. #1853C22semidirect
77. #1823C22semidirect
78. #1802C22semidirect
79. #1802C22semidirect
84. C4×#106C22semidirect
85. C4×#103C22semidirect
334. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23612
#329822
#329562
#329052
#299682
#297562
#295962
#270962
#269282
#267862
#266582
#220432
#219022
C4×#4832