Finite Group Databasebeta

← #6477864

Subgroups — #6477864

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic, complemented
2C2×#2363C211yesmaximal, characteristic
3#32985C212yesmaximal, complemented
4#32956C211yesmaximal, characteristic, complemented
5#32905C211yesmaximal, characteristic, complemented
6#29969C211yesmaximal, characteristic, complemented
7#29757C211yesmaximal, characteristic, complemented
8#29598C212yesmaximal
9#27124C214yesmaximal, complemented
10#26956C214yesmaximal, complemented
11#26814C214yesmaximal, complemented
12#26686C214yesmaximal, complemented
13#22049C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic, complemented
15#21581C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#484C2212yes—
20C2×#485C2214yescomplemented
21C2×#476C2212yescomplemented
22C2×#461C2212yescomplemented
23C2×#450C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2848C2212yes—
27#2646—21no—
28#2626—21no—
29#2589—22no—
30#2587—22no—
31#2497C2212yescomplemented
32#2492C2212yescomplemented
33#2429C2212yescomplemented
34#2423C2212yescomplemented
35#2363—22no—
36#2363—22no—
37#2363C2214yes—
38#2363C2214yes—
39#2329—21no—
40#2312—21no—
41C4×C22⋊Q8C2212yes—
42C2×#390—21no—
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#386—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2100C2212yescomplemented
51#2087C2214yescomplemented
52#2083C2212yescomplemented
53#2068C2214yescomplemented
54#2057C2212yescomplemented
55#2043C2212yescomplemented
56#2033C2214yes—
57#2033C2214yes—
58#2000C2214yes—
59#2000C2214yes—
60#1973C2214yescomplemented
61#1936C2214yescomplemented
62#1927C2214yescomplemented
63#1925C2214yescomplemented
64#1924C2214yescomplemented
65#1924C2214yescomplemented
66#1918C2214yescomplemented
67#1880C2214yescomplemented
68#1877C2214yescomplemented
69#1873C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1864C2214yescomplemented
73#1854C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1825C2214yescomplemented
77#1824C2212yes—
78#1802C2212yescomplemented
79#1802C2212yescomplemented
80#1797C2211yescharacteristic
81#1797C2211yescharacteristic
82#1786C2212yes—
83#1779C2212yes—
84#1776C2214yescomplemented
85#1763C2214yescomplemented
86#1749C2212yes—
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×#121—24no—
99C2×#121—22no—
100C2×#121—24no—
101C2×#121C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—21no—
104C2×C4×D4—21no—
105C2×C4×D4—44no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C2×#117—44no—
110C2×#117C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#485—48no—
127#484—28no—
128#476—44no—
129#461—44no—
130#450—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147#400—22no—
148#400—22no—
149#399—22no—
150#398—22no—
151#398—22no—
152#396—22no—
153#396—22no—
154#393—22no—
155#391—21no—
156#391—22no—
157#391C2312yes—
158#391C2312yes—
159#390—24no—
160#390—24no—
161#390—22no—
162#390C2314yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#386—24no—
183#386—24no—
184#385—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#380—21no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#379—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4C2314yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—24no—
203C4×C22⋊C4—24no—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4C2312yes—
207C4×C22⋊C4C2312yes—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8C2412yes—
225C22×D4—28no—
226C22×D4—24no—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—82no—
232C22×D4—82no—
233C22×D4#4114yes—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—82no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C2×SD16—28no—
256C2×SD16—44no—
257C2×SD16—44no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267#121—216no—
268#121—28no—
269#121—28no—
270C4×Q8—44no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—44no—
276C4×D4—44no—
277#117—28no—
278#117—44no—
279#117—48no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4#4114yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—42no—
337C2×C42—42no—
338C2×C42—44no—
339C2×C42—82no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2412yes—
346#107—28no—
347#107—44no—
348#106—28no—
349C4⋊C8—42no—
350C4⋊C8—42no—
351#103—28no—
352#103—28no—
353#103—44no—
354#102—28no—
355#102—44no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361#97—44no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—24no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—48no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—816no—
417C2×D4C2×#4114yes—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—816no—
466C22×C4#13714yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478SD16—88no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—82no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4C2×#4114yes—
511C4⋊C4H(5,𝔽2)14yes—
512C22⋊C4—216no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—84no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—42no—
532C42—41no—
533C42—41no—
534C42—48no—
535C42—82no—
536C42—84no—
537C42—82no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—44no—
551C23—48no—
552C23—44no—
553C23—48no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—48no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—82no—
563C23—82no—
564C23—82no—
565C23—82no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23#52512yes—
571C23C2×#13712yes—
572C23#57512yes—
573C23C23×D411yescharacteristic
574Q8—44no—
575Q8—44no—
576Q8—162no—
577D4—24no—
578D4—24no—
579D4—48no—
580D4—42no—
581D4—42no—
582D4—816no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
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—88no—
636C2×C4—88no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—84no—
640C2×C4#52512yes—
641C2×C4C2×#13712yes—
642C2×C4C2×#13712yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C22×#4111yescharacteristic
645C2×C4C2×H(5,𝔽2)12yes—
646C2×C4#58512yes—
647C8—44no—
648C8—44no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—42no—
656C22—42no—
657C22—44no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—44no—
665C22—88no—
666C22—816no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22#311614yes—
672C22C2×#52511yescharacteristic
673C22#348411yescharacteristic
674C22#348511yescharacteristic
675C22C22×#13711yescharacteristic
676C22C2×#57511yescharacteristic
677C22#358511yescharacteristic
678C22#358511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—44no—
686C4—41no—
687C4—41no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#346012yes—
695C4#350512yes—
696C2—24no—
697C2—82no—
698C2—84no—
699C2#3509111yescharacteristic
700C2C2×#311611yescharacteristic
701C2#6056411yescharacteristic
702C2#6157011yescharacteristic
703C2#6157711yescharacteristic
704C2#6171311yescharacteristic
705C2#6172111yescharacteristic
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. #32985C2semidirect
4. #32956C2semidirect
5. #32905C2semidirect
6. #29969C2semidirect
7. #29757C2semidirect
9. #27124C2semidirect
10. #26956C2semidirect
11. #26814C2semidirect
12. #26686C2semidirect
13. #22049C2semidirect
14. #21906C2semidirect
15. #21581C2semidirect
20. C2×#485C22semidirect
21. C2×#476C22semidirect
22. C2×#461C22semidirect
31. #2497C22semidirect
32. #2492C22semidirect
33. #2429C22semidirect
34. #2423C22semidirect
50. #2100C22semidirect
51. #2087C22semidirect
52. #2083C22semidirect
53. #2068C22semidirect
54. #2057C22semidirect
55. #2043C22semidirect
60. #1973C22semidirect
61. #1936C22semidirect
62. #1927C22semidirect
63. #1925C22semidirect
64. #1924C22semidirect
65. #1924C22semidirect
66. #1918C22semidirect
67. #1880C22semidirect
68. #1877C22semidirect
69. #1873C22semidirect
72. #1864C22semidirect
73. #1854C22semidirect
76. #1825C22semidirect
78. #1802C22semidirect
79. #1802C22semidirect
84. #1776C22semidirect
85. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23632
#329852
#329562
#329052
#299692
#297572
#295982
#271242
#269562
#268142
#266862
#220492
#219062
#215812