Finite Group Databasebeta

← #6624511

Subgroups — #6624511

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic, complemented
3#33111C212yesmaximal, complemented
4#32905C211yesmaximal, characteristic, complemented
5#28312C212yesmaximal
6#28120C212yesmaximal, complemented
7#27905C211yesmaximal, characteristic, complemented
8#27644C211yesmaximal, characteristic, complemented
9#27502C214yesmaximal, complemented
10#27208C214yesmaximal, complemented
11#26091C212yesmaximal, complemented
12#24627C212yesmaximal, complemented
13#23100C212yesmaximal, complemented
14#22813C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#21579C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#480C2212yes—
23C2×#461C2212yes—
24C2×#450C2212yes—
25C2×#447C2211yescharacteristic
26C2×C4×D8C2212yes—
27#2857C2211yescharacteristic
28#2850C2212yes—
29#2596—22no—
30#2571C2211yescharacteristic, complemented
31#2547C2211yescharacteristic, complemented
32#2514C2212yescomplemented
33#2510—22no—
34#2494—21no—
35#2487—21no—
36#2448C2212yescomplemented
37#2365C2211yescharacteristic, complemented
38#2354—22no—
39#2354—21no—
40#2354—21no—
41#2354C2212yes—
42#2354C2212yes—
43#2354C2214yes—
44#2354C2211yescharacteristic, complemented
45#2352—21no—
46#2347—21no—
47C4×C22⋊Q8C2212yes—
48C4×#121C2211yescharacteristic
49C2×#390C2212yes—
50C2×#388—22no—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2096C2214yescomplemented
56#2092C2214yes—
57#2083C2212yescomplemented
58#2071C2212yescomplemented
59#2065C2214yes—
60#2055C2214yes—
61#2012C2214yes—
62#1983C2214yescomplemented
63#1973C2214yes—
64#1972C2214yes—
65#1956C2212yescomplemented
66#1949C2214yescomplemented
67#1947C2212yescomplemented
68#1939C2212yescomplemented
69#1928C2214yescomplemented
70#1928C2214yescomplemented
71#1926C2214yes—
72#1926C2214yescomplemented
73#1919C2214yescomplemented
74#1917C2214yescomplemented
75#1880C2214yes—
76#1879C2212yes—
77#1877C2214yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1862C2212yes—
81#1852C2211yescharacteristic
82#1844C2212yescomplemented
83#1823C2212yes—
84#1802C2212yes—
85#1802C2212yescomplemented
86#1797C2212yes—
87#1797C2211yescharacteristic
88#1786C2211yescharacteristic
89#1776C2214yescomplemented
90#1770C2212yescomplemented
91#1763C2214yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C22×SD16C2314yes—
98C22×D8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#124C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#121C2311yescharacteristic
103C2×C4×Q8C2312yes—
104C2×C4×D4—28no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4C2311yescharacteristic
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—21no—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117—24no—
115C2×#117—48no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#483—28no—
129#483—44no—
130#480—44no—
131#461—28no—
132#450—44no—
133#447—24no—
134C4×D8—28no—
135C2×#107C2312yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155#401—22no—
156#400—22no—
157#400—22no—
158#399—22no—
159#397—22no—
160#397—22no—
161#391C2312yes—
162#391C2312yes—
163#391C2312yes—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2311yescharacteristic
168#390C2312yes—
169#390C2314yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2311yescharacteristic
176#388—22no—
177#388—22no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—22no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#386—21no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#383—22no—
199#382—21no—
200#382C2312yes—
201#380—21no—
202#380—21no—
203#380—21no—
204#380—21no—
205#380—22no—
206#380—22no—
207#380—24no—
208#380—21no—
209#380C2311yescharacteristic
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—21no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8C2412yes—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—42no—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—28no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—28no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×D8—24no—
264C2×D8—24no—
265C2×D8—42no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270#124—44no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273#121—22no—
274#121—22no—
275C4×Q8—44no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4#4118yes—
284#117—44no—
285#117—416no—
286#117—48no—
287#117—48no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—48no—
339C2×C22⋊C4—84no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42—48no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—44no—
358#106—28no—
359#106—44no—
360C4⋊C8—24no—
361C4⋊C8—44no—
362#103—28no—
363#103—28no—
364#103—44no—
365#102—24no—
366#102—24no—
367#102—24no—
368#102—42no—
369#102—44no—
370#102—44no—
371C22⋊C8—24no—
372C22⋊C8—22no—
373C22⋊C8#4114yes—
374#97—44no—
375C4×C8—24no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—24no—
396C24—22no—
397C24—21no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24C22×D411yescharacteristic
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C2×#4114yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—432no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4AGL(1,ℤ/8)14yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4C2511yescharacteristic
476SD16—88no—
477D8—48no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8C2×#4114yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4C2×#4114yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—82no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—24no—
538C42—42no—
539C42—42no—
540C42—48no—
541C42—48no—
542C42—44no—
543C42—42no—
544C42—82no—
545C42C2×#4114yes—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—24no—
551C23—48no—
552C23—48no—
553C23—48no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23#55012yes—
572C23C2×AGL(1,ℤ/8)12yes—
573C23#57712yes—
574C23C23×D411yescharacteristic
575Q8—44no—
576Q8—44no—
577D4—24no—
578D4—24no—
579D4—48no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—42no—
584D4—84no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—416no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—416no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4—84no—
638C2×C4#52512yes—
639C2×C4#52512yes—
640C2×C4#55012yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4C2×AGL(1,ℤ/8)12yes—
643C2×C4#57512yes—
644C2×C4C22×#4111yescharacteristic
645C2×C4C22×#4112yes—
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4#58512yes—
648C8—24no—
649C8—42no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—816no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22#339714yes—
673C22C2×#52511yescharacteristic
674C22C2×#55011yescharacteristic
675C22#348511yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57711yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—44no—
688C4—41no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4—84no—
693C4—81no—
694C4—82no—
695C4#311414yes—
696C4#347512yes—
697C4#349412yes—
698C2—24no—
699C2—84no—
700C2—161no—
701C2#3651711yescharacteristic
702C2C2×#311411yescharacteristic
703C2#6058511yescharacteristic
704C2C2×#339711yescharacteristic
705C2#6140311yescharacteristic
706C2#6171711yescharacteristic
707C2#6173111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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. #36586C2semidirect
2. C2×#2354C2semidirect
3. #33111C2semidirect
4. #32905C2semidirect
6. #28120C2semidirect
7. #27905C2semidirect
8. #27644C2semidirect
9. #27502C2semidirect
10. #27208C2semidirect
11. #26091C2semidirect
12. #24627C2semidirect
13. #23100C2semidirect
14. #22813C2semidirect
15. #22629C2semidirect
16. #21579C2semidirect
30. #2571C22semidirect
31. #2547C22semidirect
32. #2514C22semidirect
36. #2448C22semidirect
37. #2365C22semidirect
44. #2354C22semidirect
55. #2096C22semidirect
57. #2083C22semidirect
58. #2071C22semidirect
62. #1983C22semidirect
65. #1956C22semidirect
66. #1949C22semidirect
67. #1947C22semidirect
68. #1939C22semidirect
69. #1928C22semidirect
70. #1928C22semidirect
72. #1926C22semidirect
73. #1919C22semidirect
74. #1917C22semidirect
82. #1844C22semidirect
85. #1802C22semidirect
89. #1776C22semidirect
90. #1770C22semidirect
91. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#23542
#331112
#329052
#283122
#281202
#279052
#276442
#275022
#272082
#260912
#246272
#231002
#228132
#226292
#215792
C4×#4472