Finite Group Databasebeta

← #6519033

Subgroups — #6519033

Order 512.

2082 conjugacy classes of subgroups.

The subgroup lattice has 763 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic, complemented
3#27604C212yesmaximal, complemented
4#27603C211yesmaximal, characteristic, complemented
5#27601C211yesmaximal, characteristic, complemented
6#25019C214yesmaximal, complemented
7#25018C214yesmaximal, complemented
8#24974C214yesmaximal, complemented
9#24973C214yesmaximal, complemented
10#24009C212yesmaximal, complemented
11#24005C211yesmaximal, characteristic, complemented
12#24002C211yesmaximal, characteristic, complemented
13#21983C212yesmaximal, complemented
14#21786C211yesmaximal, characteristic, complemented
15#20733C212yesmaximal, complemented
16C22×#121C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#481C2212yescomplemented
21C2×#482C2214yescomplemented
22C2×#480C2212yescomplemented
23C2×#431C2212yes—
24#2535—22no—
25#2535—22no—
26#2535C2214yes—
27#2535C2214yes—
28C2×#397—21no—
29C2×#392—21no—
30#2353—22no—
31#2353—22no—
32#2352C2212yescomplemented
33#2352C2212yescomplemented
34#2352C2212yescomplemented
35#2352C2212yescomplemented
36#2313—21no—
37#2312—21no—
38#2308—21no—
39C2×#386—21no—
40C2×#386—21no—
41#2283C2212yes—
42C2×#382C2211yescharacteristic
43#2280C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49#2077C2214yescomplemented
50#2077C2214yescomplemented
51#2076C2214yescomplemented
52#2075C2214yescomplemented
53#2034C2214yescomplemented
54#2034C2214yescomplemented
55#2033C2214yescomplemented
56#2033C2214yescomplemented
57#1954C2214yescomplemented
58#1954C2214yescomplemented
59#1949C2214yescomplemented
60#1949C2214yescomplemented
61#1875C2214yescomplemented
62#1874C2212yescomplemented
63#1874C2212yescomplemented
64#1854C2214yescomplemented
65#1853C2212yes—
66#1853C2212yes—
67#1853C2212yescomplemented
68#1852C2212yes—
69#1852C2212yes—
70#1852C2212yescomplemented
71#1846C2214yescomplemented
72#1835C2214yescomplemented
73#1825C2214yescomplemented
74#1824C2212yes—
75#1824C2212yes—
76#1824C2212yescomplemented
77#1823C2212yescomplemented
78#1804C2214yescomplemented
79#1804C2214yescomplemented
80#1786C2212yes—
81#1786C2212yes—
82#1779C2211yescharacteristic
83#1779C2211yescharacteristic
84#1779C2212yes—
85C4×#102C2214yescomplemented
86C4×#102C2214yescomplemented
87C23×D4—21no—
88C23×D4—21no—
89C23×D4D411yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#121—21no—
95C2×#121—21no—
96C2×#121—22no—
97C2×#121—44no—
98C2×#121—44no—
99C2×#121C2312yes—
100C2×#121C2314yes—
101C2×#121C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—28no—
104C2×C4×D4—28no—
105C2×C4×D4D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4C2314yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42C2311yescharacteristic
130#482—48no—
131#481—44no—
132#480—44no—
133#431—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2314yes—
152#397—24no—
153#397—24no—
154#392—24no—
155#392—24no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—24no—
169#386—24no—
170#386—24no—
171#386—24no—
172#383—22no—
173#382—22no—
174#382C2312yes—
175#382C2312yes—
176#382C2314yes—
177#381—22no—
178#381—22no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—22no—
186#380C2312yes—
187#380C2314yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—22no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205#376—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C25C2×D411yescharacteristic
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—24no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—82no—
238C22×D4—82no—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2412yes—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4—42no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4—44no—
257C23×C4—82no—
258C23×C4—82no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C23×C4C2411yescharacteristic
270C22×C8C2414yes—
271C22×C8C2414yes—
272#121—24no—
273#121—24no—
274#121—28no—
275#121—42no—
276#121—44no—
277#121—42no—
278#121—48no—
279#121—44no—
280#121—44no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—48no—
290#117—44no—
291#117—44no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—24no—
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—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42C2×D412yes—
349C2×C42#4118yes—
350C2×C42C2412yes—
351#107—44no—
352#107—44no—
353#106—44no—
354#106—44no—
355C4⋊C8—28no—
356#102—28no—
357#102—28no—
358#102—44no—
359#102—44no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—44no—
365C22⋊C8—42no—
366C22⋊C8—42no—
367C22⋊C8—44no—
368C4×C8—28no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—41no—
394C24—41no—
395C24—41no—
396C24—41no—
397C24—41no—
398C24—41no—
399C24—41no—
400C24—41no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24#13514yes—
408C24C22×D412yes—
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—24no—
420C2×D4—416no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—416no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—42no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—84no—
435C2×D4—816no—
436C2×D4—816no—
437C2×D4H(5,𝔽2)14yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—48no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4—84no—
493C22×C4—84no—
494C22×C4—81no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—81no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—816no—
503C22×C4#12314yes—
504C22×C4#12314yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4114yes—
512C22×C4C2511yescharacteristic
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—81no—
533C4⋊C4—81no—
534C4⋊C4—81no—
535C4⋊C4—81no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—48no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—84no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42—82no—
559C42#12318yes—
560C42C2×#4114yes—
561C42C2×#4114yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—28no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—44no—
588C23—44no—
589C23—48no—
590C23—44no—
591C23—48no—
592C23—48no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—82no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23—82no—
606C23—82no—
607C23—82no—
608C23—82no—
609C23—84no—
610C23—84no—
611C23—84no—
612C23—84no—
613C23C2×#12312yes—
614C23C2×#13512yes—
615C23C2×#13512yes—
616C23C2×#13512yes—
617C23#57512yes—
618C23C23×D411yescharacteristic
619D4—28no—
620D4—48no—
621D4—48no—
622D4—48no—
623D4—44no—
624D4—88no—
625D4—88no—
626D4—816no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—28no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—28no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—84no—
673C2×C4—816no—
674C2×C4—82no—
675C2×C4—82no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—84no—
679C2×C4—162no—
680C2×C4—162no—
681C2×C4—162no—
682C2×C4—162no—
683C2×C4C2×#12312yes—
684C2×C4C2×#12314yes—
685C2×C4C2×#12314yes—
686C2×C4C22×#4112yes—
687C2×C4C2×H(5,𝔽2)12yes—
688C2×C4#58512yes—
689C8—44no—
690C8—44no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—44no—
716C22—42no—
717C22—42no—
718C22—44no—
719C22—42no—
720C22—44no—
721C22—88no—
722C22—88no—
723C22—816no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22#309114yes—
729C22C22×#12311yescharacteristic
730C22#348411yescharacteristic
731C22#348511yescharacteristic
732C22C22×#13511yescharacteristic
733C22C2×#57511yescharacteristic
734C22#358511yescharacteristic
735C22#358511yescharacteristic
736C4—24no—
737C4—24no—
738C4—22no—
739C4—42no—
740C4—41no—
741C4—41no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4#349414yes—
750C2—24no—
751C2—42no—
752C2—42no—
753C2—82no—
754C2—84no—
755C2#3554011yescharacteristic
756C2C2×#309111yescharacteristic
757C2#6051811yescharacteristic
758C2#6156111yescharacteristic
759C2#6158111yescharacteristic
760C2#6169911yescharacteristic
761C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34185C2semidirect
3. #27604C2semidirect
4. #27603C2semidirect
5. #27601C2semidirect
6. #25019C2semidirect
7. #25018C2semidirect
8. #24974C2semidirect
9. #24973C2semidirect
10. #24009C2semidirect
11. #24005C2semidirect
12. #24002C2semidirect
13. #21983C2semidirect
14. #21786C2semidirect
15. #20733C2semidirect
20. C2×#481C22semidirect
21. C2×#482C22semidirect
22. C2×#480C22semidirect
32. #2352C22semidirect
33. #2352C22semidirect
34. #2352C22semidirect
35. #2352C22semidirect
49. #2077C22semidirect
50. #2077C22semidirect
51. #2076C22semidirect
52. #2075C22semidirect
53. #2034C22semidirect
54. #2034C22semidirect
55. #2033C22semidirect
56. #2033C22semidirect
57. #1954C22semidirect
58. #1954C22semidirect
59. #1949C22semidirect
60. #1949C22semidirect
61. #1875C22semidirect
62. #1874C22semidirect
63. #1874C22semidirect
64. #1854C22semidirect
67. #1853C22semidirect
70. #1852C22semidirect
71. #1846C22semidirect
72. #1835C22semidirect
73. #1825C22semidirect
76. #1824C22semidirect
77. #1823C22semidirect
78. #1804C22semidirect
79. #1804C22semidirect
85. C4×#102C22semidirect
86. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
#341852
#276042
#276032
#276012
#250192
#250182
#249742
#249732
#240092
#240052
#240022
#219832
#217862
#207332