Finite Group Databasebeta

← #6489127

Subgroups — #6489127

Order 512.

2592 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic, complemented
3#32905C212yesmaximal, complemented
4#32902C212yesmaximal, complemented
5#30040C214yesmaximal, complemented
6#29149C212yesmaximal, complemented
7#28450C212yesmaximal, complemented
8#28157C212yesmaximal, complemented
9#28156C212yesmaximal, complemented
10#27978C212yesmaximal, complemented
11#25725C212yesmaximal, complemented
12#23936C214yesmaximal, complemented
13#22272C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C2×#529C2212yes—
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4C2212yes—
20C2×#496C2214yes—
21C2×#461C2214yescomplemented
22C2×#459C2214yescomplemented
23C2×#450C2214yes—
24C2×#447C2214yes—
25#2861C2212yescomplemented
26#2860—21no—
27#2860—21no—
28#2859C2211yescharacteristic, complemented
29#2858C2211yescharacteristic, complemented
30#2857C2211yescharacteristic
31#2856C2211yescharacteristic
32#2854—21no—
33#2854—21no—
34#2850C2212yes—
35C4×C4⋊Q8C2211yescharacteristic, complemented
36C4×#127—21no—
37C4×#127—21no—
38C4×#127—21no—
39C4×#127—21no—
40C4×#127C2212yes—
41C4×#127C2212yes—
42C4×#127C2212yes—
43C4×#127C2212yes—
44C4×#127C2211yescharacteristic, complemented
45C4×#124C2212yescomplemented
46C4×#121—21no—
47C4×#121—21no—
48#2293—21no—
49#2293—21no—
50C2×#390—21no—
51C2×#390—21no—
52C2×#390C2212yes—
53C2×#384C2211yescharacteristic
54C2×C43C2211yescharacteristic
55#2072C2214yescomplemented
56#2071C2214yescomplemented
57#2025C2218yescomplemented
58#2014C2214yes—
59#2012C2218yescomplemented
60#2011C2214yes—
61#1961C2214yes—
62#1959C2214yescomplemented
63#1880C2218yescomplemented
64#1879C2218yescomplemented
65#1879C2218yescomplemented
66#1877C2218yescomplemented
67#1874C2214yes—
68#1872C2214yes—
69#1839C2212yescomplemented
70#1836C2214yescomplemented
71#1831C2212yescomplemented
72#1817C2212yes—
73#1802C2214yescomplemented
74#1802C2214yescomplemented
75#1797C2212yes—
76#1767C2212yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C22×SD16C2318yes—
84C22×D8C2318yes—
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127—28no—
87C2×#127—21no—
88C2×#127—21no—
89C2×#127C2312yes—
90C2×#127C2311yescharacteristic
91C2×#127C2312yes—
92#529—48no—
93C2×#126—22no—
94C2×#125C2312yes—
95C2×#124C2312yes—
96C2×#124C2312yes—
97C2×#123—22no—
98C2×C22⋊Q8C2312yes—
99C2×#121—21no—
100C2×#121—21no—
101C2×#121C2312yes—
102C2×C4×Q8C2312yes—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—48no—
117C2×C4×D4—48no—
118C2×C4×D4—48no—
119C2×C4×D4C2314yes—
120C2×C4×D4C2314yes—
121C2×C4×D4C2312yes—
122C2×#117—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C22⋊C4—22no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#496—216no—
137#461—48no—
138#459—48no—
139#450—48no—
140#447—48no—
141C2×C4⋊C8C2314yes—
142C2×#103C2318yes—
143C2×#102C2318yes—
144C2×#102C2318yes—
145C2×#102C2318yes—
146C2×C22⋊C8C2314yes—
147C2×#97C2314yes—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390C2314yes—
169#390C2314yes—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#386—22no—
189#386—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2312yes—
197#380—21no—
198#380—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43D411yescharacteristic
206C43D411yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209C43C2311yescharacteristic
210C43C2311yescharacteristic
211#340C2318yes—
212#340C2318yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25—22no—
218C22×Q8C2412yes—
219C22×D4—28no—
220C22×D4—28no—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—44no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4C2412yes—
238C22×D4C2414yes—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—24no—
242C23×C4—24no—
243C23×C4—48no—
244C23×C4—48no—
245C23×C4—48no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×SD16—216no—
254C2×SD16—48no—
255C2×SD16—48no—
256C2×D8—216no—
257C2×D8—48no—
258C2×D8—48no—
259C22×C8C2418yes—
260C4⋊Q8—24no—
261#127—28no—
262#127—28no—
263#127—28no—
264#127—24no—
265#127—24no—
266#127#4118yes—
267#126—48no—
268#125—44no—
269#124—28no—
270#124—44no—
271#123—48no—
272C22⋊Q8—28no—
273#121—28no—
274#121—44no—
275#121—44no—
276C4×Q8—44no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—48no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—48no—
290C4×D4—48no—
291C4×D4—48no—
292C4×D4—48no—
293C4×D4—44no—
294C4×D4—48no—
295#117—48no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
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—22no—
334C2×C22⋊C4—48no—
335C2×C22⋊C4—48no—
336C2×C22⋊C4—48no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—42no—
344C2×C42—42no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C4⋊C8—28no—
374#103—48no—
375#102—48no—
376#102—48no—
377#102—48no—
378C22⋊C8—28no—
379#97—28no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386C24—28no—
387C24—24no—
388C24—22no—
389C24—44no—
390C24—44no—
391C24—44no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24C22×D411yescharacteristic
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—432no—
419C2×D4—432no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—44no—
435C2×D4C2×#4118yes—
436C22×C4—22no—
437C22×C4—22no—
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—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—832no—
472C22×C4AGL(1,ℤ/8)14yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4H(5,𝔽2)14yes—
487C22×C4C2511yescharacteristic
488SD16—816no—
489D8—816no—
490C2×C8—28no—
491C2×C8—28no—
492C2×C8—28no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—44no—
555C42AGL(1,ℤ/8)14yes—
556C42C2×#4114yes—
557C42H(5,𝔽2)14yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—28no—
563C23—416no—
564C23—416no—
565C23—416no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—88no—
580C23—88no—
581C23—88no—
582C23—88no—
583C23#53512yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23#57512yes—
586C23C23×D411yescharacteristic
587Q8—48no—
588D4—28no—
589D4—28no—
590D4—416no—
591D4—416no—
592D4—416no—
593D4—416no—
594D4—416no—
595D4—44no—
596D4—44no—
597D4—48no—
598D4—84no—
599D4—88no—
600D4—84no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—88no—
649C2×C4—816no—
650C2×C4—88no—
651C2×C4—816no—
652C2×C4#52512yes—
653C2×C4#52512yes—
654C2×C4#53512yes—
655C2×C4#53512yes—
656C2×C4#53512yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×AGL(1,ℤ/8)12yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4C2×AGL(1,ℤ/8)12yes—
662C2×C4#57512yes—
663C2×C4C22×#4112yes—
664C2×C4C2×H(5,𝔽2)12yes—
665C2×C4C2×H(5,𝔽2)12yes—
666C8—48no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—48no—
674C22—48no—
675C22—48no—
676C22—832no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#321414yes—
689C22C2×#52511yescharacteristic
690C22C2×#53511yescharacteristic
691C22C2×#53511yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C22×AGL(1,ℤ/8)11yescharacteristic
694C22C2×#57511yescharacteristic
695C22C2×#57511yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—84no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4#321714yes—
713C4#351914yes—
714C2—24no—
715C2—88no—
716C2—162no—
717C2#3523711yescharacteristic
718C2#6055811yescharacteristic
719C2#6055811yescharacteristic
720C2C2×#321711yescharacteristic
721C2C2×#321411yescharacteristic
722C2#6080311yescharacteristic
723C2#6080511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36631C2semidirect
2. C2×C4×#127C2semidirect
3. #32905C2semidirect
4. #32902C2semidirect
5. #30040C2semidirect
6. #29149C2semidirect
7. #28450C2semidirect
8. #28157C2semidirect
9. #28156C2semidirect
10. #27978C2semidirect
11. #25725C2semidirect
12. #23936C2semidirect
13. #22272C2semidirect
14. #20778C2semidirect
21. C2×#461C22semidirect
22. C2×#459C22semidirect
25. #2861C22semidirect
28. #2859C22semidirect
29. #2858C22semidirect
35. C4×C4⋊Q8C22semidirect
44. C4×#127C22semidirect
45. C4×#124C22semidirect
55. #2072C22semidirect
56. #2071C22semidirect
57. #2025C22semidirect
59. #2012C22semidirect
62. #1959C22semidirect
63. #1880C22semidirect
64. #1879C22semidirect
65. #1879C22semidirect
66. #1877C22semidirect
69. #1839C22semidirect
70. #1836C22semidirect
71. #1831C22semidirect
73. #1802C22semidirect
74. #1802C22semidirect
76. #1767C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366312
C2×C4×#1272
#329052
#329022
#300402
#291492
#284502
#281572
#281562
#279782
#257252
#239362
#222722
#207782