Finite Group Databasebeta

← #6495602

Subgroups — #6495602

Order 512.

2410 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#36644C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic, complemented
3#33106C212yesmaximal, complemented
4#32902C212yesmaximal, complemented
5#29980C212yesmaximal, complemented
6#29794C212yesmaximal, complemented
7#29552C214yesmaximal, complemented
8#29364C214yesmaximal, complemented
9#29328C212yesmaximal, complemented
10#28301C212yesmaximal, complemented
11#24060C212yesmaximal, complemented
12#23930C212yesmaximal, complemented
13#22145C214yesmaximal, complemented
14#20777C211yesmaximal, characteristic, complemented
15C2×#532—21no—
16C2×#532—21no—
17C2×#532C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#480C2214yes—
21C2×#459C2214yes—
22C2×#447C2214yes—
23#2868C2211yescharacteristic, complemented
24#2867C2211yescharacteristic, complemented
25#2858—21no—
26#2858—21no—
27#2858C2212yescomplemented
28#2856—21no—
29#2856—21no—
30#2856C2212yes—
31#2856C2212yes—
32C4×#127—21no—
33C4×#127—21no—
34C4×#127C2212yescomplemented
35C4×#127C2212yescomplemented
36#2293—21no—
37#2293—21no—
38#2293—21no—
39#2293—21no—
40#2293C2212yes—
41#2293C2212yes—
42#2293C2212yes—
43#2293C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52C2×C43C2211yescharacteristic
53#2099C2214yescomplemented
54#2093C2214yes—
55#2057C2214yescomplemented
56#2041C2214yescomplemented
57#2041C2214yescomplemented
58#2023C2214yes—
59#2011C2214yes—
60#2011C2214yes—
61#1982C2214yescomplemented
62#1970C2218yes—
63#1947C2214yescomplemented
64#1934C2218yescomplemented
65#1925C2218yescomplemented
66#1879C2218yes—
67#1879C2218yes—
68#1874C2214yes—
69#1874C2214yes—
70#1862C2218yescomplemented
71#1839C2212yescomplemented
72#1831C2212yescomplemented
73#1817C2212yes—
74#1802C2214yes—
75#1797C2212yes—
76C4×C4⋊C8C2212yescomplemented
77C4×C22⋊C8C2212yes—
78#1744C2212yescomplemented
79C22×#41—22no—
80C23×D4C2311yescharacteristic
81C24×C4—21no—
82C22×D8C2318yes—
83#532—48no—
84#532—48no—
85#532—48no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2312yes—
89C2×#127—21no—
90C2×#127—21no—
91C2×#127—21no—
92C2×#127—21no—
93C2×#127C2312yes—
94C2×#127C2312yes—
95C2×#127C2312yes—
96C2×#124—22no—
97C2×#124—22no—
98C2×#124—22no—
99C2×#121—22no—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121C2314yes—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2314yes—
116C2×#117—22no—
117C2×#117—22no—
118C2×#117—22no—
119C2×#117—22no—
120C2×#117—48no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42—21no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#480—48no—
140#459—216no—
141#447—48no—
142C2×#107C2318yes—
143C2×C4⋊C8C2314yes—
144C2×#102C2318yes—
145C2×#102C2318yes—
146C2×#102C2318yes—
147C2×#102C2318yes—
148C2×C22⋊C8C2314yes—
149C2×C4×C8C2314yes—
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#390C2314yes—
165#390C2314yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2312yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43D411yescharacteristic
204C43D411yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208C43C2311yescharacteristic
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215C25—21no—
216C2×#41—48no—
217C2×#41—48no—
218C2×#41—48no—
219C22×Q8—22no—
220C22×D4—24no—
221C22×D4—24no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4C2414yes—
232C22×D4C2412yes—
233C23×C4—28no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×D8—216no—
250C2×D8—48no—
251C2×D8—48no—
252C22×C8C2418yes—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256#127—28no—
257#127—44no—
258#127—44no—
259#127—44no—
260#127—44no—
261#127—44no—
262#127—44no—
263#124—48no—
264#124—48no—
265#124—48no—
266#121—28no—
267#121—28no—
268#121—48no—
269#121—48no—
270#121—48no—
271#121—48no—
272#121—48no—
273C4×D4—28no—
274C4×D4—28no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—48no—
284C4×D4—48no—
285C4×D4#4118yescomplemented
286#117—48no—
287#117—48no—
288#117—48no—
289#117—48no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
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—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42—44no—
341C2×C42D818yescomplemented
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#107—48no—
371C4⋊C8—28no—
372#102—216no—
373#102—48no—
374#102—48no—
375#102—48no—
376C22⋊C8—28no—
377C4×C8—44no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24—24no—
385C24—22no—
386C24—24no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24C22×D411yescharacteristic
394#41—832no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—416no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4C2×#4114yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—44no—
455C22×C4—432no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—48no—
470C22×C4C2×D814yes—
471C22×C4C2×D814yes—
472C22×C4C2×D814yes—
473C22×C4AGL(1,ℤ/8)14yes—
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×C4C22×D411yescharacteristic
487C22×C4H(5,𝔽2)14yes—
488C22×C4C2511yescharacteristic
489D8—816no—
490C2×C8—28no—
491C2×C8—28no—
492C2×C8—28no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4C2×#4114yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C42—22no—
528C42—22no—
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—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—44no—
551C42—44no—
552C42C2×D814yes—
553C42C2×D814yes—
554C42C2×D814yes—
555C42C2×D814yes—
556C42C2×#4114yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—24no—
562C23—48no—
563C23—48no—
564C23—48no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—84no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23#53512yes—
583C23C22×D812yes—
584C23C2×AGL(1,ℤ/8)12yes—
585C23C23×D411yescharacteristic
586Q8—88no—
587D4—28no—
588D4—48no—
589D4—44no—
590D4—48no—
591D4—48no—
592D4—88no—
593D4—88no—
594D4—88no—
595D4—88no—
596D4—88no—
597D4—816no—
598D4—88no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
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—21no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—416no—
634C2×C4—48no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—416no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—816no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4#52512yes—
651C2×C4#52512yes—
652C2×C4#53512yes—
653C2×C4C22×D812yes—
654C2×C4C22×D812yes—
655C2×C4C22×D812yes—
656C2×C4C22×D812yes—
657C2×C4C22×D812yes—
658C2×C4C22×D812yes—
659C2×C4C2×AGL(1,ℤ/8)12yes—
660C2×C4C2×AGL(1,ℤ/8)12yes—
661C2×C4#57512yes—
662C2×C4#57512yes—
663C2×C4C22×#4112yes—
664C2×C4C2×H(5,𝔽2)12yes—
665C2×C4#58512yes—
666C8—48no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—816no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22#320514yes—
689C22C2×#52511yescharacteristic
690C22C2×#53511yescharacteristic
691C22#344911yescharacteristic
692C22C23×D811yescharacteristic
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—22no—
705C4—22no—
706C4—44no—
707C4—41no—
708C4—41no—
709C4—82no—
710C4—82no—
711C4—84no—
712C4#310014yes—
713C4#349414yes—
714C2—24no—
715C2—84no—
716C2—164no—
717C2#3530111yescharacteristic
718C2C2×#310011yescharacteristic
719C2#6056111yescharacteristic
720C2C2×#320511yescharacteristic
721C2#6080511yescharacteristic
722C2#6094011yescharacteristic
723C2#6095811yescharacteristic
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. #36644C2semidirect
2. C2×#2293C2semidirect
3. #33106C2semidirect
4. #32902C2semidirect
5. #29980C2semidirect
6. #29794C2semidirect
7. #29552C2semidirect
8. #29364C2semidirect
9. #29328C2semidirect
10. #28301C2semidirect
11. #24060C2semidirect
12. #23930C2semidirect
13. #22145C2semidirect
14. #20777C2semidirect
23. #2868C22semidirect
24. #2867C22semidirect
27. #2858C22semidirect
34. C4×#127C22semidirect
35. C4×#127C22semidirect
53. #2099C22semidirect
55. #2057C22semidirect
56. #2041C22semidirect
57. #2041C22semidirect
61. #1982C22semidirect
63. #1947C22semidirect
64. #1934C22semidirect
65. #1925C22semidirect
70. #1862C22semidirect
71. #1839C22semidirect
72. #1831C22semidirect
76. C4×C4⋊C8C22semidirect
78. #1744C22semidirect
285. C4×D4#41semidirect
341. C2×C42D8semidirect

Maximal subgroups

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

subgroupindex
#366442
C2×#22932
#331062
#329022
#299802
#297942
#295522
#293642
#293282
#283012
#240602
#239302
#221452
#207772