Finite Group Databasebeta

← #6571132

Subgroups — #6571132

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36459C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic, complemented
3#30126C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic, complemented
5#29493C212yesmaximal, complemented
6#28207C212yesmaximal, complemented
7#27523C214yesmaximal, complemented
8#27229C214yesmaximal, complemented
9#26463C212yesmaximal, complemented
10#25855C211yesmaximal, characteristic, complemented
11#25321C212yesmaximal, complemented
12#24208C211yesmaximal, characteristic
13#23819C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic, complemented
15#23215C212yesmaximal, complemented
16#23093C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#486C2212yes—
21C2×#478C2214yescomplemented
22C2×#459C2212yes—
23C2×#447C2212yes—
24#2860C2212yescomplemented
25#2857C2211yescharacteristic
26#2563C2211yescharacteristic, complemented
27#2560—22no—
28#2552C2211yescharacteristic, complemented
29#2532—22no—
30#2496—21no—
31#2495C2212yes—
32#2490C2212yes—
33#2485—21no—
34#2369C2211yescharacteristic, complemented
35#2366—21no—
36#2366—21no—
37#2366C2212yes—
38#2366C2212yes—
39#2354—21no—
40#2354C2211yescharacteristic, complemented
41#2347—21no—
42D4×C22⋊C4—22no—
43D4×C22⋊C4C2214yes—
44#2328—22no—
45#2309—21no—
46C4×#127C2212yes—
47C4×#127C2212yes—
48C4×#125C2212yescomplemented
49C4×#121C2211yescharacteristic
50#2285C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2211yescharacteristic, complemented
53#2100C2212yescomplemented
54#2094C2211yescharacteristic, complemented
55#2084C2211yescharacteristic, complemented
56#2083C2212yescomplemented
57#2077C2212yes—
58#2076C2214yes—
59#2073C2214yescomplemented
60#2071C2211yescharacteristic, complemented
61#2065C2214yescomplemented
62#2054C2214yescomplemented
63#2041C2212yes—
64#2041C2212yes—
65#2041C2212yes—
66#2034C2212yes—
67#2025C2214yescomplemented
68#2011C2212yes—
69#1987C2212yescomplemented
70#1982C2212yes—
71#1970C2214yescomplemented
72#1957C2212yes—
73#1947C2212yescomplemented
74#1939C2212yescomplemented
75#1934C2214yescomplemented
76#1926C2214yescomplemented
77#1925C2214yescomplemented
78#1923C2214yes—
79#1879C2214yescomplemented
80#1879C2214yescomplemented
81#1864C2214yescomplemented
82#1862C2214yescomplemented
83#1831C2212yescomplemented
84#1804C2214yescomplemented
85#1797C2211yescharacteristic
86#1786C2214yes—
87#1786C2212yes—
88C4×#106C2212yescomplemented
89C8×C4⋊C4C2212yescomplemented
90C4×#102C2214yescomplemented
91C4×#102C2214yescomplemented
92C4×#102C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×D8C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127—28no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127C2312yes—
104C2×#127C2312yes—
105C2×#126—22no—
106C2×#125C2312yes—
107C2×C22⋊Q8—21no—
108C2×#121C2311yescharacteristic
109#518—48no—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2314yes—
119C2×C4×D4C2314yes—
120C2×#117—21no—
121C2×#117—21no—
122C2×#117D411yescharacteristic
123C2×#117D411yescharacteristic
124C22×C22⋊C4—24no—
125C22×C42C2311yescharacteristic
126#486—28no—
127#478—48no—
128#459—28no—
129#447—44no—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#394—22no—
153#393—22no—
154#391—21no—
155#391—22no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#390C2312yes—
166#390C2311yescharacteristic
167#390C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389C2311yescharacteristic
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#386—21no—
185#386—24no—
186#386—24no—
187#386—24no—
188#386—24no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#383—22no—
194#382—21no—
195#382C2312yes—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380C2312yes—
202#380C2312yes—
203#380C2311yescharacteristic
204#379—24no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C25—22no—
235C22×Q8—21no—
236C22×D4—28no—
237C22×D4—28no—
238C22×D4—28no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—44no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4—82no—
251C22×D4—82no—
252C22×D4C2411yescharacteristic
253C22×D4C2414yes—
254C23×C4—21no—
255C23×C4—24no—
256C23×C4—24no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C2×D8—28no—
261C2×D8—44no—
262C2×D8—44no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267#127—28no—
268#127—28no—
269#127—28no—
270#127—24no—
271#127#4118yes—
272#126—84no—
273#125—24no—
274#125—42no—
275C22⋊Q8—82no—
276#121—22no—
277#121—41no—
278C4×D4—24no—
279C4×D4—48no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—48no—
288C4×D4—84no—
289C4×D4—84no—
290#117—24no—
291#117—24no—
292#117—44no—
293#117—44no—
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—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—416no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—44no—
354#106—24no—
355#106—42no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#102—28no—
359#102—28no—
360#102—28no—
361#102—28no—
362#102—44no—
363#102—44no—
364#102—42no—
365#102—42no—
366#102—44no—
367C22⋊C8—42no—
368C22⋊C8—42no—
369C4×C8—24no—
370C4×C8—44no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—28no—
389C24—24no—
390C24—21no—
391C24—44no—
392C24—44no—
393C24—82no—
394C24—82no—
395C24—82no—
396C24—82no—
397C24C22×D411yescharacteristic
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—432no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—82no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—816no—
431C2×D4—81no—
432C2×D4C2×#4118yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—41no—
449C22×C4—48no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4#13514yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C2511yescharacteristic
469D8—88no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4AGL(1,ℤ/8)14yes—
506C4⋊C4H(5,𝔽2)12yes—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—42no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—42no—
536C42—42no—
537C42—44no—
538C42—44no—
539C42—82no—
540C42C2×#4114yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—416no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—88no—
563C23—88no—
564C23—164no—
565C23—164no—
566C23—164no—
567C23—164no—
568C23C2×#13512yes—
569C23C23×D411yescharacteristic
570Q8—162no—
571D4—28no—
572D4—28no—
573D4—416no—
574D4—416no—
575D4—416no—
576D4—44no—
577D4—42no—
578D4—44no—
579D4—42no—
580D4—88no—
581D4—84no—
582D4—88no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—84no—
629C2×C4—168no—
630C2×C4—168no—
631C2×C4#52512yes—
632C2×C4#52512yes—
633C2×C4C2×#13512yes—
634C2×C4C2×#13512yes—
635C2×C4C2×AGL(1,ℤ/8)14yes—
636C2×C4C2×AGL(1,ℤ/8)12yes—
637C2×C4#57612yes—
638C2×C4C22×#4112yes—
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4#58512yes—
641C2×C4#58511yescharacteristic
642C8—42no—
643C8—44no—
644C8—42no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—1616no—
667C22C2×#52511yescharacteristic
668C22#348811yescharacteristic
669C22#350011yescharacteristic
670C22C22×#13511yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57611yescharacteristic
673C22#358511yescharacteristic
674C4—24no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—42no—
679C4—42no—
680C4—41no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4#310514yes—
690C4#351412yes—
691C4#352112yes—
692C2—42no—
693C2—88no—
694C2—161no—
695C2#3607511yescharacteristic
696C2C2×#310511yescharacteristic
697C2#6057811yescharacteristic
698C2#6194911yescharacteristic
699C2#6200111yescharacteristic
700C2#6252711yescharacteristic
701C2#6257611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#24208C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36459C2semidirect
2. #34210C2semidirect
3. #30126C2semidirect
4. #29939C2semidirect
5. #29493C2semidirect
6. #28207C2semidirect
7. #27523C2semidirect
8. #27229C2semidirect
9. #26463C2semidirect
10. #25855C2semidirect
11. #25321C2semidirect
13. #23819C2semidirect
14. #23755C2semidirect
15. #23215C2semidirect
16. #23093C2semidirect
17. #22632C2semidirect
21. C2×#478C22semidirect
24. #2860C22semidirect
26. #2563C22semidirect
28. #2552C22semidirect
34. #2369C22semidirect
40. #2354C22semidirect
48. C4×#125C22semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2094C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
59. #2073C22semidirect
60. #2071C22semidirect
61. #2065C22semidirect
62. #2054C22semidirect
67. #2025C22semidirect
69. #1987C22semidirect
71. #1970C22semidirect
73. #1947C22semidirect
74. #1939C22semidirect
75. #1934C22semidirect
76. #1926C22semidirect
77. #1925C22semidirect
79. #1879C22semidirect
80. #1879C22semidirect
81. #1864C22semidirect
82. #1862C22semidirect
83. #1831C22semidirect
84. #1804C22semidirect
88. C4×#106C22semidirect
89. C8×C4⋊C4C22semidirect
90. C4×#102C22semidirect
91. C4×#102C22semidirect
92. C4×#102C22semidirect
94. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#364592
#342102
#301262
#299392
#294932
#282072
#275232
#272292
#264632
#258552
#253212
#242082
#238192
#237552
#232152
#230932
#226322