Finite Group Databasebeta

← #6795010

Subgroups — #6795010

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#32956C211yesmaximal, characteristic, complemented
4#32437C214yesmaximal, complemented
5#32353C214yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#28528C212yesmaximal, complemented
8#27728C212yesmaximal, complemented
9#27588C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25450C212yesmaximal, complemented
12#24677C212yesmaximal, complemented
13#24500C212yesmaximal
14#24299C211yesmaximal, characteristic, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21983C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#500C2212yescomplemented
21C2×#476C2212yes—
22C2×#476C2212yescomplemented
23C2×#447C2211yescharacteristic
24C2×C4×D8C2212yes—
25C2×#431C2212yes—
26#2857C2211yescharacteristic
27#2854C2212yes—
28#2571C2211yescharacteristic, complemented
29#2563—21no—
30#2563—21no—
31#2563—22no—
32#2563C2212yes—
33#2563C2214yes—
34#2563C2212yes—
35#2563C2211yescharacteristic, complemented
36#2532—22no—
37#2513C2212yescomplemented
38#2508C2212yescomplemented
39#2495—21no—
40#2488—22no—
41#2486—21no—
42C2×#393—22no—
43#2369C2211yescharacteristic, complemented
44#2354C2211yescharacteristic, complemented
45#2352—21no—
46#2346—21no—
47C4×#121C2211yescharacteristic
48C2×#390C2212yes—
49#2283C2212yes—
50C2×#382C2212yes—
51C2×#380—21no—
52C2×#380—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yescomplemented
55#2092C2214yes—
56#2092C2214yescomplemented
57#2085C2214yescomplemented
58#2073C2214yes—
59#2073C2214yescomplemented
60#2067C2214yes—
61#2058C2212yescomplemented
62#2057C2214yescomplemented
63#2054C2214yes—
64#2023C2212yescomplemented
65#1990C2212yescomplemented
66#1987C2212yescomplemented
67#1970C2214yes—
68#1949C2214yescomplemented
69#1949C2214yescomplemented
70#1939C2212yescomplemented
71#1934C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1916C2214yescomplemented
75#1879C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1864C2214yes—
79#1862C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1846C2214yescomplemented
86#1835C2214yescomplemented
87#1817C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×D8C2312yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×#121C2311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—44no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2312yes—
115C2×#117—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#500—44no—
127#476—44no—
128#476—44no—
129#447—24no—
130C4×D8—44no—
131#431—28no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×C4×C8C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#400—22no—
153#398—22no—
154#397—22no—
155#397—22no—
156#393—24no—
157#393—24no—
158#393—24no—
159#393—24no—
160#393—22no—
161#391—21no—
162#391—22no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#390C2312yes—
168#390C2312yes—
169#390C2311yescharacteristic
170#389—21no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—21no—
181#388—22no—
182#388—22no—
183#387—22no—
184#386—22no—
185#386—21no—
186#384C2311yescharacteristic
187#383—22no—
188#382—21no—
189#382—22no—
190#382—22no—
191#382—22no—
192#382C2314yes—
193#382C2314yes—
194#382C2312yes—
195#382C2312yes—
196#381—22no—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—24no—
203#380—24no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380C2312yes—
208#380C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C25—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—44no—
241C22×Q8—82no—
242C22×Q8—82no—
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—42no—
249C22×D4C2411yescharacteristic
250C22×D4C2412yes—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2411yescharacteristic
264C2×D8—24no—
265C2×D8—24no—
266C2×D8—42no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—24no—
272C4⋊Q8—24no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275C4⋊Q8—44no—
276#123—84no—
277C22⋊Q8—82no—
278#121—22no—
279#121—22no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288#117—44no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—84no—
336C2×C22⋊C4—84no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351#107—42no—
352#107—44no—
353#107—44no—
354#106—44no—
355#106—44no—
356C4⋊C8—28no—
357C4⋊C8—24no—
358#102—28no—
359#102—24no—
360#102—24no—
361#102—42no—
362#102—42no—
363#102—44no—
364#102—44no—
365C22⋊C8—24no—
366C22⋊C8—22no—
367C22⋊C8#4114yes—
368C4×C8—28no—
369C4×C8—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—24no—
392C24—22no—
393C24—21no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D411yescharacteristic
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—81no—
409C2×Q8—81no—
410C2×Q8—816no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D42−1+414yes—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—816no—
465C22×C4#13514yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C42−1+412yes—
477C22×C4C2511yescharacteristic
478D8—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8C2×#4114yes—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—82no—
524C22⋊C4—81no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—44no—
541C42—42no—
542C42—82no—
543C42H(5,𝔽2)14yes—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—24no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23#54712yes—
570C23C2×#13512yes—
571C23#57612yes—
572C23C23×D411yescharacteristic
573Q8—88no—
574Q8—88no—
575Q8—162no—
576D4—24no—
577D4—24no—
578D4—48no—
579D4—42no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—84no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
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—21no—
608C2×C4—44no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—88no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—88no—
635C2×C4—84no—
636C2×C4—88no—
637C2×C4#53512yes—
638C2×C4#53512yes—
639C2×C4#54712yes—
640C2×C4C2×#13512yes—
641C2×C4C2×#13512yes—
642C2×C4#57512yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)12yes—
645C2×C4C2×2−1+412yes—
646C2×C4C2×2−1+411yescharacteristic
647C8—24no—
648C8—44no—
649C8—42no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—816no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22#334114yes—
672C22C2×#53511yescharacteristic
673C22C2×#54711yescharacteristic
674C22#348611yescharacteristic
675C22C22×#13511yescharacteristic
676C22C2×#57511yescharacteristic
677C22C2×#57611yescharacteristic
678C22#358511yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—41no—
687C4—81no—
688C4—81no—
689C4—82no—
690C4—81no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4#321114yes—
695C4#352012yes—
696C4#353812yes—
697C2—24no—
698C2—84no—
699C2—161no—
700C2#3891711yescharacteristic
701C2C2×#321111yescharacteristic
702C2#6081111yescharacteristic
703C2C2×#334111yescharacteristic
704C2#6125611yescharacteristic
705C2#6181911yescharacteristic
706C2#6183511yescharacteristic
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. #36176C2semidirect
2. C2×#2563C2semidirect
3. #32956C2semidirect
4. #32437C2semidirect
5. #32353C2semidirect
6. #29912C2semidirect
7. #28528C2semidirect
8. #27728C2semidirect
9. #27588C2semidirect
10. #25813C2semidirect
11. #25450C2semidirect
12. #24677C2semidirect
14. #24299C2semidirect
15. #22599C2semidirect
16. #21983C2semidirect
20. C2×#500C22semidirect
22. C2×#476C22semidirect
28. #2571C22semidirect
35. #2563C22semidirect
37. #2513C22semidirect
38. #2508C22semidirect
43. #2369C22semidirect
44. #2354C22semidirect
54. #2101C22semidirect
56. #2092C22semidirect
57. #2085C22semidirect
59. #2073C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
64. #2023C22semidirect
65. #1990C22semidirect
66. #1987C22semidirect
68. #1949C22semidirect
69. #1949C22semidirect
70. #1939C22semidirect
74. #1916C22semidirect
85. #1846C22semidirect
86. #1835C22semidirect
92. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#25632
#329562
#324372
#323532
#299122
#285282
#277282
#275882
#258132
#254502
#246772
#245002
#242992
#225992
#219832
C4×#4472