Finite Group Databasebeta

← #6471063

Subgroups — #6471063

Order 512.

1990 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2C2×#2360C211yesmaximal, characteristic, complemented
3#33147C211yesmaximal, characteristic, complemented
4#33107C211yesmaximal, characteristic, complemented
5#32979C212yesmaximal, complemented
6#29951C211yesmaximal, characteristic, complemented
7#29713C211yesmaximal, characteristic, complemented
8#29593C212yesmaximal, complemented
9#27034C214yesmaximal, complemented
10#26724C214yesmaximal, complemented
11#26629C214yesmaximal, complemented
12#26551C214yesmaximal, complemented
13#22076C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#448C212yesmaximal, complemented
16C2×#538—21no—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#481C2212yes—
21C2×#481C2214yes—
22C2×#448C2212yes—
23C2×C4×SD16C2214yescomplemented
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2847C2212yes—
27#2639—21no—
28#2617—21no—
29#2559—22no—
30#2557—22no—
31#2496C2212yes—
32#2491C2212yes—
33#2428C2212yes—
34#2422C2212yes—
35#2360—22no—
36#2360—22no—
37#2360C2214yes—
38#2360C2214yes—
39#2325—21no—
40C4×#121C2212yes—
41C2×#391C2211yescharacteristic
42C2×#389—21no—
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#385—21no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4—21no—
48C2×C4×C22⋊C4C2211yescharacteristic
49#2096C2212yescomplemented
50#2072C2212yescomplemented
51#2066C2214yescomplemented
52#2055C2214yescomplemented
53#2035C2214yes—
54#2035C2214yes—
55#2024C2212yescomplemented
56#2014C2212yescomplemented
57#2000C2214yes—
58#2000C2214yes—
59#1988C2212yescomplemented
60#1982C2212yescomplemented
61#1948C2212yescomplemented
62#1948C2212yescomplemented
63#1941C2214yescomplemented
64#1934C2214yescomplemented
65#1926C2214yescomplemented
66#1923C2214yes—
67#1923C2214yes—
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1880C2214yescomplemented
71#1874C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1865C2214yescomplemented
75#1864C2214yescomplemented
76#1853C2212yes—
77#1853C2212yes—
78#1853C2214yes—
79#1824C2212yes—
80#1824C2214yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1779C2212yes—
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×SD16C2314yes—
91#538—48no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#126—22no—
95C2×#124—22no—
96C2×#123—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8C2314yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—22no—
105C2×#121C2312yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#120—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—44no—
111C2×C4×Q8C2314yes—
112C2×C4×D4—22no—
113C2×C4×D4C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—44no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4D411yescharacteristic
126C22×C22⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#481—28no—
132#481—48no—
133#448—28no—
134C4×SD16—48no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2314yes—
151#400—22no—
152#397—22no—
153#397—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#393—22no—
158#392—22no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389—22no—
170#389—24no—
171#389—24no—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#387—22no—
178#387—24no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#385—24no—
186#385—24no—
187#385—22no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#380—21no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—24no—
200C4×C22⋊C4—24no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8#4114yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—82no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—42no—
262#127—42no—
263#126—48no—
264#124—48no—
265#123—48no—
266C22⋊Q8—216no—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C22⋊Q8—44no—
270#121—24no—
271#121—24no—
272#121—44no—
273#121—44no—
274#121—48no—
275#121—44no—
276#121—44no—
277#120—48no—
278C4×Q8—28no—
279C4×Q8—44no—
280C4×Q8—48no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
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—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4SD1618yescomplemented
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—42no—
338C2×C42—42no—
339C2×C42—44no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#106—28no—
348#106—28no—
349#106—44no—
350C4⋊C8—42no—
351C4⋊C8—42no—
352#103—28no—
353#103—44no—
354#103—44no—
355#102—28no—
356#102—28no—
357#102—44no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—44no—
361C22⋊C8—44no—
362C4×C8—44no—
363#95—24no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24C2×SD1614yes—
388C24C22×D412yes—
389C24C22×D411yescharacteristic
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—48no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—81no—
400C2×Q8—81no—
401C2×Q8C2×#4114yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—81no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—84no—
470C22×C4—816no—
471C22×C4C2×SD1614yes—
472C22×C4C2×SD1614yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484SD16—416no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C4⋊C4—24no—
492C4⋊C4—216no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—84no—
511C4⋊C4—81no—
512C4⋊C4C2×#4114yes—
513C4⋊C4H(5,𝔽2)14yes—
514C22⋊C4—216no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4C2×SD1618yes—
529C22⋊C4C2×SD1618yes—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—41no—
534C42—41no—
535C42—42no—
536C42—48no—
537C42—44no—
538C42—82no—
539C42—82no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—28no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23#52512yes—
565C23C22×SD1614yes—
566C23C22×SD1612yes—
567C23#57512yes—
568C23C23×D411yescharacteristic
569Q8—28no—
570Q8—48no—
571Q8—44no—
572Q8—162no—
573D4—44no—
574D4—44no—
575D4—88no—
576D4—162no—
577D4—162no—
578D4—164no—
579D4—164no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—48no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—816no—
635C2×C4—84no—
636C2×C4—164no—
637C2×C4#52512yes—
638C2×C4C22×SD1614yes—
639C2×C4C22×SD1612yes—
640C2×C4C22×SD1612yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×H(5,𝔽2)12yes—
644C2×C4#58512yes—
645C8—44no—
646C8—44no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—416no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22#310314yes—
669C22C2×#52511yescharacteristic
670C22#348411yescharacteristic
671C22#348511yescharacteristic
672C22C23×SD1611yescharacteristic
673C22C2×#57511yescharacteristic
674C22#358511yescharacteristic
675C22#358611yescharacteristic
676C4—22no—
677C4—22no—
678C4—24no—
679C4—22no—
680C4—41no—
681C4—41no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4#345912yes—
692C4#350612yes—
693C2—24no—
694C2—44no—
695C2—162no—
696C2#3498611yescharacteristic
697C2C2×#310311yescharacteristic
698C2#6056511yescharacteristic
699C2#6156211yescharacteristic
700C2#6157611yescharacteristic
701C2#6169711yescharacteristic
702C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2360C2semidirect
3. #33147C2semidirect
4. #33107C2semidirect
5. #32979C2semidirect
6. #29951C2semidirect
7. #29713C2semidirect
8. #29593C2semidirect
9. #27034C2semidirect
10. #26724C2semidirect
11. #26629C2semidirect
12. #26551C2semidirect
13. #22076C2semidirect
14. #21902C2semidirect
15. C4×#448C2semidirect
23. C2×C4×SD16C22semidirect
49. #2096C22semidirect
50. #2072C22semidirect
51. #2066C22semidirect
52. #2055C22semidirect
55. #2024C22semidirect
56. #2014C22semidirect
59. #1988C22semidirect
60. #1982C22semidirect
61. #1948C22semidirect
62. #1948C22semidirect
63. #1941C22semidirect
64. #1934C22semidirect
65. #1926C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
70. #1880C22semidirect
74. #1865C22semidirect
75. #1864C22semidirect
85. C4×#103C22semidirect
328. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23602
#331472
#331072
#329792
#299512
#297132
#295932
#270342
#267242
#266292
#265512
#220762
#219022
C4×#4482