Finite Group Databasebeta

← #6635482

Subgroups — #6635482

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2C2×#2421C211yesmaximal, characteristic, complemented
3#32909C212yesmaximal, complemented
4#31943C214yesmaximal, complemented
5#31939C214yesmaximal, complemented
6#30998C214yesmaximal, complemented
7#30996C214yesmaximal, complemented
8#29760C212yesmaximal, complemented
9#28306C212yesmaximal, complemented
10#27977C212yesmaximal, complemented
11#24003C211yesmaximal, characteristic, complemented
12#24002C211yesmaximal, characteristic
13#23921C211yesmaximal, characteristic, complemented
14#23919C211yesmaximal, characteristic
15#21902C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic, complemented
20C2×#483C2212yescomplemented
21C2×#480C2212yes—
22C2×#464C2214yescomplemented
23C2×#451C2212yescomplemented
24C2×#451C2212yescomplemented
25C2×#448C2212yes—
26C2×#450C2212yes—
27#2850C2212yes—
28#2850C2212yes—
29#2639—21no—
30#2639—21no—
31#2591—22no—
32#2561—22no—
33#2504C2212yes—
34#2503C2212yes—
35#2436C2212yes—
36#2435C2212yes—
37#2421—22no—
38#2421—22no—
39#2421C2214yes—
40#2421C2214yescomplemented
41C2×#400—21no—
42C2×#397—21no—
43#2308—21no—
44#2308—21no—
45C2×#391C2211yescharacteristic, complemented
46C2×#389C2211yescharacteristic, complemented
47C2×#388—21no—
48C2×#387—21no—
49C2×#382C2211yescharacteristic, complemented
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#2097C2214yescomplemented
53#2085C2214yescomplemented
54#2074C2214yescomplemented
55#2073C2214yescomplemented
56#2068C2214yescomplemented
57#2065C2214yescomplemented
58#2042C2214yescomplemented
59#2035C2214yes—
60#2033C2214yes—
61#2015C2214yescomplemented
62#2001C2214yes—
63#2000C2214yes—
64#1969C2214yescomplemented
65#1968C2214yescomplemented
66#1958C2214yescomplemented
67#1923C2214yes—
68#1923C2214yescomplemented
69#1880C2214yescomplemented
70#1878C2214yescomplemented
71#1874C2212yes—
72#1874C2212yes—
73#1872C2212yes—
74#1872C2212yescomplemented
75#1872C2212yescomplemented
76#1872C2212yescomplemented
77#1853C2212yescomplemented
78#1852C2212yes—
79#1835C2214yescomplemented
80#1805C2214yescomplemented
81#1797C2212yescomplemented
82#1787C2212yescomplemented
83#1786C2212yes—
84#1779C2211yescharacteristic
85#1779C2211yescharacteristic
86C4×C22⋊C8C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—44no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121—44no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×C4×Q8—22no—
112C2×C4×Q8—44no—
113C2×C4×Q8C2314yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#483—44no—
133#480—44no—
134#464—48no—
135#451—44no—
136#451—44no—
137#450—44no—
138#448—44no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C4×C8C2314yes—
157#400—22no—
158#400—22no—
159#400—24no—
160#400—24no—
161#397—24no—
162#397—24no—
163#397—22no—
164#397—22no—
165#394—22no—
166#394—22no—
167#393—22no—
168#393—22no—
169#391—22no—
170#391C2312yes—
171#391C2312yes—
172#391C2314yes—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389C2314yes—
177#388—24no—
178#388—24no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#382—22no—
194#382C2312yes—
195#382C2314yes—
196#382C2312yes—
197#381—22no—
198#381—22no—
199#380—21no—
200#380—21no—
201#380C2312yes—
202#380C2312yes—
203#379—22no—
204#379—22no—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4D412yes—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95D411yescharacteristic
216C2×#95D411yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C25C2×D411yescharacteristic
225C22×Q8—24no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—24no—
229C22×Q8—42no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4—44no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—44no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273#124—44no—
274#124—44no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—42no—
280C22⋊Q8—42no—
281C22⋊Q8—42no—
282C22⋊Q8—42no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286C22⋊Q8—44no—
287#121—24no—
288#121—24no—
289#121—42no—
290#121—42no—
291#121—44no—
292#121—44no—
293C4×Q8—28no—
294C4×Q8—44no—
295C4×Q8—48no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
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—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—42no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—24no—
348C2×C42—42no—
349C2×C42—42no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355#107—44no—
356#106—44no—
357C4⋊C8—44no—
358#103—44no—
359#103—44no—
360#103—44no—
361#103—44no—
362#103—44no—
363#102—44no—
364#102—44no—
365#102—44no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C22⋊C8—42no—
369C22⋊C8—42no—
370C22⋊C8—42no—
371C22⋊C8—42no—
372C4×C8—44no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—41no—
396C24—41no—
397C24—41no—
398C24—41no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—81no—
403C24—81no—
404C24—81no—
405C24—81no—
406C24#13514yes—
407C24C22×D411yescharacteristic
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—416no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—42no—
422C2×Q8—81no—
423C2×Q8—81no—
424C2×Q8—88no—
425C2×Q8H(5,𝔽2)14yes—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—88no—
440C2×D4—816no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
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—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—88no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4H(5,𝔽2)14yes—
507C22×C4C2511yescharacteristic
508Q16—88no—
509SD16—88no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—24no—
556C42—24no—
557C42—42no—
558C42—42no—
559C42—48no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—28no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—82no—
587C23—82no—
588C23—82no—
589C23—82no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—162no—
594C23—162no—
595C23—162no—
596C23—162no—
597C23#53512yes—
598C23C2×#13512yes—
599C23C2×#13512yes—
600C23C2×#13512yes—
601C23#57512yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—48no—
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—84no—
609D4—44no—
610D4—44no—
611D4—84no—
612D4—88no—
613D4—88no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—48no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—82no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—82no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4#53512yes—
673C2×C4C2×H(5,𝔽2)12yes—
674C2×C4C2×H(5,𝔽2)12yes—
675C2×C4#58512yes—
676C8—44no—
677C8—44no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22—88no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—168no—
707C22#321114yes—
708C22C2×#53511yescharacteristic
709C22#348411yescharacteristic
710C22#348511yescharacteristic
711C22C22×#13511yescharacteristic
712C22C2×#57511yescharacteristic
713C22#358611yescharacteristic
714C22#358611yescharacteristic
715C4—22no—
716C4—42no—
717C4—42no—
718C4—41no—
719C4—41no—
720C4—44no—
721C4—42no—
722C4—42no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4#353414yes—
730C2—24no—
731C2—42no—
732C2—42no—
733C2—162no—
734C2#3672111yescharacteristic
735C2C2×#321111yescharacteristic
736C2#6080911yescharacteristic
737C2#6156411yescharacteristic
738C2#6158311yescharacteristic
739C2#6169611yescharacteristic
740C2#6171211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#2421C2semidirect
3. #32909C2semidirect
4. #31943C2semidirect
5. #31939C2semidirect
6. #30998C2semidirect
7. #30996C2semidirect
8. #29760C2semidirect
9. #28306C2semidirect
10. #27977C2semidirect
11. #24003C2semidirect
13. #23921C2semidirect
15. #21902C2semidirect
19. C22×C4×Q8C22semidirect
20. C2×#483C22semidirect
22. C2×#464C22semidirect
23. C2×#451C22semidirect
24. C2×#451C22semidirect
40. #2421C22semidirect
45. C2×#391C22semidirect
46. C2×#389C22semidirect
49. C2×#382C22semidirect
52. #2097C22semidirect
53. #2085C22semidirect
54. #2074C22semidirect
55. #2073C22semidirect
56. #2068C22semidirect
57. #2065C22semidirect
58. #2042C22semidirect
61. #2015C22semidirect
64. #1969C22semidirect
65. #1968C22semidirect
66. #1958C22semidirect
68. #1923C22semidirect
69. #1880C22semidirect
70. #1878C22semidirect
74. #1872C22semidirect
75. #1872C22semidirect
76. #1872C22semidirect
77. #1853C22semidirect
79. #1835C22semidirect
80. #1805C22semidirect
81. #1797C22semidirect
82. #1787C22semidirect
86. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#350192
C2×#24212
#329092
#319432
#319392
#309982
#309962
#297602
#283062
#279772
#240032
#240022
#239212
#239192
#219022