Finite Group Databasebeta

← #6500800

Subgroups — #6500800

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2415C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27616C212yesmaximal, complemented
4#27601C211yesmaximal, characteristic, complemented
5#27600C211yesmaximal, characteristic, complemented
6#27589C212yesmaximal, complemented
7#26205C214yesmaximal, complemented
8#26201C214yesmaximal, complemented
9#26133C214yesmaximal, complemented
10#26129C214yesmaximal, complemented
11#24006C211yesmaximal, characteristic, complemented
12#24003C211yesmaximal, characteristic, complemented
13#21762C212yesmaximal, complemented
14#21738C211yesmaximal, characteristic
15#20843C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#484C2212yescomplemented
21C2×#483C2212yescomplemented
22C2×#427C2214yescomplemented
23C22×C22⋊C8C2212yes—
24#2415—22no—
25#2415—22no—
26#2415C2214yes—
27#2415C2214yes—
28C2×#397—21no—
29C2×#394—21no—
30#2353—22no—
31#2353—22no—
32#2352C2212yes—
33#2352C2212yes—
34#2352C2212yes—
35#2352C2212yes—
36#2313—21no—
37#2312—21no—
38#2308—21no—
39C2×#391C2211yescharacteristic
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#388—21no—
43#2283C2212yes—
44#2280C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49#2101C2214yescomplemented
50#2097C2214yescomplemented
51#2085C2214yescomplemented
52#2077C2214yescomplemented
53#2075C2214yescomplemented
54#2074C2214yescomplemented
55#2036C2214yescomplemented
56#2035C2214yescomplemented
57#1928C2214yescomplemented
58#1927C2214yescomplemented
59#1921C2214yescomplemented
60#1920C2214yescomplemented
61#1892C2214yescomplemented
62#1892C2214yescomplemented
63#1873C2214yescomplemented
64#1873C2214yescomplemented
65#1872C2212yescomplemented
66#1872C2212yescomplemented
67#1869C2214yes—
68#1869C2214yes—
69#1854C2214yescomplemented
70#1854C2214yescomplemented
71#1853C2212yes—
72#1853C2212yes—
73#1853C2212yescomplemented
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yescomplemented
77#1824C2212yescomplemented
78#1823C2212yes—
79#1823C2212yes—
80#1823C2212yescomplemented
81#1817C2212yes—
82#1787C2212yes—
83#1787C2212yes—
84#1779C2211yescharacteristic
85#1779C2211yescharacteristic
86#1735C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C23×C8C2314yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C4×D4—28no—
101C2×C4×D4D412yes—
102C2×C4×D4D412yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×#117—22no—
106C2×#117—44no—
107C2×#117C2314yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
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
131#484—44no—
132#483—44no—
133#427—48no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×C22⋊C8—28no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150#397—24no—
151#397—24no—
152#394—24no—
153#394—24no—
154#391—22no—
155#391C2314yes—
156#389—24no—
157#389—24no—
158#389—22no—
159#389C2314yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—24no—
165#388—24no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#383—22no—
175#382C2312yes—
176#382C2312yes—
177#381—22no—
178#381—22no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380C2312yes—
186#380C2312yes—
187#380C2312yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#379—22no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203#376—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25C2×D411yescharacteristic
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—24no—
229C22×D4—42no—
230C22×D4C2×D412yes—
231C22×D4C2×D412yes—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4#4118yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—24no—
265C22⋊Q8—24no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—48no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277#117—28no—
278#117—44no—
279#117—48no—
280#117—44no—
281#117—44no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
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—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D412yes—
347C2×C42C2412yes—
348#107—28no—
349#107—44no—
350#106—28no—
351#106—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355#103—44no—
356C22⋊C8—24no—
357C22⋊C8—24no—
358C22⋊C8—28no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24#12314yes—
393C24#13514yes—
394C24C22×D412yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—88no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—416no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—416no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
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—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—88no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4#12314yes—
477C22×C4#12318yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—416no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—48no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4H(5,𝔽2)14yes—
516C22⋊C4—24no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—48no—
538C42—82no—
539C42—82no—
540C42—82no—
541C42—82no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—22no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—21no—
559C23—21no—
560C23—22no—
561C23—21no—
562C23—21no—
563C23—22no—
564C23—28no—
565C23—28no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23—82no—
577C23—82no—
578C23C2×#12314yes—
579C23C2×#12312yes—
580C23C2×#13512yes—
581C23C2×#13512yes—
582C23C2×#13512yes—
583C23#57512yes—
584C23C23×D411yescharacteristic
585Q8—44no—
586Q8—44no—
587Q8—84no—
588D4—48no—
589D4—48no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—48no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
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—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—84no—
644C2×C4—84no—
645C2×C4—84no—
646C2×C4—162no—
647C2×C4—162no—
648C2×C4—162no—
649C2×C4—162no—
650C2×C4C2×#12312yes—
651C2×C4C2×#12314yes—
652C2×C4C22×#4112yes—
653C2×C4C2×H(5,𝔽2)12yes—
654C2×C4#58512yes—
655C8—44no—
656C8—44no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—24no—
666C22—24no—
667C22—22no—
668C22—22no—
669C22—24no—
670C22—22no—
671C22—22no—
672C22—42no—
673C22—41no—
674C22—42no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—42no—
681C22—41no—
682C22—42no—
683C22—41no—
684C22—88no—
685C22—88no—
686C22#309114yes—
687C22C22×#12311yescharacteristic
688C22#348411yescharacteristic
689C22#348511yescharacteristic
690C22C22×#13511yescharacteristic
691C22C2×#57511yescharacteristic
692C22#358611yescharacteristic
693C22#358611yescharacteristic
694C4—22no—
695C4—24no—
696C4—41no—
697C4—41no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4#350514yes—
709C2—24no—
710C2—42no—
711C2—42no—
712C2—44no—
713C2#3535111yescharacteristic
714C2C2×#309111yescharacteristic
715C2#6051911yescharacteristic
716C2#6156411yescharacteristic
717C2#6157511yescharacteristic
718C2#6169611yescharacteristic
719C2#6171211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27616C2semidirect
4. #27601C2semidirect
5. #27600C2semidirect
6. #27589C2semidirect
7. #26205C2semidirect
8. #26201C2semidirect
9. #26133C2semidirect
10. #26129C2semidirect
11. #24006C2semidirect
12. #24003C2semidirect
13. #21762C2semidirect
15. #20843C2semidirect
20. C2×#484C22semidirect
21. C2×#483C22semidirect
22. C2×#427C22semidirect
49. #2101C22semidirect
50. #2097C22semidirect
51. #2085C22semidirect
52. #2077C22semidirect
53. #2075C22semidirect
54. #2074C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
57. #1928C22semidirect
58. #1927C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1892C22semidirect
62. #1892C22semidirect
63. #1873C22semidirect
64. #1873C22semidirect
65. #1872C22semidirect
66. #1872C22semidirect
69. #1854C22semidirect
70. #1854C22semidirect
73. #1853C22semidirect
76. #1852C22semidirect
77. #1824C22semidirect
80. #1823C22semidirect
86. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24152
#341852
#276162
#276012
#276002
#275892
#262052
#262012
#261332
#261292
#240062
#240032
#217622
#217382
#208432