Finite Group Databasebeta

← #6482927

Subgroups — #6482927

Order 512.

1886 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#33100C214yesmaximal, complemented
4#29894C214yesmaximal
5#27892C212yesmaximal, complemented
6#27889C212yesmaximal, complemented
7#27877C212yesmaximal, complemented
8#27876C212yesmaximal, complemented
9#27631C214yesmaximal, complemented
10#27630C214yesmaximal, complemented
11#21982C211yesmaximal, characteristic, complemented
12#21981C211yesmaximal, characteristic, complemented
13#21860C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#20714C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2855—21no—
26#2855C2212yes—
27#2851—21no—
28#2851—21no—
29#2848—21no—
30#2848C2212yes—
31C4×C22⋊Q8—21no—
32C4×C22⋊Q8—21no—
33C4×C22⋊Q8C2212yes—
34C4×C22⋊Q8C2212yes—
35#2296—21no—
36#2285—21no—
37#2285—21no—
38#2285—21no—
39#2285—21no—
40#2285C2211yescharacteristic
41#2285C2212yes—
42#2285C2212yes—
43#2285C2212yes—
44#2285C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#382—21no—
48#2280—21no—
49C2×#380—21no—
50C2×#380C2212yes—
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2212yes—
53#2077C2218yes—
54#2075C2218yes—
55#2036C2218yes—
56#2035C2218yes—
57#1990C2212yescomplemented
58#1990C2212yescomplemented
59#1956C2212yescomplemented
60#1956C2212yescomplemented
61#1948C2214yescomplemented
62#1947C2214yescomplemented
63#1928C2218yescomplemented
64#1927C2218yescomplemented
65#1919C2218yescomplemented
66#1918C2218yescomplemented
67#1872C2214yes—
68#1872C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1845C2212yescomplemented
72#1844C2212yescomplemented
73#1834C2212yescomplemented
74#1834C2212yescomplemented
75#1817C2212yes—
76#1803C2214yescomplemented
77#1803C2214yescomplemented
78#1797C2212yes—
79#1797C2212yes—
80#1779C2212yes—
81C4×C22⋊C8C2212yes—
82C4×C22⋊C8C2212yes—
83C24×C4D411yescharacteristic
84C24×C4D411yescharacteristic
85C24×C4C2311yescharacteristic
86#533—24no—
87#533—24no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#125—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8—21no—
98C2×C4×Q8C2312yes—
99C2×#117—22no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—48no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—22no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#431—24no—
129#431—24no—
130C2×#107C2318yes—
131C2×#106C2318yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#391—21no—
143#391—21no—
144#391C2312yes—
145#391C2312yes—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389C2312yes—
153#389C2312yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#385—22no—
159#385—22no—
160#385—22no—
161#385—22no—
162#385—22no—
163#385—22no—
164#384—21no—
165#384—21no—
166#384—21no—
167#384—21no—
168#384C2312yes—
169#384C2312yes—
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2312yes—
173#384C2312yes—
174#382—21no—
175#382—22no—
176#382—22no—
177#382—22no—
178#382—22no—
179#381—22no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196#380C2314yes—
197#380C2312yes—
198#379—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C2×#95—22no—
214C2×#95—22no—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#340C2318yes—
219#336C2314yes—
220#336C2314yes—
221C25C2×D411yescharacteristic
222C22×Q8—21no—
223C22×Q8C2412yes—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260#125—44no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—44no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—44no—
271#117—48no—
272#117—48no—
273#117—48no—
274#117—44no—
275#117—44no—
276#117—48no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—48no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—48no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42#4114yes—
354C2×C42#4114yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—48no—
360#106—48no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363#103—48no—
364#103—48no—
365C22⋊C8—24no—
366C22⋊C8—24no—
367C22⋊C8—44no—
368C22⋊C8—44no—
369C4×C8—24no—
370C4×C8—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—41no—
391C24—41no—
392C24C22×D411yescharacteristic
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C22×C4—232no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
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—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4#12314yes—
468C22×C4#12314yes—
469C22×C4#13514yes—
470C22×C4AGL(1,ℤ/8)14yes—
471C22×C4#13714yes—
472C22×C4#13714yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2511yescharacteristic
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—48no—
545C42—44no—
546C42—44no—
547C42—48no—
548C42#12314yes—
549C42#12314yes—
550C42C2×#4112yes—
551C42C2×#4112yes—
552C42C2×#4112yes—
553C42C2×#4112yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—28no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23C2×#12312yes—
578C23C2×#13512yes—
579C23C2×AGL(1,ℤ/8)12yes—
580C23C2×#13712yes—
581C23C2×#13712yes—
582C23C23×D411yescharacteristic
583Q8—44no—
584Q8—44no—
585Q8—84no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
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—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4#485116yes—
647C2×C4C2×#12312yes—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4C2×#13512yes—
653C2×C4C2×#13512yes—
654C2×C4C2×AGL(1,ℤ/8)12yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×#13712yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4#58512yes—
663C2×C4#58512yes—
664C8—44no—
665C8—44no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—416no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22C2×#48514yes—
688C22C22×#12311yescharacteristic
689C22#344911yescharacteristic
690C22#344911yescharacteristic
691C22C22×#13511yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C22×#13711yescharacteristic
694C22C22×#13711yescharacteristic
695C4—22no—
696C4—22no—
697C4—22no—
698C4—22no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—82no—
704C4—84no—
705C4—82no—
706C4—84no—
707C4—82no—
708C4—82no—
709C4C2×#48514yes—
710C4C2×#48514yes—
711C4#309214yes—
712C4#349312yes—
713C4#349312yes—
714C2—24no—
715C2—42no—
716C2—42no—
717C2#3515711yescharacteristic
718C2C22×#48511yescharacteristic
719C2C2×#309211yescharacteristic
720C2#6094611yescharacteristic
721C2#6094711yescharacteristic
722C2#6096911yescharacteristic
723C2#6097111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33100C2semidirect
5. #27892C2semidirect
6. #27889C2semidirect
7. #27877C2semidirect
8. #27876C2semidirect
9. #27631C2semidirect
10. #27630C2semidirect
11. #21982C2semidirect
12. #21981C2semidirect
13. #21860C2semidirect
14. #21860C2semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1956C22semidirect
60. #1956C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1928C22semidirect
64. #1927C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
71. #1845C22semidirect
72. #1844C22semidirect
73. #1834C22semidirect
74. #1834C22semidirect
76. #1803C22semidirect
77. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22852
#331002
#298942
#278922
#278892
#278772
#278762
#276312
#276302
#219822
#219812
#218602
#207142