Finite Group Databasebeta

← #6500826

Subgroups — #6500826

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2417C211yesmaximal, characteristic
2#34185C211yesmaximal, characteristic
3#27617C212yesmaximal, complemented
4#27603C211yesmaximal, characteristic, complemented
5#27601C211yesmaximal, characteristic, complemented
6#27589C212yesmaximal, complemented
7#26221C214yesmaximal, complemented
8#26217C214yesmaximal, complemented
9#26149C214yesmaximal, complemented
10#26145C214yesmaximal, complemented
11#24006C211yesmaximal, characteristic, complemented
12#24003C211yesmaximal, characteristic, complemented
13#21768C212yesmaximal, complemented
14#21742C211yesmaximal, characteristic
15#20844C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#484C2212yescomplemented
21C2×#483C2212yescomplemented
22C2×#428C2214yescomplemented
23C2×#407C2212yes—
24#2417—22no—
25#2417—22no—
26#2417C2214yes—
27#2417C2214yes—
28C2×#398—21no—
29C2×#396—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×#389C2211yescharacteristic
41C2×#386—21no—
42#2283C2212yes—
43#2280C2212yes—
44C2×#380—21no—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49#2101C2214yescomplemented
50#2097C2214yescomplemented
51#2085C2214yescomplemented
52#2077C2214yescomplemented
53#2076C2214yescomplemented
54#2074C2214yescomplemented
55#2036C2214yescomplemented
56#2035C2214yescomplemented
57#1928C2214yescomplemented
58#1927C2214yescomplemented
59#1919C2214yescomplemented
60#1918C2214yescomplemented
61#1894C2214yescomplemented
62#1894C2214yescomplemented
63#1873C2214yescomplemented
64#1873C2214yescomplemented
65#1872C2212yescomplemented
66#1872C2212yescomplemented
67#1870C2214yes—
68#1870C2214yes—
69#1853C2212yes—
70#1853C2212yes—
71#1853C2212yescomplemented
72#1853C2214yescomplemented
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yescomplemented
76#1852C2214yescomplemented
77#1824C2212yes—
78#1824C2212yes—
79#1824C2212yescomplemented
80#1823C2212yescomplemented
81#1817C2212yes—
82#1787C2212yes—
83#1787C2212yes—
84#1779C2211yescharacteristic
85#1779C2211yescharacteristic
86#1736C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2314yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8C2314yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C4×D4—28no—
102C2×C4×D4D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—44no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
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⋊C4—21no—
128C22×C22⋊C4D411yescharacteristic
129C22×C22⋊C4D411yescharacteristic
130C22×C42C2311yescharacteristic
131#484—44no—
132#483—44no—
133#428—48no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142#407—28no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150#398—24no—
151#398—24no—
152#396—24no—
153#396—24no—
154#391—22no—
155#391C2314yes—
156#389—22no—
157#389C2314yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#386—24no—
171#386—24no—
172#383—22no—
173#382C2312yes—
174#382C2312yes—
175#381—22no—
176#381—22no—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380—21no—
183#380—24no—
184#380—24no—
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—24no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4#4118yes—
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2412yes—
261C2×M16—28no—
262C2×M16—44no—
263C2×M16—44no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
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—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350#107—28no—
351#107—44no—
352#106—28no—
353#106—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#103—44no—
358C22⋊C8—24no—
359C22⋊C8—24no—
360C22⋊C8—28no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C22⋊C8—42no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—41no—
387C24—41no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24—41no—
393C24—41no—
394C24#12314yes—
395C24#13714yes—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398C24C22×D411yescharacteristic
399C24C22×D411yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—88no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—416no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
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—22no—
440C22×C4—22no—
441C22×C4—416no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—41no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—816no—
482C22×C4#12314yes—
483C22×C4#12318yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2511yescharacteristic
494M16—416no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—48no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—84no—
522C4⋊C4H(5,𝔽2)14yes—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—82no—
544C42—82no—
545C42—82no—
546C42—82no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—21no—
556C23—21no—
557C23—28no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—21no—
568C23—22no—
569C23—21no—
570C23—28no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—82no—
580C23—82no—
581C23—82no—
582C23—82no—
583C23C2×#12314yes—
584C23C2×#12312yes—
585C23C2×#13712yes—
586C23C2×#13712yes—
587C23C2×#13712yes—
588C23#57512yes—
589C23C23×D411yescharacteristic
590Q8—44no—
591Q8—44no—
592Q8—84no—
593D4—48no—
594D4—48no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—24no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—48no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
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—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—82no—
647C2×C4—82no—
648C2×C4—816no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—162no—
653C2×C4—162no—
654C2×C4—162no—
655C2×C4—162no—
656C2×C4C2×#12312yes—
657C2×C4C2×#12314yes—
658C2×C4C22×#4112yes—
659C2×C4C2×H(5,𝔽2)12yes—
660C2×C4#58512yes—
661C8—44no—
662C8—44no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—24no—
672C22—24no—
673C22—22no—
674C22—22no—
675C22—24no—
676C22—22no—
677C22—22no—
678C22—416no—
679C22—42no—
680C22—41no—
681C22—42no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—42no—
688C22—42no—
689C22—41no—
690C22—41no—
691C22—88no—
692C22#309514yes—
693C22C22×#12311yescharacteristic
694C22#348411yescharacteristic
695C22#348511yescharacteristic
696C22C22×#13711yescharacteristic
697C22C2×#57511yescharacteristic
698C22#358511yescharacteristic
699C22#358611yescharacteristic
700C4—22no—
701C4—24no—
702C4—41no—
703C4—41no—
704C4—42no—
705C4—42no—
706C4—44no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4#350514yes—
715C2—24no—
716C2—44no—
717C2—42no—
718C2—42no—
719C2#3535111yescharacteristic
720C2C2×#309511yescharacteristic
721C2#6052011yescharacteristic
722C2#6157411yescharacteristic
723C2#6158111yescharacteristic
724C2#6170911yescharacteristic
725C2#6171611yescharacteristic
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. #27617C2semidirect
4. #27603C2semidirect
5. #27601C2semidirect
6. #27589C2semidirect
7. #26221C2semidirect
8. #26217C2semidirect
9. #26149C2semidirect
10. #26145C2semidirect
11. #24006C2semidirect
12. #24003C2semidirect
13. #21768C2semidirect
15. #20844C2semidirect
20. C2×#484C22semidirect
21. C2×#483C22semidirect
22. C2×#428C22semidirect
49. #2101C22semidirect
50. #2097C22semidirect
51. #2085C22semidirect
52. #2077C22semidirect
53. #2076C22semidirect
54. #2074C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
57. #1928C22semidirect
58. #1927C22semidirect
59. #1919C22semidirect
60. #1918C22semidirect
61. #1894C22semidirect
62. #1894C22semidirect
63. #1873C22semidirect
64. #1873C22semidirect
65. #1872C22semidirect
66. #1872C22semidirect
71. #1853C22semidirect
72. #1853C22semidirect
75. #1852C22semidirect
76. #1852C22semidirect
79. #1824C22semidirect
80. #1823C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24172
#341852
#276172
#276032
#276012
#275892
#262212
#262172
#261492
#261452
#240062
#240032
#217682
#217422
#208442