Finite Group Databasebeta

← #6532013

Subgroups — #6532013

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34196C211yesmaximal, characteristic, complemented
3#27617C212yesmaximal, complemented
4#27611C212yesmaximal, complemented
5#27588C212yesmaximal
6#27585C212yesmaximal
7#25073C214yesmaximal, complemented
8#25059C212yesmaximal, complemented
9#25058C212yesmaximal, complemented
10#25006C214yesmaximal, complemented
11#24991C212yesmaximal, complemented
12#24990C212yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21981C211yesmaximal, characteristic, complemented
15#21820C211yesmaximal, characteristic, complemented
16#20738C211yesmaximal, characteristic
17#20733C211yesmaximal, characteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2538—21no—
24#2538—22no—
25#2538—21no—
26#2538C2212yes—
27#2538C2212yes—
28#2538C2214yes—
29C2×#396—22no—
30#2355C2211yescharacteristic, complemented
31#2355C2212yescomplemented
32#2355C2211yescharacteristic, complemented
33#2355C2212yescomplemented
34Q8×C4⋊C4C2211yescharacteristic, complemented
35Q8×C4⋊C4C2211yescharacteristic, complemented
36#2344—22no—
37#2344—21no—
38Q8×C22⋊C4—21no—
39#2320—22no—
40#2320—21no—
41C2×#390C2212yes—
42C2×#389C2212yes—
43#2285C2211yescharacteristic
44#2285C2212yes—
45C2×#384C2211yescharacteristic
46#2283C2212yes—
47#2283C2211yescharacteristic
48C2×#380—21no—
49#2276—21no—
50#2101C2214yescomplemented
51#2099C2214yescomplemented
52#2094C2214yes—
53#2092C2214yes—
54#2085C2214yescomplemented
55#2084C2214yescomplemented
56#2073C2214yes—
57#2071C2214yes—
58#1985C2214yescomplemented
59#1984C2212yescomplemented
60#1984C2212yescomplemented
61#1948C2212yescomplemented
62#1948C2214yescomplemented
63#1947C2212yescomplemented
64#1926C2214yes—
65#1926C2214yes—
66#1926C2214yes—
67#1925C2214yes—
68#1920C2214yescomplemented
69#1920C2214yescomplemented
70#1919C2214yescomplemented
71#1918C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1853C2212yes—
75#1852C2212yes—
76#1847C2212yescomplemented
77#1844C2212yescomplemented
78#1836C2212yescomplemented
79#1834C2212yescomplemented
80#1824C2212yes—
81#1824C2211yescharacteristic
82#1824C2211yescharacteristic
83#1817C2212yes—
84#1817C2212yes—
85#1804C2212yes—
86#1804C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1786C2212yes—
90#1786C2211yescharacteristic
91#1786C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94C4×#102C2212yes—
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C2×C22⋊Q8—22no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—48no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8D412yes—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4—24no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C2×#117—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D412yes—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—22no—
129C22×C22⋊C4—22no—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#432—24no—
133#431—24no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2311yescharacteristic
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151#396—24no—
152#396—24no—
153#396—24no—
154#396—24no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391—22no—
159#390—22no—
160#390—22no—
161#390C2314yes—
162#390C2314yes—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#384C2312yes—
184#384C2312yes—
185#383—22no—
186#383—21no—
187#382—22no—
188#382—21no—
189#382C2311yescharacteristic
190#382C2311yescharacteristic
191#382C2312yes—
192#382C2312yes—
193#381—22no—
194#381—21no—
195#380—22no—
196#380—21no—
197#380—22no—
198#380—24no—
199#380—22no—
200#380C2311yescharacteristic
201#380C2312yes—
202#380C2311yescharacteristic
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218#376—22no—
219#376—21no—
220C2×#95—22no—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235C25—21no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—42no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2×D412yes—
241C22×Q8C2×D411yescharacteristic
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—42no—
247C22×D4C2412yes—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2411yescharacteristic
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2414yes—
263C22⋊Q8—48no—
264C22⋊Q8—416no—
265C22⋊Q8—48no—
266C4×Q8—24no—
267C4×Q8—28no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—44no—
282C4×D4—44no—
283#117—82no—
284#117—84no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—48no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—84no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42#4114yes—
350C2×C42#4114yes—
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#106—44no—
357#106—44no—
358#106—44no—
359C4⋊C8—24no—
360C4⋊C8—24no—
361#102—28no—
362#102—24no—
363#102—24no—
364#102—42no—
365#102—44no—
366#102—42no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—24no—
370#97—24no—
371C4×C8—24no—
372#95—24no—
373#95—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392C24—24no—
393C24—22no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D411yescharacteristic
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—416no—
410C2×Q8AGL(1,ℤ/8)14yes—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D42−1+414yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—816no—
455C22×C4#12314yes—
456C22×C4#12314yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2511yescharacteristic
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C22⋊C4—48no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—44no—
527C42—44no—
528C42—42no—
529C42—42no—
530C42—82no—
531C42—82no—
532C42—82no—
533C42#12314yes—
534C42#12314yes—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—24no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23C2×#12312yes—
558C23#57712yes—
559C23#57712yes—
560C23C23×D411yescharacteristic
561Q8—28no—
562Q8—48no—
563Q8—44no—
564D4—24no—
565D4—24no—
566D4—48no—
567D4—42no—
568D4—42no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4C2×#12312yes—
623C2×C4C2×#12312yes—
624C2×C4C2×#12312yes—
625C2×C4C2×#12312yes—
626C2×C4C2×#12312yes—
627C2×C4C2×AGL(1,ℤ/8)14yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4#57512yes—
630C2×C4C22×#4111yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×2−1+412yes—
633C2×C4#58512yes—
634C8—42no—
635C8—42no—
636C8—44no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—816no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22#309914yes—
652C22C22×#12311yescharacteristic
653C22#348611yescharacteristic
654C22#348711yescharacteristic
655C22C22×AGL(1,ℤ/8)11yescharacteristic
656C22C2×#57511yescharacteristic
657C22C2×#57711yescharacteristic
658C22C2×#57711yescharacteristic
659C4—22no—
660C4—22no—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—84no—
675C4#309414yes—
676C4#349512yes—
677C4#349512yes—
678C2—24no—
679C2—84no—
680C2#3567211yescharacteristic
681C2C2×#309411yescharacteristic
682C2C2×#309911yescharacteristic
683C2#6184111yescharacteristic
684C2#6184311yescharacteristic
685C2#6189911yescharacteristic
686C2#6190111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34196C2semidirect
3. #27617C2semidirect
4. #27611C2semidirect
7. #25073C2semidirect
8. #25059C2semidirect
9. #25058C2semidirect
10. #25006C2semidirect
11. #24991C2semidirect
12. #24990C2semidirect
13. #21987C2semidirect
14. #21981C2semidirect
15. #21820C2semidirect
30. #2355C22semidirect
31. #2355C22semidirect
32. #2355C22semidirect
33. #2355C22semidirect
34. Q8×C4⋊C4C22semidirect
35. Q8×C4⋊C4C22semidirect
50. #2101C22semidirect
51. #2099C22semidirect
54. #2085C22semidirect
55. #2084C22semidirect
58. #1985C22semidirect
59. #1984C22semidirect
60. #1984C22semidirect
61. #1948C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
68. #1920C22semidirect
69. #1920C22semidirect
70. #1919C22semidirect
71. #1918C22semidirect
76. #1847C22semidirect
77. #1844C22semidirect
78. #1836C22semidirect
79. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341962
#276172
#276112
#275882
#275852
#250732
#250592
#250582
#250062
#249912
#249902
#219872
#219812
#218202
#207382
#207332