Finite Group Databasebeta

← #6593971

Subgroups — #6593971

Order 512.

2059 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#33007C212yesmaximal
4#32953C211yesmaximal, characteristic
5#27960C211yesmaximal, characteristic, complemented
6#27647C211yesmaximal, characteristic, complemented
7#27616C212yesmaximal, complemented
8#27566C214yesmaximal, complemented
9#27312C214yesmaximal, complemented
10#26107C212yesmaximal, complemented
11#24816C212yesmaximal, complemented
12#24609C212yesmaximal, complemented
13#22968C212yesmaximal, complemented
14#22762C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#450C211yesmaximal, characteristic
17#20739C212yesmaximal
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4—21no—
21C2×#483C2212yes—
22C2×#474C2212yes—
23C2×#450C2211yescharacteristic
24C2×C4×SD16C2212yes—
25C2×#432C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2571C2211yescharacteristic, complemented
29#2568C2211yescharacteristic, complemented
30#2533—22no—
31#2514—21no—
32#2514C2212yescomplemented
33#2509C2212yescomplemented
34#2503—21no—
35#2493—22no—
36#2365C2211yescharacteristic, complemented
37#2360—21no—
38#2356—21no—
39#2355C2211yescharacteristic, complemented
40D4×C4⋊C4—21no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4C2214yes—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#389C2212yes—
48C2×#385—22no—
49C2×#384C2211yescharacteristic
50#2283C2212yes—
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2101C2214yescomplemented
54#2099C2214yescomplemented
55#2088C2214yescomplemented
56#2085C2214yescomplemented
57#2084C2212yescomplemented
58#2072C2212yescomplemented
59#2068C2214yescomplemented
60#1990C2212yescomplemented
61#1948C2212yescomplemented
62#1943C2214yescomplemented
63#1939C2212yescomplemented
64#1935C2214yescomplemented
65#1921C2214yescomplemented
66#1920C2214yescomplemented
67#1917C2214yescomplemented
68#1877C2212yes—
69#1877C2214yes—
70#1877C2214yes—
71#1872C2211yescharacteristic
72#1864C2212yes—
73#1864C2214yes—
74#1864C2214yes—
75#1862C2214yes—
76#1853C2211yescharacteristic
77#1852C2212yes—
78#1834C2212yescomplemented
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2212yes—
83#1804C2214yes—
84#1804C2214yes—
85#1803C2212yescomplemented
86#1802C2212yes—
87#1797C2211yescharacteristic
88#1786C2211yescharacteristic
89#1786C2212yes—
90#1770C2212yescomplemented
91#1762C2214yes—
92#1762C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C24×C4—21no—
97C24×C4—22no—
98C22×SD16C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C4×Q8—21no—
103C2×C4×Q8C2311yescharacteristic
104C2×C4×D4—28no—
105C2×C4×D4—28no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—44no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2314yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—48no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—22no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#483—44no—
131#474—28no—
132#450—24no—
133C4×SD16—28no—
134#432—44no—
135C2×#107C2314yes—
136C2×#106C2312yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2311yescharacteristic
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#401—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#400—22no—
157#400—22no—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2311yescharacteristic
164#391C2312yes—
165#389—21no—
166#389—22no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389C2312yes—
171#389C2314yes—
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—21no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—21no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#383—22no—
195#383—22no—
196#382—21no—
197#382—21no—
198#382C2312yes—
199#382C2312yes—
200#381—22no—
201#381—22no—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—24no—
206#380C2312yes—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D412yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221#376—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2312yes—
237C25—21no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4#4114yes—
245C22×D4C2412yes—
246C23×C4—28no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—44no—
254C23×C4—48no—
255C23×C4—48no—
256C23×C4—44no—
257C23×C4—84no—
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×SD16—24no—
264C2×SD16—24no—
265C2×SD16—24no—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—44no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—48no—
285C4×D4—48no—
286C4×D4#4118yes—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—416no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—48no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—82no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—48no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352#107—28no—
353#106—24no—
354#106—44no—
355C4⋊C8—24no—
356C4⋊C8—44no—
357#103—24no—
358#103—44no—
359#102—24no—
360#102—24no—
361#102—28no—
362#102—28no—
363#102—24no—
364C22⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—24no—
367#97—44no—
368C4×C8—24no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—24no—
395C24—24no—
396C24—24no—
397C24—22no—
398C24—42no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24C22×D411yescharacteristic
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—216no—
413C2×D4—216no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4C2×#4114yes—
419C2×D4C2×#4114yes—
420C2×D42−1+414yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—41no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—432no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—832no—
462C22×C4AGL(1,ℤ/8)14yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4C2511yescharacteristic
475SD16—48no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—42no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4C2×#4114yes—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—44no—
535C42—42no—
536C42—42no—
537C42—82no—
538C42C2×#4114yes—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—216no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—48no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—42no—
553C23—42no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23#54612yes—
558C23C2×AGL(1,ℤ/8)12yes—
559C23#57712yes—
560C23C23×D411yescharacteristic
561Q8—42no—
562Q8—42no—
563Q8—84no—
564D4—216no—
565D4—24no—
566D4—24no—
567D4—24no—
568D4#55218yes—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
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—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—416no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—42no—
600C2×C4—416no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—84no—
623C2×C4—816no—
624C2×C4—816no—
625C2×C4—88no—
626C2×C4#52512yes—
627C2×C4#52512yes—
628C2×C4#54612yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57512yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C22×#4112yes—
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+412yes—
636C8—42no—
637C8—42no—
638C8—44no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—28no—
646C22—432no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22#332414yes—
654C22C2×#52511yescharacteristic
655C22C2×#54611yescharacteristic
656C22#348611yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57511yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358611yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—44no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—81no—
675C4—84no—
676C4—81no—
677C4#311414yes—
678C4C2×#55212yes—
679C4#349212yes—
680C2—24no—
681C2—48no—
682C2#3616311yescharacteristic
683C2C2×#311411yescharacteristic
684C2#6058511yescharacteristic
685C2C2×#332411yescharacteristic
686C2#6115911yescharacteristic
687C2#6182711yescharacteristic
688C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36177C2semidirect
5. #27960C2semidirect
6. #27647C2semidirect
7. #27616C2semidirect
8. #27566C2semidirect
9. #27312C2semidirect
10. #26107C2semidirect
11. #24816C2semidirect
12. #24609C2semidirect
13. #22968C2semidirect
14. #22762C2semidirect
15. #22603C2semidirect
28. #2571C22semidirect
29. #2568C22semidirect
32. #2514C22semidirect
33. #2509C22semidirect
36. #2365C22semidirect
39. #2355C22semidirect
53. #2101C22semidirect
54. #2099C22semidirect
55. #2088C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
59. #2068C22semidirect
60. #1990C22semidirect
61. #1948C22semidirect
62. #1943C22semidirect
63. #1939C22semidirect
64. #1935C22semidirect
65. #1921C22semidirect
66. #1920C22semidirect
67. #1917C22semidirect
78. #1834C22semidirect
85. #1803C22semidirect
90. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×D4×C4⋊C42
#330072
#329532
#279602
#276472
#276162
#275662
#273122
#261072
#248162
#246092
#229682
#227622
#226032
C4×#4502
#207392