Finite Group Databasebeta

← #6792521

Subgroups — #6792521

Order 512.

2043 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36133C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#32905C211yesmaximal, characteristic
4#31636C214yesmaximal, complemented
5#31520C214yesmaximal, complemented
6#29908C212yesmaximal, complemented
7#28400C212yesmaximal, complemented
8#27813C212yesmaximal, complemented
9#25902C211yesmaximal, characteristic, complemented
10#25320C212yesmaximal, complemented
11#24813C212yesmaximal, complemented
12#24501C212yesmaximal
13#24301C211yesmaximal, characteristic, complemented
14#23933C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21860C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C2×#533C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8—21no—
21C22×C22⋊Q8—21no—
22C22×C4×D4C2211yescharacteristic
23C2×#487C2212yescomplemented
24C2×#475C2212yescomplemented
25C2×#461C2212yes—
26C2×#450C2211yescharacteristic
27C2×#450C2212yes—
28C2×C4×SD16C2212yes—
29C2×#431C2212yes—
30#2857C2211yescharacteristic
31#2855C2212yes—
32#2551C2211yescharacteristic, complemented
33#2546C2211yescharacteristic, complemented
34#2542—21no—
35#2542—22no—
36#2542—21no—
37#2542C2214yes—
38#2542C2212yes—
39#2542C2212yes—
40#2506—21no—
41#2462—22no—
42#2451C2212yescomplemented
43#2446C2212yescomplemented
44#2440—21no—
45#2438—22no—
46C2×#399—22no—
47#2365—21no—
48#2365C2211yescharacteristic, complemented
49Q8×C4⋊C4C2211yescharacteristic, complemented
50#2344—21no—
51C4×C22⋊Q8C2211yescharacteristic
52C2×#390C2212yes—
53C2×#384C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2099C2214yescomplemented
56#2096C2214yescomplemented
57#2084C2214yescomplemented
58#2072C2214yescomplemented
59#2065C2214yes—
60#2058C2212yescomplemented
61#2055C2214yes—
62#2024C2214yescomplemented
63#2024C2212yescomplemented
64#2023C2214yes—
65#2013C2214yes—
66#1990C2214yescomplemented
67#1990C2212yescomplemented
68#1988C2212yescomplemented
69#1968C2214yes—
70#1956C2214yescomplemented
71#1939C2212yescomplemented
72#1934C2214yes—
73#1916C2214yescomplemented
74#1880C2214yes—
75#1880C2214yes—
76#1877C2214yes—
77#1877C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1872C2212yes—
83#1864C2212yes—
84#1853C2212yes—
85#1853C2211yescharacteristic
86#1817C2212yes—
87#1803C2214yescomplemented
88#1803C2214yescomplemented
89#1802C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×SD16C2314yes—
100#533—28no—
101#533—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#125—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—48no—
120C2×C22⋊Q8—48no—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2311yescharacteristic
123C2×C4×Q8—22no—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—44no—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C2×#117—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4D411yescharacteristic
137C22×C4⋊C4D411yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2312yes—
140C22×C22⋊C4—21no—
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C42C2311yescharacteristic
144C22×C42C2312yes—
145#487—44no—
146#475—44no—
147#461—44no—
148#450—24no—
149#450—44no—
150C4×SD16—44no—
151#431—44no—
152C2×#106C2314yes—
153C2×#106C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2311yescharacteristic
167C2×C22⋊C8C2311yescharacteristic
168C2×C4×C8C2312yes—
169C2×C4×C8C2314yes—
170#400—22no—
171#399—24no—
172#399—24no—
173#399—24no—
174#399—24no—
175#398—22no—
176#398—22no—
177#398—22no—
178#395—22no—
179#395—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2311yescharacteristic
185#390—22no—
186#390—22no—
187#390C2314yes—
188#390C2314yes—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—22no—
196#389C2311yescharacteristic
197#389C2312yes—
198#389C2312yes—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—21no—
204#387—22no—
205#387—22no—
206#385—22no—
207#385—21no—
208#384—21no—
209#384—22no—
210#384—22no—
211#384C2311yescharacteristic
212#384C2314yes—
213#384C2314yes—
214#381—22no—
215#380—21no—
216#380—21no—
217#380C2311yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C2×#95D412yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#336C2312yes—
238#336C2314yes—
239#336C2314yes—
240#336C2312yes—
241C25—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—42no—
255C22×D4C2412yes—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×SD16—24no—
269C2×SD16—24no—
270C2×SD16—28no—
271C2×SD16—42no—
272C2×SD16—44no—
273C2×SD16—44no—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8—44no—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C4⋊Q8—44no—
286C4⋊Q8—44no—
287C4⋊Q8—44no—
288#125—48no—
289C22⋊Q8—22no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—48no—
293C22⋊Q8—48no—
294C22⋊Q8—416no—
295C22⋊Q8—416no—
296C22⋊Q8—48no—
297C22⋊Q8—48no—
298C22⋊Q8—48no—
299C22⋊Q8#4114yes—
300C4×Q8—24no—
301C4×Q8—24no—
302C4×Q8—24no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—48no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—44no—
311C4×D4—44no—
312#117—48no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—48no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—48no—
342C2×C4⋊C4SD1618yescomplemented
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—44no—
365C2×C22⋊C4—48no—
366C2×C22⋊C4—48no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382#106—42no—
383#106—44no—
384C4⋊C8—24no—
385C4⋊C8—44no—
386#103—42no—
387#103—44no—
388#103—44no—
389#103—44no—
390#103—44no—
391#102—24no—
392#102—24no—
393#102—42no—
394#102—44no—
395C22⋊C8—22no—
396C22⋊C8—22no—
397C22⋊C8—24no—
398C4×C8—24no—
399C4×C8—44no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24—24no—
418C24—22no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24C22×D411yescharacteristic
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—82no—
442C2×Q8—816no—
443C2×Q8C2×#4112yes—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—48no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D42−1+414yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—816no—
488C22×C4C2×SD1618yes—
489C22×C4C2×SD1614yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4C2511yescharacteristic
502SD16—48no—
503SD16—88no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—48no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—48no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—81no—
545C4⋊C4—81no—
546C4⋊C4—82no—
547C4⋊C4C2×SD1618yes—
548C4⋊C4C2×SD1614yes—
549C4⋊C4C2×SD1614yes—
550C4⋊C4AGL(1,ℤ/8)14yes—
551C4⋊C4C2×#4112yes—
552C22⋊C4—24no—
553C22⋊C4—44no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C42—22no—
562C42—22no—
563C42—24no—
564C42—24no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—44no—
571C42—44no—
572C42—42no—
573C42—42no—
574C42H(5,𝔽2)14yes—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—24no—
580C23—48no—
581C23—48no—
582C23—48no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23#54612yes—
594C23C22×SD1612yes—
595C23#57812yes—
596C23C23×D411yescharacteristic
597Q8—24no—
598Q8—42no—
599Q8—44no—
600Q8—44no—
601Q8—88no—
602Q8—88no—
603Q8—88no—
604D4—24no—
605D4—24no—
606D4—48no—
607D4—42no—
608D4—42no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—216no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—44no—
634C2×C4—48no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—88no—
658C2×C4—84no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4#53512yes—
662C2×C4#53512yes—
663C2×C4#54612yes—
664C2×C4C22×SD1614yes—
665C2×C4C22×SD1614yes—
666C2×C4C22×SD1612yes—
667C2×C4C22×SD1612yes—
668C2×C4C2×AGL(1,ℤ/8)14yes—
669C2×C4C2×AGL(1,ℤ/8)12yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)12yes—
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C2×C4C2×2−1+412yes—
674C8—42no—
675C8—42no—
676C8—44no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—816no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22#329814yes—
692C22C2×#53511yescharacteristic
693C22C2×#54611yescharacteristic
694C22#348611yescharacteristic
695C22C23×SD1611yescharacteristic
696C22C22×AGL(1,ℤ/8)11yescharacteristic
697C22C2×#57811yescharacteristic
698C22#358611yescharacteristic
699C4—22no—
700C4—22no—
701C4—24no—
702C4—24no—
703C4—24no—
704C4—22no—
705C4—41no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—82no—
710C4—84no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4#320814yes—
716C4#352012yes—
717C4#353612yes—
718C2—24no—
719C2—84no—
720C2#3890511yescharacteristic
721C2C2×#320811yescharacteristic
722C2#6081411yescharacteristic
723C2C2×#329811yescharacteristic
724C2#6116111yescharacteristic
725C2#6181811yescharacteristic
726C2#6183711yescharacteristic
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. #36133C2semidirect
4. #31636C2semidirect
5. #31520C2semidirect
6. #29908C2semidirect
7. #28400C2semidirect
8. #27813C2semidirect
9. #25902C2semidirect
10. #25320C2semidirect
11. #24813C2semidirect
13. #24301C2semidirect
15. #22599C2semidirect
16. #21860C2semidirect
23. C2×#487C22semidirect
24. C2×#475C22semidirect
32. #2551C22semidirect
33. #2546C22semidirect
42. #2451C22semidirect
43. #2446C22semidirect
48. #2365C22semidirect
49. Q8×C4⋊C4C22semidirect
55. #2099C22semidirect
56. #2096C22semidirect
57. #2084C22semidirect
58. #2072C22semidirect
60. #2058C22semidirect
62. #2024C22semidirect
63. #2024C22semidirect
66. #1990C22semidirect
67. #1990C22semidirect
68. #1988C22semidirect
70. #1956C22semidirect
71. #1939C22semidirect
73. #1916C22semidirect
87. #1803C22semidirect
88. #1803C22semidirect
92. C8×C4⋊C4C22semidirect
342. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361332
C2×#25422
#329052
#316362
#315202
#299082
#284002
#278132
#259022
#253202
#248132
#245012
#243012
#239332
#225992
#218602
C4×#4502