Finite Group Databasebeta

← #6487674

Subgroups — #6487674

Order 512.

2452 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3114C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
4#32908C212yesmaximal, complemented
5#32905C211yesmaximal, characteristic, complemented
6#32902C211yesmaximal, characteristic
7#29760C212yesmaximal, complemented
8#29756C211yesmaximal, characteristic, complemented
9#29754C211yesmaximal, characteristic, complemented
10#24825C214yesmaximal, complemented
11#24823C214yesmaximal, complemented
12#24740C214yesmaximal, complemented
13#24738C214yesmaximal, complemented
14#21955C211yesmaximal, characteristic, complemented
15#21867C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4—21no—
21#3114—216no—
22C2×#483C2212yes—
23C2×#480C2212yes—
24C2×#478C2214yes—
25C2×#463C2214yes—
26C2×#463C2214yes—
27C2×#461C2212yes—
28C2×#459C2212yes—
29C2×#450C2212yes—
30C2×#447C2212yes—
31C2×#442C2214yes—
32C2×#440C2214yes—
33C2×#432C2212yes—
34#2857C2211yescharacteristic
35#2856C2211yescharacteristic
36#2850C2212yes—
37#2618—21no—
38#2618—21no—
39#2601—22no—
40#2601—22no—
41#2566C2212yescomplemented
42#2565C2212yescomplemented
43#2545C2212yescomplemented
44#2544C2212yescomplemented
45D4×C4⋊C4—22no—
46D4×C4⋊C4—22no—
47D4×C4⋊C4C2214yes—
48D4×C4⋊C4C2214yes—
49#2325—21no—
50#2325—21no—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#385—21no—
54C2×#385—21no—
55C2×#384C2211yescharacteristic
56C2×#380—21no—
57C2×C4×C4⋊C4C2211yescharacteristic
58#2085C2214yescomplemented
59#2084C2212yescomplemented
60#2083C2212yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2066C2214yescomplemented
64#2065C2214yescomplemented
65#2043C2212yescomplemented
66#2042C2214yescomplemented
67#2041C2212yescomplemented
68#1992C2214yescomplemented
69#1959C2214yescomplemented
70#1940C2214yescomplemented
71#1940C2214yescomplemented
72#1879C2214yes—
73#1879C2214yes—
74#1877C2214yes—
75#1877C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1804C2214yes—
81#1803C2214yescomplemented
82#1802C2212yes—
83#1802C2212yes—
84#1802C2214yescomplemented
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1797C2212yes—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4—21no—
91C24×C4—21no—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96#529—28no—
97#529—28no—
98C2×#126—22no—
99C2×#125C2312yes—
100C2×#124C2312yes—
101C2×#124C2312yes—
102C2×#123—22no—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—44no—
115C2×C4×D4—44no—
116C2×C4×D4D418yescomplemented
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2314yes—
121C2×C4×D4C2314yes—
122C2×#117—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—24no—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—48no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#483—28no—
142#480—28no—
143#478—216no—
144#463—216no—
145#463—48no—
146#461—44no—
147#459—44no—
148#450—28no—
149#447—28no—
150#442—216no—
151#440—216no—
152#432—28no—
153C2×#107C2314yes—
154C2×#106C2314yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2312yes—
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2312yes—
169C2×#97C2314yes—
170#398—22no—
171#398—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#394—22no—
177#394—22no—
178#391C2312yes—
179#391C2312yes—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389C2314yes—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—22no—
197#387—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#385—22no—
203#385—22no—
204#385—24no—
205#385—24no—
206#385—24no—
207#385—24no—
208#384—22no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2314yes—
212#382C2312yes—
213#380—21no—
214#380—21no—
215#380—24no—
216#380—24no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C2×#95—21no—
222C2×#95—21no—
223C2×#95D411yescharacteristic
224C2×#95D411yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232C25—21no—
233C22×Q8C2412yes—
234C22×D4—24no—
235C22×D4—24no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4C2×D414yes—
242C22×D4C2412yes—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—28no—
246C23×C4—24no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—82no—
260C23×C4—82no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×SD16—28no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×D8—28no—
272C2×D8—28no—
273C2×D8—28no—
274C22×C8C2414yes—
275C22×C8C2414yes—
276C4⋊Q8—42no—
277#127—42no—
278#126—48no—
279#125—24no—
280#125—24no—
281#124—24no—
282#124—24no—
283#124—44no—
284#123—48no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287#121—24no—
288#121—24no—
289#121—44no—
290#121—44no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—28no—
296C4×D4—216no—
297C4×D4—24no—
298C4×D4—28no—
299C4×D4—24no—
300C4×D4—44no—
301C4×D4—42no—
302C4×D4—42no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—48no—
306C4×D4—48no—
307C4×D4C2×D4116yes—
308#117—48no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—42no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—416no—
332C2×C4⋊C4—416no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—44no—
365C2×C22⋊C4—44no—
366C2×C42—22no—
367C2×C42—42no—
368C2×C42—42no—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2×D414yes—
374C2×C42#4114yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2412yes—
379#107—28no—
380#106—28no—
381C4⋊C8—28no—
382C4⋊C8—42no—
383C4⋊C8—42no—
384C4⋊C8—44no—
385#103—28no—
386#103—44no—
387#103—44no—
388#102—28no—
389#102—28no—
390#102—28no—
391#102—44no—
392#102—44no—
393C22⋊C8—24no—
394C22⋊C8—24no—
395#97—28no—
396#95—24no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—24no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—22no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24C22×D411yescharacteristic
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—216no—
426C2×D4—216no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—42no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—81no—
443C2×D4—81no—
444C2×D4C22×D414yes—
445C2×D4H(5,𝔽2)14yes—
446C2×D42−1+414yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—216no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—48no—
473C22×C4—432no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—84no—
495C22×C4—816no—
496C22×C4—816no—
497C22×C4#12114yes—
498C22×C4#12114yes—
499C22×C4AGL(1,ℤ/8)14yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4112yes—
511C22×C4C2511yescharacteristic
512SD16—416no—
513D8—416no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—81no—
546C4⋊C4—81no—
547C4⋊C4—84no—
548C4⋊C4—81no—
549C4⋊C4—81no—
550C4⋊C4C22×D414yes—
551C22⋊C4—216no—
552C22⋊C4—48no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—41no—
570C42—41no—
571C42—42no—
572C42—44no—
573C42#12118yes—
574C42C22×D414yes—
575C42C2×#4114yes—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—216no—
581C23—24no—
582C23—24no—
583C23—24no—
584C23—48no—
585C23—48no—
586C23—48no—
587C23—48no—
588C23—48no—
589C23—42no—
590C23—42no—
591C23—48no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23C2×#12112yes—
602C23C2×AGL(1,ℤ/8)12yes—
603C23#57512yes—
604C23C23×D411yescharacteristic
605Q8—44no—
606Q8—44no—
607D4—216no—
608D4—24no—
609D4—24no—
610D4—24no—
611D4—44no—
612D4—44no—
613D4—88no—
614D4—162no—
615D4—162no—
616D4#53718yes—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—28no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—22no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—432no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—88no—
666C2×C4—88no—
667C2×C4—88no—
668C2×C4—82no—
669C2×C4—82no—
670C2×C4—816no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4C2×#12114yes—
674C2×C4C2×#12112yes—
675C2×C4C2×#12114yes—
676C2×C4C2×AGL(1,ℤ/8)12yes—
677C2×C4C2×AGL(1,ℤ/8)12yes—
678C2×C4C23×D411yescharacteristic
679C2×C4C22×#4111yescharacteristic
680C2×C4C2×H(5,𝔽2)12yes—
681C2×C4C2×2−1+412yes—
682C8—44no—
683C8—44no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—28no—
691C22—432no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22#303614yes—
706C22C22×#12111yescharacteristic
707C22#348411yescharacteristic
708C22#348611yescharacteristic
709C22C22×AGL(1,ℤ/8)11yescharacteristic
710C22C2×#57511yescharacteristic
711C22#358511yescharacteristic
712C22#358611yescharacteristic
713C4—24no—
714C4—22no—
715C4—22no—
716C4—22no—
717C4—24no—
718C4—42no—
719C4—42no—
720C4—41no—
721C4—41no—
722C4—44no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4C2×#53712yes—
729C4#349212yes—
730C2—24no—
731C2—48no—
732C2—162no—
733C2#3522311yescharacteristic
734C2C2×#303611yescharacteristic
735C2#6042511yescharacteristic
736C2#6157111yescharacteristic
737C2#6157511yescharacteristic
738C2#6182711yescharacteristic
739C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
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. C2×#3114C2semidirect
2. #35103C2semidirect
3. C2×D4×C4⋊C4C2semidirect
4. #32908C2semidirect
5. #32905C2semidirect
7. #29760C2semidirect
8. #29756C2semidirect
9. #29754C2semidirect
10. #24825C2semidirect
11. #24823C2semidirect
12. #24740C2semidirect
13. #24738C2semidirect
14. #21955C2semidirect
15. #21867C2semidirect
41. #2566C22semidirect
42. #2565C22semidirect
43. #2545C22semidirect
44. #2544C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2083C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2066C22semidirect
64. #2065C22semidirect
65. #2043C22semidirect
66. #2042C22semidirect
67. #2041C22semidirect
68. #1992C22semidirect
69. #1959C22semidirect
70. #1940C22semidirect
71. #1940C22semidirect
81. #1803C22semidirect
84. #1802C22semidirect
116. C2×C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31142
#351032
C2×D4×C4⋊C42
#329082
#329052
#329022
#297602
#297562
#297542
#248252
#248232
#247402
#247382
#219552
#218672