Finite Group Databasebeta

← #6482040

Subgroups — #6482040

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32983C212yesmaximal, complemented
4#32982C212yesmaximal, complemented
5#29894C212yesmaximal, complemented
6#29894C212yesmaximal, complemented
7#26298C212yesmaximal, complemented
8#26296C212yesmaximal, complemented
9#26236C212yesmaximal, complemented
10#26234C212yesmaximal, complemented
11#25024C214yesmaximal, complemented
12#24979C214yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#21788C211yesmaximal, characteristic
16#21394C211yesmaximal, characteristic, complemented
17#21361C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C2×#484C2214yescomplemented
23C2×#483C2214yescomplemented
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2848C2212yes—
27#2848C2212yes—
28#2633—22no—
29#2547C2211yescharacteristic
30#2547C2211yescharacteristic
31#2542C2211yescharacteristic
32#2542C2211yescharacteristic
33#2536C2212yes—
34#2536C2212yes—
35#2502—21no—
36#2482—22no—
37#2435—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4—21no—
40D4×C4⋊C4—22no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2214yes—
44#2326—21no—
45#2326—21no—
46C2×#389—21no—
47C2×#385—22no—
48C2×#384C2211yescharacteristic
49#2283C2211yescharacteristic
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2075C2214yescomplemented
55#2075C2214yescomplemented
56#2036C2214yescomplemented
57#2036C2214yescomplemented
58#2035C2214yescomplemented
59#2035C2214yescomplemented
60#1990C2212yescomplemented
61#1988C2212yescomplemented
62#1981C2212yescomplemented
63#1956C2212yescomplemented
64#1955C2214yes—
65#1954C2212yes—
66#1954C2212yes—
67#1949C2212yescomplemented
68#1949C2212yescomplemented
69#1948C2212yescomplemented
70#1947C2212yescomplemented
71#1872C2212yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2214yescomplemented
76#1872C2214yescomplemented
77#1853C2211yescharacteristic
78#1853C2211yescharacteristic
79#1853C2212yes—
80#1853C2214yescomplemented
81#1852C2211yescharacteristic
82#1852C2211yescharacteristic
83#1852C2212yes—
84#1852C2214yescomplemented
85#1847C2212yescomplemented
86#1836C2212yescomplemented
87#1824C2214yescomplemented
88#1823C2214yescomplemented
89#1803C2214yescomplemented
90#1779C2212yes—
91#1779C2212yes—
92C4×#107C2212yescomplemented
93C4×#106C2212yescomplemented
94#1763C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4—22no—
97C24×C4C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#124—21no—
104C2×#124—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C22⋊Q8C2312yes—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—24no—
117C2×C4×D4—24no—
118C2×C4×D4—48no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D412yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—48no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2314yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C22⋊C4D412yes—
140C22×C42—21no—
141C22×C42C2311yescharacteristic
142#484—48no—
143#483—48no—
144#432—24no—
145#431—24no—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#106C2312yes—
150C2×#106C2314yes—
151C2×#106C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×#97C2312yes—
166C2×C4×C8C2312yes—
167#399—22no—
168#399—22no—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#394—22no—
174#394—22no—
175#391—21no—
176#391—21no—
177#390—21no—
178#390—21no—
179#389—22no—
180#389—22no—
181#389—24no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384—21no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384C2312yes—
201#384C2312yes—
202#383—21no—
203#382C2311yescharacteristic
204#382C2311yescharacteristic
205#381—21no—
206#380—22no—
207#380—22no—
208#380C2312yes—
209#380C2312yes—
210#380C2314yes—
211#380C2312yes—
212#380C2312yes—
213#380C2314yes—
214#380C2311yescharacteristic
215#380C2311yescharacteristic
216#379—22no—
217#379—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225#376—21no—
226C2×#95—21no—
227C2×#95—22no—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238C25C2×D412yes—
239C22×Q8—21no—
240C22×Q8C2412yes—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—24no—
244C22×D4C2×D412yes—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C23×C4—21no—
248C23×C4—24no—
249C23×C4—24no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—48no—
257C23×C4—48no—
258C23×C4—84no—
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2412yes—
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C22×C8C2412yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274#125—44no—
275#124—44no—
276#124—44no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—416no—
297C4×D4—48no—
298C4×D4—48no—
299#117—44no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—48no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—48no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42#4114yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—24no—
374#107—42no—
375#107—44no—
376#106—24no—
377#106—42no—
378#106—44no—
379C4⋊C8—24no—
380C4⋊C8—24no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—44no—
385C22⋊C8—22no—
386C22⋊C8—22no—
387C22⋊C8—24no—
388C22⋊C8—44no—
389C22⋊C8—44no—
390C22⋊C8—44no—
391C22⋊C8—44no—
392#97—24no—
393C4×C8—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—24no—
411C24—24no—
412C24—24no—
413C24—24no—
414C24—24no—
415C24—42no—
416C24—42no—
417C24C22×D412yes—
418C24C22×D412yes—
419C24C22×D411yescharacteristic
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—232no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4AGL(1,ℤ/8)14yes—
436C2×D4#13714yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—832no—
487C22×C4#12314yes—
488C22×C4#12314yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2511yescharacteristic
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4H(5,𝔽2)14yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C42—44no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—42no—
563C42#12314yes—
564C42#12314yes—
565C42C2×#4112yes—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C42C2×#4112yes—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—216no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—48no—
587C23—48no—
588C23—48no—
589C23—48no—
590C23—48no—
591C23—48no—
592C23C2×#12312yes—
593C23C2×AGL(1,ℤ/8)14yes—
594C23C2×#13714yes—
595C23#57712yes—
596C23#57812yes—
597C23C23×D411yescharacteristic
598Q8—44no—
599Q8—44no—
600Q8—84no—
601D4—216no—
602D4—28no—
603D4—28no—
604C2×C4—28no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—44no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—816no—
658C2×C4—816no—
659C2×C4—84no—
660C2×C4—88no—
661C2×C4C2×#12312yes—
662C2×C4C2×#12312yes—
663C2×C4C2×#12312yes—
664C2×C4C2×#12312yes—
665C2×C4C2×#12312yes—
666C2×C4C2×AGL(1,ℤ/8)12yes—
667C2×C4C2×#13712yes—
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)12yes—
671C2×C4#58512yes—
672C8—42no—
673C8—42no—
674C8—44no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—432no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22#309714yes—
697C22C22×#12311yescharacteristic
698C22#348411yescharacteristic
699C22#348511yescharacteristic
700C22C22×AGL(1,ℤ/8)11yescharacteristic
701C22C22×#13711yescharacteristic
702C22C2×#57711yescharacteristic
703C22C2×#57811yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—42no—
709C4—42no—
710C4—44no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—81no—
717C4—84no—
718C4—81no—
719C4#309214yes—
720C4#346012yes—
721C4#350612yes—
722C2—24no—
723C2—44no—
724C2—44no—
725C2#3515211yescharacteristic
726C2C2×#309211yescharacteristic
727C2C2×#309711yescharacteristic
728C2#6160011yescharacteristic
729C2#6160211yescharacteristic
730C2#6173511yescharacteristic
731C2#6173711yescharacteristic
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. #32983C2semidirect
4. #32982C2semidirect
5. #29894C2semidirect
6. #29894C2semidirect
7. #26298C2semidirect
8. #26296C2semidirect
9. #26236C2semidirect
10. #26234C2semidirect
11. #25024C2semidirect
12. #24979C2semidirect
13. #21987C2semidirect
14. #21860C2semidirect
16. #21394C2semidirect
17. #21361C2semidirect
22. C2×#484C22semidirect
23. C2×#483C22semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2075C22semidirect
55. #2075C22semidirect
56. #2036C22semidirect
57. #2036C22semidirect
58. #2035C22semidirect
59. #2035C22semidirect
60. #1990C22semidirect
61. #1988C22semidirect
62. #1981C22semidirect
63. #1956C22semidirect
67. #1949C22semidirect
68. #1949C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
75. #1872C22semidirect
76. #1872C22semidirect
80. #1853C22semidirect
84. #1852C22semidirect
85. #1847C22semidirect
86. #1836C22semidirect
87. #1824C22semidirect
88. #1823C22semidirect
89. #1803C22semidirect
92. C4×#107C22semidirect
93. C4×#106C22semidirect
94. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×D4×C4⋊C42
#329832
#329822
#298942
#262982
#262962
#262362
#262342
#250242
#249792
#219872
#218602
#217882
#213942
#213612