Finite Group Databasebeta

← #6856367

Subgroups — #6856367

Order 512.

1921 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36580C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic, complemented
3#32902C211yesmaximal, characteristic
4#32605C214yesmaximal, complemented
5#32550C214yesmaximal, complemented
6#29718C212yesmaximal, complemented
7#29330C212yesmaximal, complemented
8#29238C212yesmaximal, complemented
9#28186C212yesmaximal, complemented
10#27973C211yesmaximal, characteristic, complemented
11#27681C211yesmaximal, characteristic, complemented
12#25292C212yesmaximal, complemented
13#24993C212yesmaximal, complemented
14#24186C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22040C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic, complemented
19C22×C22⋊Q8—21no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#459C2212yes—
23C2×#450C2211yescharacteristic
24C2×#447C2212yes—
25C2×C8×Q8C2212yescomplemented
26C2×C4×SD16C2212yes—
27#2856C2211yescharacteristic
28#2850C2212yes—
29#2592C2212yescomplemented
30#2582C2211yescharacteristic, complemented
31#2580C2212yescomplemented
32#2548C2211yescharacteristic, complemented
33#2538—21no—
34#2538—21no—
35#2538—22no—
36#2538C2212yes—
37#2538C2214yes—
38#2538C2212yes—
39#2458—22no—
40#2448—21no—
41#2443—22no—
42#2436—21no—
43C2×#396—22no—
44#2367—21no—
45#2361—21no—
46#2355C2211yescharacteristic, complemented
47C4×C22⋊Q8C2211yescharacteristic
48C4×#121C2212yes—
49C2×#390C2212yes—
50C2×#389C2212yes—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53#2101C2214yescomplemented
54#2100C2212yescomplemented
55#2095C2212yescomplemented
56#2088C2214yescomplemented
57#2073C2214yes—
58#2071C2214yescomplemented
59#2067C2214yescomplemented
60#2055C2214yes—
61#2054C2214yes—
62#2042C2214yescomplemented
63#2012C2214yes—
64#1984C2214yescomplemented
65#1984C2212yescomplemented
66#1970C2214yes—
67#1957C2212yescomplemented
68#1947C2212yescomplemented
69#1934C2214yes—
70#1934C2214yescomplemented
71#1933C2214yescomplemented
72#1925C2214yes—
73#1921C2214yescomplemented
74#1879C2214yes—
75#1879C2214yes—
76#1877C2212yes—
77#1874C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1864C2212yes—
83#1864C2214yescomplemented
84#1853C2211yescharacteristic
85#1852C2212yes—
86#1834C2212yescomplemented
87#1803C2212yescomplemented
88#1802C2212yes—
89#1797C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×D8C2314yes—
100#559—44no—
101C2×#127C2311yescharacteristic
102C2×#124—21no—
103C2×#124—21no—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—48no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×#121—21no—
112C2×#121—21no—
113C2×#121C2312yes—
114C2×C4×Q8—44no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D412yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—44no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C22×C4⋊C4—22no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4D412yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142#459—44no—
143#450—24no—
144#447—28no—
145C8×Q8—44no—
146C4×SD16—44no—
147C2×#107C2314yes—
148C2×#106C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×C4×C8C2312yes—
164C2×C4×C8C2314yes—
165#399—22no—
166#398—22no—
167#397—22no—
168#396—24no—
169#396—24no—
170#396—24no—
171#396—24no—
172#394—22no—
173#393—22no—
174#393—22no—
175#391—21no—
176#391—21no—
177#391C2312yes—
178#391C2311yescharacteristic
179#390—21no—
180#390—21no—
181#390—21no—
182#390—22no—
183#390—22no—
184#390C2312yes—
185#390C2314yes—
186#390C2312yes—
187#390C2312yes—
188#390C2314yes—
189#389—21no—
190#389—22no—
191#389—22no—
192#389C2314yes—
193#389C2311yescharacteristic
194#389C2312yes—
195#389C2314yes—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#387—22no—
202#387—21no—
203#386—22no—
204#386—22no—
205#386—22no—
206#385—21no—
207#385—22no—
208#385—22no—
209#384—21no—
210#384—21no—
211#384C2311yescharacteristic
212#384C2312yes—
213#384C2312yes—
214#382C2312yes—
215#380—22no—
216#380—22no—
217#380—24no—
218#380C2311yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C2×#95—22no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C25—21no—
238C22×Q8—24no—
239C22×Q8—24no—
240C22×Q8—42no—
241C22×Q8C2×D411yescharacteristic
242C22×Q8C2×D412yes—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2411yescharacteristic
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4#4114yes—
253C22×D4C2412yes—
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—24no—
269C2×SD16—42no—
270C2×SD16—42no—
271C2×D8—28no—
272C2×D8—44no—
273C2×D8—44no—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277#127—24no—
278#124—44no—
279#124—44no—
280#124—44no—
281#123—48no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—48no—
285C22⋊Q8—416no—
286C22⋊Q8—48no—
287#121—24no—
288#121—24no—
289#121—44no—
290#121—82no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—42no—
297C4×Q8—42no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—84no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—48no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4—48no—
361C2×C22⋊C4—84no—
362C2×C42—24no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375#107—44no—
376#106—42no—
377C4⋊C8—24no—
378C4⋊C8—44no—
379#103—44no—
380#103—42no—
381#102—28no—
382#102—24no—
383#102—44no—
384#102—42no—
385#102—44no—
386#102—42no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8—22no—
390C22⋊C8—22no—
391C4×C8—24no—
392C4×C8—44no—
393#95—24no—
394#95—24no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—21no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406C24—24no—
407C24—22no—
408C24—22no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24C22×D411yescharacteristic
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—416no—
429C2×Q8AGL(1,ℤ/8)14yes—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—24no—
440C2×D4—48no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—48no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4C2×#4114yes—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4—816no—
490C22×C4#13514yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C2×#4112yes—
501C22×C42−1+412yes—
502C22×C4C2511yescharacteristic
503SD16—48no—
504D8—88no—
505C2×C8—24no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—24no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—48no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—48no—
540C4⋊C4—81no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—81no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—48no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—82no—
559C22⋊C4—84no—
560C22⋊C4—84no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—22no—
564C42—22no—
565C42—24no—
566C42—22no—
567C42—24no—
568C42—42no—
569C42—42no—
570C42—42no—
571C42—42no—
572C42—82no—
573C42H(5,𝔽2)14yes—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—24no—
579C23—48no—
580C23—48no—
581C23—42no—
582C23—48no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23#54812yes—
600C23C2×#13512yes—
601C23#57612yes—
602C23C23×D411yescharacteristic
603Q8—28no—
604Q8—48no—
605Q8—44no—
606Q8—42no—
607Q8—42no—
608D4—24no—
609D4—24no—
610D4—48no—
611D4—44no—
612D4—42no—
613D4—42no—
614D4—44no—
615D4—88no—
616D4—84no—
617D4—162no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—44no—
664C2×C4—88no—
665C2×C4—84no—
666C2×C4—88no—
667C2×C4—84no—
668C2×C4—88no—
669C2×C4—164no—
670C2×C4#53412yes—
671C2×C4#53412yes—
672C2×C4#54812yes—
673C2×C4C2×#13512yes—
674C2×C4C2×#13512yes—
675C2×C4C2×AGL(1,ℤ/8)14yes—
676C2×C4C2×AGL(1,ℤ/8)12yes—
677C2×C4C22×#4111yescharacteristic
678C2×C4C2×H(5,𝔽2)12yes—
679C2×C4C2×2−1+411yescharacteristic
680C2×C4#58512yes—
681C8—44no—
682C8—42no—
683C8—42no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—816no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22#336114yes—
706C22C2×#53411yescharacteristic
707C22C2×#54811yescharacteristic
708C22#348711yescharacteristic
709C22C22×#13511yescharacteristic
710C22C22×AGL(1,ℤ/8)11yescharacteristic
711C22C2×#57611yescharacteristic
712C22#358611yescharacteristic
713C4—22no—
714C4—22no—
715C4—24no—
716C4—22no—
717C4—42no—
718C4—41no—
719C4—42no—
720C4—42no—
721C4—81no—
722C4—82no—
723C4—81no—
724C4—82no—
725C4—81no—
726C4—81no—
727C4—84no—
728C4#318814yes—
729C4#350512yes—
730C4#354112yes—
731C2—24no—
732C2—84no—
733C2—162no—
734C2#3972311yescharacteristic
735C2C2×#318811yescharacteristic
736C2#6076211yescharacteristic
737C2C2×#336111yescharacteristic
738C2#6131711yescharacteristic
739C2#6188211yescharacteristic
740C2#6190711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
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
1. #36580C2semidirect
2. C2×#2538C2semidirect
4. #32605C2semidirect
5. #32550C2semidirect
6. #29718C2semidirect
7. #29330C2semidirect
8. #29238C2semidirect
9. #28186C2semidirect
10. #27973C2semidirect
11. #27681C2semidirect
12. #25292C2semidirect
13. #24993C2semidirect
14. #24186C2semidirect
15. Q8×C22⋊C8C2semidirect
16. #22040C2semidirect
17. C4×#450C2semidirect
18. C2×#559C22semidirect
25. C2×C8×Q8C22semidirect
29. #2592C22semidirect
30. #2582C22semidirect
31. #2580C22semidirect
32. #2548C22semidirect
46. #2355C22semidirect
53. #2101C22semidirect
54. #2100C22semidirect
55. #2095C22semidirect
56. #2088C22semidirect
58. #2071C22semidirect
59. #2067C22semidirect
62. #2042C22semidirect
64. #1984C22semidirect
65. #1984C22semidirect
67. #1957C22semidirect
68. #1947C22semidirect
70. #1934C22semidirect
71. #1933C22semidirect
73. #1921C22semidirect
83. #1864C22semidirect
86. #1834C22semidirect
87. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#365802
C2×#25382
#329022
#326052
#325502
#297182
#293302
#292382
#281862
#279732
#276812
#252922
#249932
#241862
Q8×C22⋊C82
#220402
C4×#4502