Finite Group Databasebeta

← #6540314

Subgroups — #6540314

Order 512.

2001 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic, complemented
2#34310C211yesmaximal, characteristic
3#30172C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#29109C212yesmaximal, complemented
7#28160C211yesmaximal, characteristic, complemented
8#26675C214yesmaximal, complemented
9#24971C212yesmaximal, complemented
10#24921C214yesmaximal, complemented
11#24240C212yesmaximal
12#23933C212yesmaximal, complemented
13#23677C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#23003C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22759C212yesmaximal, complemented
18C2×D42C2211yescharacteristic
19C2×#533C2212yescomplemented
20C2×#518—21no—
21C2×#460C2212yes—
22C2×#450C2214yes—
23C2×#440C2214yescomplemented
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic, complemented
26#2857C2212yescomplemented
27#2552—21no—
28#2542—21no—
29#2538C2212yes—
30#2441—22no—
31#2429—22no—
32#2423C2214yescomplemented
33#2366C2211yescharacteristic
34#2365—21no—
35#2363—21no—
36#2361C2212yes—
37#2358—22no—
38#2354—21no—
39#2354C2211yescharacteristic
40Q8×C4⋊C4C2212yescomplemented
41D4×C4⋊C4C2211yescharacteristic
42D4×C22⋊C4—21no—
43D4×C22⋊C4C2212yes—
44#2324—21no—
45#2321—22no—
46C4×C4⋊Q8C2211yescharacteristic, complemented
47C4×#127C2211yescharacteristic
48C4×#121C2212yes—
49#2293C2211yescharacteristic
50C2×#390C2212yes—
51C42×D4C2211yescharacteristic
52#2099C2212yescomplemented
53#2093C2212yes—
54#2084C2211yescharacteristic, complemented
55#2076C2214yescomplemented
56#2071C2211yescharacteristic
57#2071C2212yescomplemented
58#2066C2212yescomplemented
59#2057C2212yescomplemented
60#2055C2212yes—
61#2043C2211yescharacteristic, complemented
62#2036C2214yescomplemented
63#2034C2214yes—
64#2023C2212yes—
65#2023C2212yescomplemented
66#2023C2212yescomplemented
67#2013C2211yescharacteristic
68#2013C2212yescomplemented
69#2000C2214yes—
70#1957C2212yes—
71#1954C2212yes—
72#1947C2212yescomplemented
73#1936C2214yes—
74#1936C2214yescomplemented
75#1924C2214yescomplemented
76#1919C2214yescomplemented
77#1880C2214yes—
78#1880C2214yescomplemented
79#1877C2214yescomplemented
80#1872C2214yescomplemented
81#1862C2214yescomplemented
82#1834C2212yescomplemented
83#1831C2211yescharacteristic, complemented
84#1824C2214yescomplemented
85#1817C2212yescomplemented
86#1803C2212yescomplemented
87#1802C2212yes—
88#1802C2214yescomplemented
89#1786C2214yes—
90#1770C2212yescomplemented
91C4×C4⋊C8C2211yescharacteristic, complemented
92#1763C2214yescomplemented
93C4×#103C2212yescomplemented
94#1749C2212yescomplemented
95C23×D4—21no—
96C23×D4C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99D42—24no—
100D42—24no—
101#533—48no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2314yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×#120—22no—
114#518—48no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2314yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C2×#117—21no—
132C2×#117—21no—
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42C2312yes—
138#460—28no—
139#450—48no—
140#440—48no—
141C4×SD16—28no—
142C2×#107C2314yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2312yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#399—24no—
160#398—24no—
161#396—24no—
162#391—22no—
163#391C2312yes—
164#390—22no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2311yescharacteristic
172#390C2311yescharacteristic
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—24no—
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2314yes—
184#388—24no—
185#388—21no—
186#388—22no—
187#388—21no—
188#388—22no—
189#388—22no—
190#387—24no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#385—22no—
197#385—22no—
198#384—22no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#380—22no—
206#380—21no—
207#380C2312yes—
208#380C2314yes—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2312yes—
236C25—22no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—44no—
247C22×D4—44no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C2×SD16—28no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—28no—
264C2×SD16—44no—
265C2×SD16—42no—
266C22×C8C2412yes—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273#127—22no—
274#127#4114yes—
275#125—48no—
276C22⋊Q8—28no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—84no—
280#121—28no—
281#121—24no—
282#121—24no—
283#120—216no—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—44no—
288C4×Q8—48no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—82no—
302C4×D4#4114yes—
303#117—44no—
304#117—48no—
305#117—44no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—48no—
352C2×C22⋊C4—48no—
353C2×C22⋊C4—84no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—44no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—44no—
373#106—24no—
374C4⋊C8—24no—
375C4⋊C8—42no—
376#103—24no—
377#103—24no—
378#103—42no—
379#103—44no—
380#102—24no—
381#102—28no—
382#102—44no—
383#102—44no—
384#102—44no—
385C22⋊C8—44no—
386C22⋊C8—44no—
387#97—44no—
388C4×C8—24no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—28no—
402C24—21no—
403C24—22no—
404C24—44no—
405C24—44no—
406C24—82no—
407C24—82no—
408C24—82no—
409C24—82no—
410C24C22×D412yes—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8H(5,𝔽2)14yes—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—432no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—88no—
442C2×D4—88no—
443C2×D4#13714yes—
444C2×D4C2×#4112yes—
445C2×D4C2×#4114yes—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2511yescharacteristic
485SD16—48no—
486SD16—416no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—82no—
521C4⋊C4—81no—
522C4⋊C4—84no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4C2×#4112yes—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—42no—
539C22⋊C4—44no—
540C22⋊C4—81no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—84no—
546C42—22no—
547C42—22no—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—42no—
552C42—42no—
553C42—41no—
554C42—48no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42AGL(1,ℤ/8)14yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—416no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—41no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23—41no—
581C23—42no—
582C23—88no—
583C23—88no—
584C23—164no—
585C23—164no—
586C23—164no—
587C23—164no—
588C23C2×#13714yes—
589C23C23×D411yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—88no—
595D4—28no—
596D4—24no—
597D4—28no—
598D4—416no—
599D4—48no—
600D4—44no—
601D4—42no—
602D4—84no—
603D4—88no—
604C2×C4—21no—
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—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—84no—
650C2×C4—164no—
651C2×C4—164no—
652C2×C4—164no—
653C2×C4#52512yes—
654C2×C4#52512yes—
655C2×C4C2×AGL(1,ℤ/8)12yes—
656C2×C4C2×AGL(1,ℤ/8)12yes—
657C2×C4C2×AGL(1,ℤ/8)12yes—
658C2×C4C2×#13712yes—
659C2×C4#57712yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C2×H(5,𝔽2)12yes—
663C2×C4#58512yes—
664C8—42no—
665C8—42no—
666C8—44no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—42no—
677C22—44no—
678C22—44no—
679C22—42no—
680C22—82no—
681C22—81no—
682C22—82no—
683C22—81no—
684C22—82no—
685C22—81no—
686C22—82no—
687C22—81no—
688C22—1616no—
689C22C2×#52511yescharacteristic
690C22#348411yescharacteristic
691C22#350011yescharacteristic
692C22C22×AGL(1,ℤ/8)11yescharacteristic
693C22C22×#13711yescharacteristic
694C22C2×#57711yescharacteristic
695C22#358511yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—22no—
700C4—41no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—81no—
708C4—81no—
709C4—84no—
710C4—82no—
711C4#311014yes—
712C4#349412yes—
713C4#351312yes—
714C2—44no—
715C2—84no—
716C2—82no—
717C2#3573011yescharacteristic
718C2C2×#311011yescharacteristic
719C2#6059711yescharacteristic
720C2#6158711yescharacteristic
721C2#6159311yescharacteristic
722C2#6253911yescharacteristic
723C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36134C2semidirect
3. #30172C2semidirect
4. #29980C2semidirect
5. #29139C2semidirect
6. #29109C2semidirect
7. #28160C2semidirect
8. #26675C2semidirect
9. #24971C2semidirect
10. #24921C2semidirect
12. #23933C2semidirect
13. #23677C2semidirect
14. #23574C2semidirect
15. #23003C2semidirect
16. #22839C2semidirect
17. #22759C2semidirect
19. C2×#533C22semidirect
23. C2×#440C22semidirect
25. #2868C22semidirect
26. #2857C22semidirect
32. #2423C22semidirect
40. Q8×C4⋊C4C22semidirect
46. C4×C4⋊Q8C22semidirect
52. #2099C22semidirect
54. #2084C22semidirect
55. #2076C22semidirect
57. #2071C22semidirect
58. #2066C22semidirect
59. #2057C22semidirect
61. #2043C22semidirect
62. #2036C22semidirect
65. #2023C22semidirect
66. #2023C22semidirect
68. #2013C22semidirect
72. #1947C22semidirect
74. #1936C22semidirect
75. #1924C22semidirect
76. #1919C22semidirect
78. #1880C22semidirect
79. #1877C22semidirect
80. #1872C22semidirect
81. #1862C22semidirect
82. #1834C22semidirect
83. #1831C22semidirect
84. #1824C22semidirect
85. #1817C22semidirect
86. #1803C22semidirect
88. #1802C22semidirect
90. #1770C22semidirect
91. C4×C4⋊C8C22semidirect
92. #1763C22semidirect
93. C4×#103C22semidirect
94. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#361342
#343102
#301722
#299802
#291392
#291092
#281602
#266752
#249712
#249212
#242402
#239332
#236772
#235742
#230032
#228392
#227592