Finite Group Databasebeta

← #6793018

Subgroups — #6793018

Order 512.

1819 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#31613C214yesmaximal, complemented
5#31557C214yesmaximal, complemented
6#29942C212yesmaximal, complemented
7#29218C211yesmaximal, characteristic, complemented
8#28280C211yesmaximal, characteristic, complemented
9#28140C212yesmaximal, complemented
10#26107C212yesmaximal, complemented
11#25339C212yesmaximal, complemented
12#24836C212yesmaximal, complemented
13#24812C212yesmaximal, complemented
14#24152C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21888C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#476C2212yes—
21C2×#476C2212yes—
22C2×#459C2211yescharacteristic
23C2×C4×D8C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2857C2212yes—
28#2572—22no—
29#2571C2211yescharacteristic
30#2563C2211yescharacteristic
31#2563C2211yescharacteristic
32#2547—21no—
33#2537—21no—
34#2537C2212yes—
35#2537C2211yescharacteristic
36#2495—21no—
37#2495C2212yes—
38#2432—22no—
39#2422—22no—
40#2422C2214yes—
41#2420—21no—
42#2420C2212yes—
43#2398—22no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#121—21no—
49C4×#121C2212yes—
50#2283—21no—
51#2283C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2100C2214yes—
56#2088C2214yes—
57#2085C2214yes—
58#2077C2214yes—
59#2075C2214yes—
60#2065C2214yes—
61#2065C2214yes—
62#2034C2214yes—
63#1999C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1982C2212yes—
69#1982C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1925C2212yes—
76#1893C2214yes—
77#1893C2214yes—
78#1893C2214yes—
79#1879C2212yes—
80#1879C2214yes—
81#1864C2214yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1807C2214yes—
86#1804C2214yes—
87#1804C2214yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×D8C2312yes—
97C23×C8C2314yes—
98#547—48no—
99#532—28no—
100#532—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#124—22no—
107C2×#123—24no—
108C2×C22⋊Q8—22no—
109C2×#121—21no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121C2314yes—
113C2×#121C2312yes—
114C2×C4×Q8—21no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2314yes—
126C2×C4×D4C2311yescharacteristic
127C22×C4⋊C4—22no—
128C22×C42—21no—
129C22×C42C2312yes—
130#476—44no—
131#476—44no—
132#459—24no—
133C4×D8—44no—
134C8×D4—44no—
135C2×#107C2312yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8—44no—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C4×C8C2312yes—
154#400—22no—
155#400—22no—
156#397—22no—
157#397—22no—
158#397—24no—
159#394—22no—
160#394—22no—
161#394—24no—
162#394—24no—
163#393—22no—
164#393—22no—
165#391—21no—
166#391—22no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390C2312yes—
171#390C2312yes—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#386—22no—
181#386—22no—
182#385—24no—
183#385—24no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#383—22no—
191#383—22no—
192#382—21no—
193#382—21no—
194#382C2312yes—
195#382C2312yes—
196#381—22no—
197#381—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—21no—
204#380C2312yes—
205#380C2312yes—
206#380C2314yes—
207#380C2312yes—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#336C2314yes—
236#336C2312yes—
237C2×#41—48no—
238C2×#41—48no—
239C2×#41—84no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×D4—22no—
243C22×D4—24no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2412yes—
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—24no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—42no—
260C22×C8—44no—
261C22×C8—44no—
262C22×C8—44no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—44no—
271C4⋊Q8—82no—
272#127—22no—
273#127#4114yes—
274#124—48no—
275#123—416no—
276C22⋊Q8—48no—
277#121—28no—
278#121—24no—
279#121—42no—
280#121—44no—
281#121—48no—
282#121—44no—
283#121—48no—
284C4×Q8—82no—
285C4×D4—24no—
286C4×D4—28no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—41no—
298C4×D4—44no—
299C4×D4—48no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4—84no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—42no—
363#106—44no—
364#106—44no—
365C4⋊C8—22no—
366C4⋊C8—22no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370C4⋊C8—42no—
371#102—24no—
372#102—42no—
373#102—44no—
374#102—42no—
375#102—42no—
376#102—44no—
377#102—44no—
378C22⋊C8—44no—
379C4×C8—24no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24—22no—
400C24C22×D412yes—
401#41—832no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—21no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—48no—
427C2×D4—44no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—82no—
431C2×D4—84no—
432C2×D4AGL(1,ℤ/8)14yes—
433C2×D4C2×#4114yes—
434C2×D42−1+412yes—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—81no—
468C22×C4—84no—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C2511yescharacteristic
479D8—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—22no—
489C2×C8—88no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—48no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—82no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—42no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—42no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—21no—
546C42—41no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—44no—
551C42#13514yes—
552C42C2×#4112yes—
553C42H(5,𝔽2)12yes—
554C42H(5,𝔽2)12yes—
555C422−1+412yes—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23—42no—
576C23C2×AGL(1,ℤ/8)14yes—
577C23C23×D411yescharacteristic
578Q8—88no—
579Q8—84no—
580D4—28no—
581D4—24no—
582D4—22no—
583D4—24no—
584D4—22no—
585D4—48no—
586D4—41no—
587D4—41no—
588D4—42no—
589D4—42no—
590D4—48no—
591D4—88no—
592D4—816no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—48no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—84no—
639C2×C4—816no—
640C2×C4—84no—
641C2×C4—88no—
642C2×C4—84no—
643C2×C4#53712yes—
644C2×C4#53712yes—
645C2×C4#54612yes—
646C2×C4#54612yes—
647C2×C4C2×#13512yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4C2×AGL(1,ℤ/8)12yes—
651C2×C4#57712yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C2×C4#58511yescharacteristic
658C8—42no—
659C8—44no—
660C8—42no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—81no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22—81no—
682C22C2×#53711yescharacteristic
683C22C2×#54611yescharacteristic
684C22#348911yescharacteristic
685C22C22×#13511yescharacteristic
686C22C22×AGL(1,ℤ/8)11yescharacteristic
687C22C2×#57711yescharacteristic
688C22#358511yescharacteristic
689C4—22no—
690C4—22no—
691C4—22no—
692C4—41no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—82no—
697C4—81no—
698C4—82no—
699C4—81no—
700C4—81no—
701C4—82no—
702C4—84no—
703C4#324714yes—
704C4#330314yes—
705C4#351511yescharacteristic
706C4#352011yescharacteristic
707C4#354611yescharacteristic
708C4#355211yescharacteristic
709C2—44no—
710C2—82no—
711C2—84no—
712C2#3890811yescharacteristic
713C2C2×#324711yescharacteristic
714C2#6092911yescharacteristic
715C2C2×#330311yescharacteristic
716C2#6115511yescharacteristic
717C2#6209811yescharacteristic
718C2#6212811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #36127C2semidirect
2. #34943C2semidirect
3. #33197C2semidirect
4. #31613C2semidirect
5. #31557C2semidirect
6. #29942C2semidirect
7. #29218C2semidirect
8. #28280C2semidirect
9. #28140C2semidirect
10. #26107C2semidirect
11. #25339C2semidirect
12. #24836C2semidirect
13. #24812C2semidirect
14. #24152C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21888C2semidirect
17. C4×#459C2semidirect

Maximal subgroups

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

subgroupindex
#361272
#349432
#331972
#316132
#315572
#299422
#292182
#282802
#281402
#261072
#253392
#248362
#248122
#241522
D4×C4⋊C82
#218882
C4×#4592