Finite Group Databasebeta

← #6795850

Subgroups — #6795850

Order 512.

1675 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic, complemented
3#33197C211yesmaximal, characteristic, complemented
4#32206C214yesmaximal, complemented
5#30673C214yesmaximal, complemented
6#29925C212yesmaximal, complemented
7#29208C211yesmaximal, characteristic, complemented
8#28280C211yesmaximal, characteristic, complemented
9#28227C212yesmaximal, complemented
10#27627C212yesmaximal, complemented
11#26087C212yesmaximal, complemented
12#25339C212yesmaximal, complemented
13#24750C212yesmaximal, complemented
14#24677C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21962C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#555—21no—
19C2×#476C2212yes—
20C2×#476C2212yes—
21C2×#466C2214yes—
22C2×#459C2211yescharacteristic
23C2×C4×D8C2212yes—
24C2×C8×D4C2212yes—
25C22×C4⋊C8C2212yes—
26#2868C2211yescharacteristic
27#2856C2212yes—
28#2572—22no—
29#2571C2211yescharacteristic
30#2563—21no—
31#2563C2211yescharacteristic
32#2563C2211yescharacteristic
33#2537—21no—
34#2537C2211yescharacteristic
35#2537C2212yes—
36#2500—22no—
37#2495—21no—
38#2495C2212yes—
39#2491—22no—
40#2491C2214yes—
41#2420—21no—
42#2420C2212yes—
43#2398—22no—
44#2354—21no—
45#2354C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#121—21no—
49C4×#121C2212yes—
50#2285C2212yes—
51#2283—21no—
52#2283C2212yes—
53C42×D4C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2092C2214yes—
56#2077C2214yes—
57#2073C2214yes—
58#2067C2214yes—
59#2065C2214yes—
60#2057C2214yes—
61#2054C2214yes—
62#2034C2214yes—
63#2033C2214yes—
64#2033C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1982C2212yes—
68#1982C2211yescharacteristic
69#1949C2214yes—
70#1949C2214yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1947C2211yescharacteristic
77#1939C2212yes—
78#1926C2214yes—
79#1925C2212yes—
80#1921C2214yes—
81#1893C2214yes—
82#1879C2212yes—
83#1850C2214yes—
84#1845C2212yes—
85#1842C2211yescharacteristic
86#1808C2214yes—
87#1807C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×D8C2312yes—
96C23×C8C2314yes—
97#555—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#124—24no—
104C2×#121—21no—
105C2×#121—22no—
106C2×#121—22no—
107C2×#121—22no—
108C2×#121C2312yes—
109C2×#121C2314yes—
110C2×C4×Q8—21no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42—21no—
124C22×C42C2312yes—
125#476—44no—
126#476—44no—
127#466—48no—
128#459—24no—
129C4×D8—44no—
130C8×D4—44no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8—28no—
137C2×C4⋊C8C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×C22⋊C8C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#400—22no—
154#397—22no—
155#397—22no—
156#394—22no—
157#394—22no—
158#394—24no—
159#394—24no—
160#393—22no—
161#393—24no—
162#393—22no—
163#391—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2312yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—24no—
179#388—24no—
180#386—22no—
181#386—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#383—22no—
189#383—22no—
190#382—21no—
191#382—21no—
192#382—22no—
193#382C2312yes—
194#382C2312yes—
195#381—22no—
196#381—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2312yes—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376—22no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×D8—24no—
253C2×D8—24no—
254C2×D8—42no—
255C22×C8—28no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—22no—
262C4⋊Q8—22no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—82no—
267#127—22no—
268#127—22no—
269#124—416no—
270#121—24no—
271#121—42no—
272#121—48no—
273#121—44no—
274#121—48no—
275#121—44no—
276#121—44no—
277#121—84no—
278C4×Q8—82no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—22no—
288C4×D4—24no—
289C4×D4—22no—
290C4×D4—41no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—84no—
294#117—48no—
295#117—48no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#107—42no—
357#107—44no—
358#106—44no—
359#106—44no—
360C4⋊C8—28no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363C4⋊C8—22no—
364C4⋊C8—42no—
365C4⋊C8#4114yes—
366#102—28no—
367#102—24no—
368#102—42no—
369#102—42no—
370#102—42no—
371#102—44no—
372#102—44no—
373C22⋊C8—44no—
374C4×C8—24no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—24no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—21no—
394C24—22no—
395C24C22×D412yes—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—82no—
401C2×Q8—82no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—82no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4AGL(1,ℤ/8)14yes—
428C2×D4H(5,𝔽2)12yes—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2511yescharacteristic
470D8—48no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—88no—
480C2×C8C2×#4114yes—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—82no—
506C4⋊C4—81no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—42no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—84no—
526C22⋊C4—84no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C42—24no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—21no—
535C42—22no—
536C42—21no—
537C42—41no—
538C42—48no—
539C42—42no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42#13514yes—
544C42C2×#4112yes—
545C42H(5,𝔽2)12yes—
546C422−1+412yes—
547C422−1+412yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23C2×AGL(1,ℤ/8)14yes—
569C23C23×D411yescharacteristic
570Q8—84no—
571Q8—164no—
572D4—28no—
573D4—22no—
574D4—22no—
575D4—41no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—41no—
580D4—42no—
581D4—84no—
582D4—88no—
583D4—84no—
584D4—168no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—88no—
634C2×C4#53712yes—
635C2×C4#53712yes—
636C2×C4#54712yes—
637C2×C4#54712yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13512yes—
640C2×C4C2×#13512yes—
641C2×C4C2×AGL(1,ℤ/8)12yes—
642C2×C4#57612yes—
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×2−1+411yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4#58511yescharacteristic
649C8—24no—
650C8—44no—
651C8—42no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22C2×#53711yescharacteristic
674C22C2×#54711yescharacteristic
675C22#348811yescharacteristic
676C22C22×#13511yescharacteristic
677C22C22×AGL(1,ℤ/8)11yescharacteristic
678C22C2×#57611yescharacteristic
679C22#358611yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—41no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—82no—
689C4—84no—
690C4—81no—
691C4—81no—
692C4—82no—
693C4—81no—
694C4#324714yes—
695C4#334114yes—
696C4#351811yescharacteristic
697C4#351911yescharacteristic
698C4#354411yescharacteristic
699C4#354711yescharacteristic
700C2—44no—
701C2—82no—
702C2—162no—
703C2#3892211yescharacteristic
704C2C2×#324711yescharacteristic
705C2#6092811yescharacteristic
706C2C2×#334111yescharacteristic
707C2#6125211yescharacteristic
708C2#6195011yescharacteristic
709C2#6197911yescharacteristic
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. #36187C2semidirect
2. #34943C2semidirect
3. #33197C2semidirect
4. #32206C2semidirect
5. #30673C2semidirect
6. #29925C2semidirect
7. #29208C2semidirect
8. #28280C2semidirect
9. #28227C2semidirect
10. #27627C2semidirect
11. #26087C2semidirect
12. #25339C2semidirect
13. #24750C2semidirect
14. #24677C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21962C2semidirect
17. C4×#459C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#349432
#331972
#322062
#306732
#299252
#292082
#282802
#282272
#276272
#260872
#253392
#247502
#246772
D4×C4⋊C82
#219622
C4×#4592