Finite Group Databasebeta

← #6571147

Subgroups — #6571147

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30128C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic, complemented
5#29498C212yesmaximal, complemented
6#28201C212yesmaximal, complemented
7#27541C214yesmaximal, complemented
8#27245C214yesmaximal, complemented
9#26460C212yesmaximal, complemented
10#25855C211yesmaximal, characteristic
11#25322C212yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23822C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#23223C212yesmaximal, complemented
16#23101C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#486C2212yes—
21C2×#474C2214yescomplemented
22C2×#460C2212yes—
23C2×#450C2212yes—
24#2863C2212yescomplemented
25#2857C2211yescharacteristic
26#2605—22no—
27#2567C2211yescharacteristic, complemented
28#2551C2211yescharacteristic, complemented
29#2509—21no—
30#2497C2212yes—
31#2492C2212yes—
32#2465—22no—
33#2439—21no—
34#2369C2211yescharacteristic, complemented
35#2366—21no—
36#2366—21no—
37#2366C2212yes—
38#2366C2212yes—
39#2365—21no—
40#2355C2211yescharacteristic, complemented
41#2347—21no—
42D4×C22⋊C4—22no—
43D4×C22⋊C4C2214yes—
44#2328—22no—
45#2309—21no—
46C4×C4⋊Q8C2212yescomplemented
47C4×#127C2212yes—
48C4×#127C2212yes—
49C4×C22⋊Q8C2211yescharacteristic
50#2285C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2211yescharacteristic, complemented
53#2100C2212yescomplemented
54#2095C2211yescharacteristic, complemented
55#2083C2211yescharacteristic, complemented
56#2083C2212yescomplemented
57#2076C2214yes—
58#2076C2212yes—
59#2072C2211yescharacteristic, complemented
60#2071C2214yescomplemented
61#2067C2214yescomplemented
62#2055C2214yescomplemented
63#2041C2212yes—
64#2041C2212yes—
65#2041C2212yes—
66#2036C2212yes—
67#2023C2214yescomplemented
68#2014C2212yes—
69#1987C2212yescomplemented
70#1982C2212yes—
71#1973C2214yescomplemented
72#1957C2212yes—
73#1947C2212yescomplemented
74#1939C2212yescomplemented
75#1936C2214yescomplemented
76#1925C2214yescomplemented
77#1925C2214yescomplemented
78#1924C2214yes—
79#1877C2214yescomplemented
80#1877C2214yescomplemented
81#1862C2214yescomplemented
82#1862C2214yescomplemented
83#1831C2212yescomplemented
84#1802C2214yescomplemented
85#1797C2211yescharacteristic
86#1786C2214yes—
87#1786C2212yes—
88C4×#107C2212yescomplemented
89#1770C2212yescomplemented
90#1762C2214yescomplemented
91#1762C2214yescomplemented
92C4×#103C2212yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127—28no—
103C2×#127—21no—
104C2×#127—21no—
105C2×#127C2312yes—
106C2×#127C2312yes—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8C2311yescharacteristic
109#518—48no—
110C2×C4×Q8—22no—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2314yes—
119C2×C4×D4C2314yes—
120C2×#117—21no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C22⋊C4—24no—
124C22×C42C2311yescharacteristic
125#486—28no—
126#474—48no—
127#460—28no—
128#450—44no—
129C2×#107C2312yes—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#400—22no—
149#396—22no—
150#395—22no—
151#395—22no—
152#393—22no—
153#391—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2311yescharacteristic
159#391C2312yes—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2314yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#386—24no—
187#386—24no—
188#386—24no—
189#386—24no—
190#385—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#383—22no—
196#382—21no—
197#382—21no—
198#380—21no—
199#380—21no—
200#380C2312yes—
201#380C2312yes—
202#380C2311yescharacteristic
203#379—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234C25—22no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C2411yescharacteristic
238C22×D4—28no—
239C22×D4—28no—
240C22×D4—28no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—44no—
247C22×D4—42no—
248C22×D4—42no—
249C22×D4—44no—
250C22×D4—44no—
251C22×D4—82no—
252C22×D4—82no—
253C22×D4C2414yes—
254C23×C4—24no—
255C23×C4—24no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C2×SD16—28no—
260C2×SD16—44no—
261C2×SD16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269#127—28no—
270#127—28no—
271#127—28no—
272#127—24no—
273#127#4118yes—
274C22⋊Q8—22no—
275C22⋊Q8—41no—
276C22⋊Q8—82no—
277C4×Q8—24no—
278C4×Q8—84no—
279C4×D4—48no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—48no—
287C4×D4—84no—
288C4×D4—84no—
289#117—24no—
290#117—24no—
291#117—44no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—416no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—24no—
349#107—42no—
350#107—44no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#103—42no—
354#103—42no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—44no—
360#102—44no—
361#102—44no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364#97—44no—
365C4×C8—24no—
366#95—21no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—28no—
388C24—24no—
389C24—44no—
390C24—44no—
391C24—82no—
392C24—82no—
393C24—82no—
394C24—82no—
395C24C22×D411yescharacteristic
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—82no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×D4—28no—
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—432no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—816no—
430C2×D4C2×#4118yes—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—41no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4#13714yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2511yescharacteristic
465SD16—88no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4AGL(1,ℤ/8)14yes—
505C4⋊C4H(5,𝔽2)12yes—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—81no—
517C22⋊C4—82no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—42no—
528C42—42no—
529C42—44no—
530C42—42no—
531C42—82no—
532C42—82no—
533C42C2×#4114yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—416no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—88no—
549C23—88no—
550C23—164no—
551C23—164no—
552C23—164no—
553C23—164no—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—42no—
557Q8—42no—
558Q8—88no—
559Q8—162no—
560D4—28no—
561D4—28no—
562D4—416no—
563D4—416no—
564D4—416no—
565D4—44no—
566D4—44no—
567D4—88no—
568D4—88no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—168no—
618C2×C4—168no—
619C2×C4#52512yes—
620C2×C4#52512yes—
621C2×C4C2×AGL(1,ℤ/8)14yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4#57612yes—
626C2×C4C22×#4112yes—
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4#58512yes—
629C2×C4#58511yescharacteristic
630C8—42no—
631C8—44no—
632C8—42no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22—1616no—
648C22C2×#52511yescharacteristic
649C22#348811yescharacteristic
650C22#350011yescharacteristic
651C22C22×AGL(1,ℤ/8)11yescharacteristic
652C22C22×#13711yescharacteristic
653C22C2×#57611yescharacteristic
654C22#358511yescharacteristic
655C4—24no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—41no—
664C4—82no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4#311014yes—
672C4#351412yes—
673C4#352112yes—
674C2—42no—
675C2—88no—
676C2#3607511yescharacteristic
677C2C2×#311011yescharacteristic
678C2#6057811yescharacteristic
679C2#6196511yescharacteristic
680C2#6197911yescharacteristic
681C2#6253711yescharacteristic
682C2#6258811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36480C2semidirect
3. #30128C2semidirect
4. #29939C2semidirect
5. #29498C2semidirect
6. #28201C2semidirect
7. #27541C2semidirect
8. #27245C2semidirect
9. #26460C2semidirect
11. #25322C2semidirect
13. #23822C2semidirect
14. #23757C2semidirect
15. #23223C2semidirect
16. #23101C2semidirect
17. #22636C2semidirect
21. C2×#474C22semidirect
24. #2863C22semidirect
27. #2567C22semidirect
28. #2551C22semidirect
34. #2369C22semidirect
40. #2355C22semidirect
46. C4×C4⋊Q8C22semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
55. #2083C22semidirect
56. #2083C22semidirect
59. #2072C22semidirect
60. #2071C22semidirect
61. #2067C22semidirect
62. #2055C22semidirect
67. #2023C22semidirect
69. #1987C22semidirect
71. #1973C22semidirect
73. #1947C22semidirect
74. #1939C22semidirect
75. #1936C22semidirect
76. #1925C22semidirect
77. #1925C22semidirect
79. #1877C22semidirect
80. #1877C22semidirect
81. #1862C22semidirect
82. #1862C22semidirect
83. #1831C22semidirect
84. #1802C22semidirect
88. C4×#107C22semidirect
89. #1770C22semidirect
90. #1762C22semidirect
91. #1762C22semidirect
92. C4×#103C22semidirect
94. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#364802
#342102
#301282
#299392
#294982
#282012
#275412
#272452
#264602
#258552
#253222
#242112
#238222
#237572
#232232
#231012
#226362