Finite Group Databasebeta

← #6571129

Subgroups — #6571129

Order 512.

1905 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36450C211yesmaximal, characteristic, complemented
2#34210C211yesmaximal, characteristic
3#30128C212yesmaximal, complemented
4#29937C211yesmaximal, characteristic, complemented
5#29492C212yesmaximal, complemented
6#28207C212yesmaximal, complemented
7#27525C214yesmaximal, complemented
8#27227C214yesmaximal, complemented
9#26467C212yesmaximal, complemented
10#25855C211yesmaximal, characteristic
11#25325C212yesmaximal, complemented
12#24211C211yesmaximal, characteristic
13#23819C212yesmaximal, complemented
14#23757C211yesmaximal, characteristic, complemented
15#23216C212yesmaximal, complemented
16#23095C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C22×#127C2211yescharacteristic
19C2×#518—21no—
20C2×#486C2212yes—
21C2×#478C2214yescomplemented
22C2×#460C2212yes—
23C2×#450C2212yes—
24#2863C2212yescomplemented
25#2857C2211yescharacteristic
26#2577—22no—
27#2568C2211yescharacteristic, complemented
28#2552C2211yescharacteristic, complemented
29#2528—22no—
30#2514—21no—
31#2499—21no—
32#2494C2212yes—
33#2489C2212yes—
34#2366—21no—
35#2366—21no—
36#2366C2212yes—
37#2366C2212yes—
38#2365C2211yescharacteristic, complemented
39#2362—21no—
40#2355—21no—
41#2355C2211yescharacteristic, complemented
42D4×C22⋊C4—22no—
43D4×C22⋊C4C2214yes—
44#2328—22no—
45#2309—21no—
46C4×#127C2212yes—
47C4×#127C2212yes—
48C4×#125C2212yescomplemented
49C4×C22⋊Q8C2211yescharacteristic
50#2285C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2211yescharacteristic, complemented
53#2100C2212yescomplemented
54#2095C2211yescharacteristic, complemented
55#2084C2212yescomplemented
56#2083C2211yescharacteristic, complemented
57#2077C2214yes—
58#2076C2212yes—
59#2073C2214yescomplemented
60#2072C2211yescharacteristic, complemented
61#2067C2214yescomplemented
62#2055C2214yescomplemented
63#2041C2212yes—
64#2041C2212yes—
65#2041C2212yes—
66#2036C2212yes—
67#2025C2214yescomplemented
68#2014C2212yes—
69#1988C2212yescomplemented
70#1982C2212yes—
71#1968C2214yescomplemented
72#1957C2212yes—
73#1948C2212yescomplemented
74#1939C2212yescomplemented
75#1934C2214yescomplemented
76#1926C2214yescomplemented
77#1925C2214yescomplemented
78#1923C2214yes—
79#1877C2214yescomplemented
80#1877C2214yescomplemented
81#1864C2214yescomplemented
82#1862C2214yescomplemented
83#1831C2212yescomplemented
84#1804C2214yescomplemented
85#1797C2211yescharacteristic
86#1786C2214yes—
87#1786C2212yes—
88C4×#107C2212yescomplemented
89C8×C4⋊C4C2212yescomplemented
90C4×#103C2212yescomplemented
91C4×#102C2214yescomplemented
92C4×#102C2214yescomplemented
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yescomplemented
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127—28no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127C2312yes—
104C2×#127C2312yes—
105C2×#125C2312yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8C2311yescharacteristic
108#518—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4C2314yes—
118C2×C4×D4C2314yes—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C22⋊C4—24no—
123C22×C42C2311yescharacteristic
124#486—28no—
125#478—48no—
126#460—28no—
127#450—44no—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2314yes—
146#401—22no—
147#400—22no—
148#400—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391C2312yes—
157#391C2311yescharacteristic
158#391C2312yes—
159#390—22no—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#389—21no—
168#389—21no—
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—24no—
185#386—24no—
186#386—24no—
187#386—24no—
188#385—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#383—22no—
194#382—21no—
195#382—21no—
196#382—22no—
197#382C2312yes—
198#380—21no—
199#380—21no—
200#380C2311yescharacteristic
201#380C2312yes—
202#380C2312yes—
203#379—24no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
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⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
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#336C2311yescharacteristic
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C25—22no—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×D4—28no—
238C22×D4—28no—
239C22×D4—28no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—44no—
246C22×D4—42no—
247C22×D4—42no—
248C22×D4—44no—
249C22×D4—44no—
250C22×D4—82no—
251C22×D4—82no—
252C22×D4C2414yes—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×SD16—28no—
259C2×SD16—44no—
260C2×SD16—44no—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265#127—28no—
266#127—28no—
267#127—28no—
268#127—24no—
269#127#4118yes—
270#125—24no—
271#125—42no—
272C22⋊Q8—22no—
273C22⋊Q8—41no—
274C22⋊Q8—44no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×D4—48no—
278C4×D4—48no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—48no—
284C4×D4—48no—
285C4×D4—84no—
286C4×D4—84no—
287#117—24no—
288#117—24no—
289#117—44no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—416no—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—24no—
346#107—42no—
347#106—44no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#103—42no—
351#103—42no—
352#102—28no—
353#102—28no—
354#102—28no—
355#102—28no—
356#102—44no—
357#102—44no—
358#102—44no—
359C22⋊C8—42no—
360C22⋊C8—42no—
361C4×C8—24no—
362C4×C8—44no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—28no—
387C24—24no—
388C24—44no—
389C24—44no—
390C24—82no—
391C24—82no—
392C24—82no—
393C24—82no—
394C24C22×D411yescharacteristic
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—432no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—816no—
426C2×D4C2×#4118yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—41no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4#13514yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2511yescharacteristic
461SD16—88no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4AGL(1,ℤ/8)14yes—
500C4⋊C4H(5,𝔽2)12yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—81no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—44no—
527C42—82no—
528C42C2×#4114yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—28no—
536C23—416no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—88no—
544C23—88no—
545C23—164no—
546C23—164no—
547C23—164no—
548C23—164no—
549C23C2×#13512yes—
550C23C23×D411yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—84no—
554D4—28no—
555D4—28no—
556D4—416no—
557D4—416no—
558D4—416no—
559D4—44no—
560D4—44no—
561D4—88no—
562D4—88no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—168no—
612C2×C4—168no—
613C2×C4#52512yes—
614C2×C4#52512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×AGL(1,ℤ/8)14yes—
618C2×C4C2×AGL(1,ℤ/8)12yes—
619C2×C4#57712yes—
620C2×C4C22×#4112yes—
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4#58512yes—
623C2×C4#58511yescharacteristic
624C8—42no—
625C8—44no—
626C8—42no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22—84no—
641C22—1616no—
642C22C2×#52511yescharacteristic
643C22#348811yescharacteristic
644C22#350011yescharacteristic
645C22C22×#13511yescharacteristic
646C22C22×AGL(1,ℤ/8)11yescharacteristic
647C22C2×#57711yescharacteristic
648C22#358511yescharacteristic
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—41no—
658C4—82no—
659C4—82no—
660C4—81no—
661C4—82no—
662C4—82no—
663C4—81no—
664C4—81no—
665C4#310514yes—
666C4#351412yes—
667C4#352112yes—
668C2—42no—
669C2—88no—
670C2#3607511yescharacteristic
671C2C2×#310511yescharacteristic
672C2#6057911yescharacteristic
673C2#6195311yescharacteristic
674C2#6199911yescharacteristic
675C2#6253111yescharacteristic
676C2#6257611yescharacteristic
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. #36450C2semidirect
3. #30128C2semidirect
4. #29937C2semidirect
5. #29492C2semidirect
6. #28207C2semidirect
7. #27525C2semidirect
8. #27227C2semidirect
9. #26467C2semidirect
11. #25325C2semidirect
13. #23819C2semidirect
14. #23757C2semidirect
15. #23216C2semidirect
16. #23095C2semidirect
17. #22632C2semidirect
21. C2×#478C22semidirect
24. #2863C22semidirect
27. #2568C22semidirect
28. #2552C22semidirect
38. #2365C22semidirect
41. #2355C22semidirect
48. C4×#125C22semidirect
52. #2100C22semidirect
53. #2100C22semidirect
54. #2095C22semidirect
55. #2084C22semidirect
56. #2083C22semidirect
59. #2073C22semidirect
60. #2072C22semidirect
61. #2067C22semidirect
62. #2055C22semidirect
67. #2025C22semidirect
69. #1988C22semidirect
71. #1968C22semidirect
73. #1948C22semidirect
74. #1939C22semidirect
75. #1934C22semidirect
76. #1926C22semidirect
77. #1925C22semidirect
79. #1877C22semidirect
80. #1877C22semidirect
81. #1864C22semidirect
82. #1862C22semidirect
83. #1831C22semidirect
84. #1804C22semidirect
88. C4×#107C22semidirect
89. C8×C4⋊C4C22semidirect
90. C4×#103C22semidirect
91. C4×#102C22semidirect
92. C4×#102C22semidirect
94. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#364502
#342102
#301282
#299372
#294922
#282072
#275252
#272272
#264672
#258552
#253252
#242112
#238192
#237572
#232162
#230952
#226322