Finite Group Databasebeta

← #6543859

Subgroups — #6543859

Order 512.

1755 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30039C212yesmaximal, complemented
4#29663C211yesmaximal, characteristic
5#29390C212yesmaximal, complemented
6#29347C212yesmaximal, complemented
7#27972C211yesmaximal, characteristic, complemented
8#27647C211yesmaximal, characteristic, complemented
9#26722C214yesmaximal, complemented
10#26624C214yesmaximal, complemented
11#26024C212yesmaximal, complemented
12#24630C212yesmaximal, complemented
13#23678C212yesmaximal, complemented
14#23570C211yesmaximal, characteristic
15#22996C212yesmaximal, complemented
16#22770C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#483C2212yes—
21C2×#461C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2214yes—
24#2862C2212yes—
25#2857C2211yescharacteristic
26#2612—22no—
27#2574—22no—
28#2551C2211yescharacteristic
29#2546C2211yescharacteristic
30#2508—21no—
31#2492C2212yes—
32#2439—21no—
33#2429C2212yes—
34#2369—21no—
35#2365C2211yescharacteristic
36#2361—22no—
37#2361C2214yes—
38#2355C2211yescharacteristic
39#2354—21no—
40#2354C2212yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43#2346—21no—
44#2326—22no—
45#2311—21no—
46C4×C4⋊Q8C2212yes—
47C4×#127C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49#2295C2212yes—
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2084C2212yes—
53#2072C2212yes—
54#2072C2212yes—
55#2072C2212yes—
56#2065C2214yes—
57#2058C2211yescharacteristic
58#2055C2214yes—
59#2041C2211yescharacteristic
60#2034C2212yes—
61#2024C2211yescharacteristic
62#2024C2212yes—
63#2023C2214yes—
64#2014C2211yescharacteristic
65#2014C2212yes—
66#2013C2214yes—
67#2000C2212yes—
68#2000C2214yes—
69#1987C2214yes—
70#1981C2214yes—
71#1956C2212yes—
72#1948C2212yes—
73#1939C2212yes—
74#1933C2214yes—
75#1933C2214yes—
76#1927C2214yes—
77#1923C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1877C2214yes—
81#1877C2214yes—
82#1872C2214yes—
83#1865C2214yes—
84#1839C2212yes—
85#1834C2212yes—
86#1823C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#102C2212yes—
93C4×#102C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—21no—
96C22×SD16C2314yes—
97C22×SD16C2312yes—
98#532—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121C2312yes—
110#520—48no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×D4—24no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2314yes—
120C2×#117—22no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128#483—44no—
129#461—28no—
130C4×SD16—28no—
131C4×SD16—48no—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#400—22no—
150#398—22no—
151#398—22no—
152#396—22no—
153#395—22no—
154#393—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2311yescharacteristic
162#391C2312yes—
163#391C2312yes—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389C2314yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—24no—
181#388—24no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—24no—
188#385—21no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#382—21no—
197#382—22no—
198#382—22no—
199#380—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2411yescharacteristic
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—24no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—24no—
252C2×SD16—28no—
253C2×SD16—42no—
254C2×SD16—44no—
255C2×SD16—42no—
256C2×SD16—44no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265#127—24no—
266#124—28no—
267C22⋊Q8—22no—
268C22⋊Q8—41no—
269C22⋊Q8—44no—
270#121—28no—
271#121—24no—
272#121—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—48no—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—416no—
282C4×D4—48no—
283C4×D4—84no—
284C4×D4#4118yes—
285#117—42no—
286#117—48no—
287#117SD1618yescomplemented
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
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—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349#106—24no—
350#106—28no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#103—24no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358#102—24no—
359#102—24no—
360#102—44no—
361#102—42no—
362C22⋊C8—42no—
363C22⋊C8—42no—
364C4×C8—24no—
365C4×C8—44no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—22no—
372#95—24no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—22no—
388C24C22×D411yescharacteristic
389#41—432no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C2×#4114yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—41no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C2×SD1614yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—48no—
457SD16—416no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×SD1614yes—
497C4⋊C4C2×SD1614yes—
498C4⋊C4C2×#4114yes—
499C4⋊C4H(5,𝔽2)12yes—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—81no—
508C22⋊C4—82no—
509C22⋊C4C2×SD1618yes—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—48no—
522C42—42no—
523C42—82no—
524C42C2×SD1618yes—
525C42C2×#4114yes—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C22×SD1612yes—
540C23C23×D411yescharacteristic
541Q8—48no—
542Q8—42no—
543Q8—44no—
544Q8—44no—
545Q8—42no—
546Q8—164no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—48no—
551D4—416no—
552D4—42no—
553D4—42no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—168no—
603C2×C4—164no—
604C2×C4#52512yes—
605C2×C4#52512yes—
606C2×C4C22×SD1614yes—
607C2×C4C22×SD1612yes—
608C2×C4C22×SD1614yes—
609C2×C4C22×SD1612yes—
610C2×C4#57512yes—
611C2×C4#57812yes—
612C2×C4C22×#4112yes—
613C2×C4C2×H(5,𝔽2)11yescharacteristic
614C2×C4#58511yescharacteristic
615C2×C4#58512yes—
616C8—42no—
617C8—42no—
618C8—44no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22C2×#52511yescharacteristic
634C22#348811yescharacteristic
635C22#350011yescharacteristic
636C22C23×SD1611yescharacteristic
637C22C2×#57511yescharacteristic
638C22C2×#57811yescharacteristic
639C22#358511yescharacteristic
640C4—24no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—24no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—41no—
650C4—82no—
651C4—81no—
652C4—81no—
653C4—82no—
654C4—81no—
655C4—84no—
656C4#310114yes—
657C4#349312yes—
658C4#350612yes—
659C2—42no—
660C2—84no—
661C2#3575611yescharacteristic
662C2C2×#310111yescharacteristic
663C2#6058911yescharacteristic
664C2#6195111yescharacteristic
665C2#6198511yescharacteristic
666C2#6252611yescharacteristic
667C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #36288C2semidirect
3. #30039C2semidirect
5. #29390C2semidirect
6. #29347C2semidirect
7. #27972C2semidirect
8. #27647C2semidirect
9. #26722C2semidirect
10. #26624C2semidirect
11. #26024C2semidirect
12. #24630C2semidirect
13. #23678C2semidirect
15. #22996C2semidirect
16. #22770C2semidirect
17. #22639C2semidirect
287. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362882
#342072
#300392
#296632
#293902
#293472
#279722
#276472
#267222
#266242
#260242
#246302
#236782
#235702
#229962
#227702
#226392