Finite Group Databasebeta

← #6544576

Subgroups — #6544576

Order 512.

1659 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36450C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30127C212yesmaximal, complemented
4#29687C211yesmaximal, characteristic
5#29520C212yesmaximal, complemented
6#29280C212yesmaximal, complemented
7#27960C211yesmaximal, characteristic, complemented
8#27680C211yesmaximal, characteristic, complemented
9#27048C214yesmaximal, complemented
10#26736C214yesmaximal, complemented
11#26021C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#23824C212yesmaximal, complemented
14#23570C211yesmaximal, characteristic
15#22968C212yesmaximal, complemented
16#22762C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#474C2212yes—
21C2×C4×SD16C2212yes—
22#2863C2212yes—
23#2857C2211yescharacteristic
24#2577—22no—
25#2568C2211yescharacteristic
26#2552C2211yescharacteristic
27#2528—22no—
28#2514—21no—
29#2499—21no—
30#2494C2212yes—
31#2489C2212yes—
32#2365C2211yescharacteristic
33#2362—21no—
34#2361—22no—
35#2361C2214yes—
36#2355—21no—
37#2355C2211yescharacteristic
38#2354—21no—
39#2354C2212yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2212yes—
42#2326—22no—
43#2311—21no—
44C4×C4⋊Q8C2212yes—
45C4×#127C2212yes—
46C4×#125C2212yes—
47C4×C22⋊Q8C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2099C2212yes—
53#2096C2212yes—
54#2073C2214yes—
55#2065C2214yes—
56#2058C2211yescharacteristic
57#2057C2212yes—
58#2055C2214yes—
59#2041C2211yescharacteristic
60#2041C2212yes—
61#2034C2212yes—
62#2033C2214yes—
63#2025C2214yes—
64#2024C2211yescharacteristic
65#2014C2211yescharacteristic
66#2000C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1949C2214yes—
70#1948C2212yes—
71#1943C2214yes—
72#1939C2212yes—
73#1935C2214yes—
74#1928C2214yes—
75#1924C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1877C2214yes—
80#1872C2212yes—
81#1864C2214yes—
82#1853C2212yes—
83#1834C2212yes—
84#1831C2212yes—
85#1823C2214yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89#1770C2212yes—
90#1762C2214yes—
91#1762C2214yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97#532—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#125C2312yes—
102C2×#124—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2311yescharacteristic
105C2×#121—21no—
106C2×#121C2312yes—
107#520—48no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—24no—
112C2×C4×D4—21no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C2×#117—22no—
118C2×#117D411yescharacteristic
119C2×#117D411yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#474—28no—
126C4×SD16—28no—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143#401—22no—
144#400—22no—
145#400—22no—
146#398—22no—
147#397—22no—
148#397—22no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—21no—
154#391C2312yes—
155#391C2311yescharacteristic
156#391C2312yes—
157#390—22no—
158#390—22no—
159#390C2314yes—
160#389—21no—
161#389—22no—
162#389—21no—
163#389C2311yescharacteristic
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—22no—
170#388—24no—
171#388—24no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—21no—
178#387—22no—
179#387—22no—
180#387—22no—
181#386—24no—
182#385—21no—
183#385—24no—
184#385—24no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#383—22no—
192#382—21no—
193#382—22no—
194#382C2312yes—
195#380—21no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2311yescharacteristic
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—21no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216#376D412yes—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2314yes—
231#336C2311yescharacteristic
232C2×#41—28no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8C2411yescharacteristic
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×SD16—24no—
249C2×SD16—42no—
250C2×SD16—42no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256#127—24no—
257#125—24no—
258#125—42no—
259#124—28no—
260C22⋊Q8—22no—
261C22⋊Q8—41no—
262C22⋊Q8—44no—
263#121—28no—
264#121—24no—
265#121—24no—
266C4×Q8—24no—
267C4×Q8—48no—
268C4×Q8—44no—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—48no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—416no—
275C4×D4—84no—
276C4×D4#4118yes—
277#117—24no—
278#117—42no—
279#117—48no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—28no—
340#106—24no—
341#106—44no—
342C4⋊C8—24no—
343C4⋊C8—44no—
344#103—24no—
345#103—44no—
346#102—24no—
347#102—28no—
348#102—24no—
349#102—44no—
350#102—42no—
351C22⋊C8—42no—
352C22⋊C8—42no—
353#97—44no—
354C4×C8—24no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—24no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—22no—
379C24C22×D411yescharacteristic
380#41—432no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×D4—24no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4C2×#4114yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—41no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4AGL(1,ℤ/8)14yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2511yescharacteristic
443SD16—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4C2×#4114yes—
485C4⋊C42−1+412yes—
486C22⋊C4—28no—
487C22⋊C4—24no—
488C22⋊C4—28no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—81no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—22no—
502C42—24no—
503C42—22no—
504C42—42no—
505C42—42no—
506C42—48no—
507C42—44no—
508C42—42no—
509C42—42no—
510C42—82no—
511C42C2×#4114yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—24no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23C2×AGL(1,ℤ/8)12yes—
526C23C23×D411yescharacteristic
527Q8—48no—
528Q8—42no—
529Q8—42no—
530Q8—84no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—48no—
535D4—416no—
536D4—42no—
537D4—42no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—416no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—168no—
589C2×C4—164no—
590C2×C4#52512yes—
591C2×C4#52512yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4#57512yes—
595C2×C4#57712yes—
596C2×C4C22×#4112yes—
597C2×C4C2×2−1+411yescharacteristic
598C2×C4#58511yescharacteristic
599C2×C4#58512yes—
600C8—44no—
601C8—42no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C2×#52511yescharacteristic
618C22#348911yescharacteristic
619C22#350011yescharacteristic
620C22C22×AGL(1,ℤ/8)11yescharacteristic
621C22C2×#57511yescharacteristic
622C22C2×#57711yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—81no—
639C4—81no—
640C4#311414yes—
641C4#349312yes—
642C4#350712yes—
643C2—42no—
644C2—84no—
645C2#3575911yescharacteristic
646C2C2×#311411yescharacteristic
647C2#6058511yescharacteristic
648C2#6211111yescharacteristic
649C2#6213411yescharacteristic
650C2#6253911yescharacteristic
651C2#6259111yescharacteristic
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. #36450C2semidirect
3. #30127C2semidirect
5. #29520C2semidirect
6. #29280C2semidirect
7. #27960C2semidirect
8. #27680C2semidirect
9. #27048C2semidirect
10. #26736C2semidirect
11. #26021C2semidirect
12. #24681C2semidirect
13. #23824C2semidirect
15. #22968C2semidirect
16. #22762C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#364502
#342072
#301272
#296872
#295202
#292802
#279602
#276802
#270482
#267362
#260212
#246812
#238242
#235702
#229682
#227622
#226362