Finite Group Databasebeta

← #6534716

Subgroups — #6534716

Order 512.

1719 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30077C212yesmaximal, complemented
4#29687C211yesmaximal, characteristic
5#29391C212yesmaximal, complemented
6#29315C212yesmaximal, complemented
7#27948C211yesmaximal, characteristic, complemented
8#27648C211yesmaximal, characteristic, complemented
9#27031C214yesmaximal, complemented
10#26555C214yesmaximal, complemented
11#26091C212yesmaximal, complemented
12#24681C212yesmaximal, complemented
13#23678C212yesmaximal, complemented
14#23570C211yesmaximal, characteristic
15#22962C212yesmaximal, complemented
16C4⋊C4×SD16C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#484C2212yes—
21C2×#474C2212yes—
22C2×C4×SD16C2212yes—
23C2×C4×SD16C2214yes—
24#2864C2212yes—
25#2857C2211yescharacteristic
26#2571C2211yescharacteristic
27#2568C2211yescharacteristic
28#2533—22no—
29#2530—22no—
30#2511—21no—
31#2499—21no—
32#2495C2212yes—
33#2421C2212yes—
34#2365C2211yescharacteristic
35#2362—21no—
36#2361—22no—
37#2361C2214yes—
38#2356—21no—
39#2354—21no—
40#2354C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44#2326—22no—
45#2311—21no—
46C4×C4⋊Q8C2212yes—
47C4×#127C2212yes—
48C4×C22⋊Q8C2211yescharacteristic
49#2295C2212yes—
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2092C2214yes—
53#2084C2212yes—
54#2084C2212yes—
55#2083C2212yes—
56#2073C2214yes—
57#2072C2212yes—
58#2065C2214yes—
59#2058C2211yescharacteristic
60#2057C2212yes—
61#2055C2214yes—
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2034C2212yes—
65#2033C2214yes—
66#2024C2211yescharacteristic
67#2014C2211yescharacteristic
68#2000C2212yes—
69#1990C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1947C2212yes—
73#1942C2214yes—
74#1941C2214yes—
75#1939C2212yes—
76#1928C2214yes—
77#1923C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1865C2214yes—
81#1864C2214yes—
82#1862C2214yes—
83#1853C2214yes—
84#1844C2212yes—
85#1839C2212yes—
86#1824C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1797C2211yescharacteristic
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#102C2212yes—
93C4×#102C2214yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—21no—
96C22×SD16C2312yes—
97#532—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#124—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×#121—21no—
105C2×#121C2312yes—
106#520—48no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
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#484—44no—
126#474—28no—
127C4×SD16—28no—
128C4×SD16—48no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#400—22no—
149#400—22no—
150#400—22no—
151#397—22no—
152#397—22no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—21no—
158#391C2312yes—
159#391C2311yescharacteristic
160#390—22no—
161#390—22no—
162#390C2314yes—
163#389—21no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389C2314yes—
168#389C2312yes—
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#388—22no—
174#388—22no—
175#388—24no—
176#388—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#386—24no—
187#385—21no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#383—22no—
197#382—21no—
198#382C2312yes—
199#382C2312yes—
200#380—21no—
201#380—22no—
202#380—22no—
203#380C2311yescharacteristic
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
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#336C2314yes—
232#336C2311yescharacteristic
233#336C2314yes—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—24no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—24no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258#127—24no—
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—24no—
268C4×Q8—42no—
269C4×Q8—48no—
270C4×Q8—44no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—416no—
277C4×D4—84no—
278C4×D4#4118yes—
279#117—24no—
280#117—24no—
281#117—48no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342#107—28no—
343#107—44no—
344#106—24no—
345#106—44no—
346#106—44no—
347C4⋊C8—28no—
348C4⋊C8—24no—
349#103—28no—
350#103—24no—
351#102—24no—
352#102—28no—
353#102—24no—
354#102—42no—
355#102—44no—
356C22⋊C8—24no—
357C22⋊C8—42no—
358C4×C8—24no—
359C4×C8—44no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24C22×D411yescharacteristic
385#41—432no—
386C2×Q8—24no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8H(5,𝔽2)12yes—
396C2×D4—24no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C2×#4114yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4#13514yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448SD16—216no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C82−1+414yes—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4AGL(1,ℤ/8)14yes—
490C4⋊C4C2×#4114yes—
491C22⋊C4—28no—
492C22⋊C4—24no—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—42no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—81no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—42no—
511C42—44no—
512C42—48no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42C2×#4114yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—24no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C2×#13512yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—48no—
534Q8—42no—
535Q8—84no—
536D4—24no—
537D4—24no—
538D4—48no—
539D4—48no—
540D4—416no—
541D4—42no—
542D4—42no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—416no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—168no—
593C2×C4—164no—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C2×AGL(1,ℤ/8)14yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4#57712yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+412yes—
604C2×C4#58511yescharacteristic
605C8—24no—
606C8—44no—
607C8—42no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22C2×#52511yescharacteristic
623C22#348811yescharacteristic
624C22#349911yescharacteristic
625C22C22×#13511yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C2×#57711yescharacteristic
628C22#358611yescharacteristic
629C4—24no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—44no—
639C4—82no—
640C4—81no—
641C4—84no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4#310514yes—
646C4#349212yes—
647C4#350712yes—
648C2—42no—
649C2—84no—
650C2#3568611yescharacteristic
651C2C2×#310511yescharacteristic
652C2#6058511yescharacteristic
653C2#6195411yescharacteristic
654C2#6198311yescharacteristic
655C2#6247011yescharacteristic
656C2#6250911yescharacteristic
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. #36314C2semidirect
3. #30077C2semidirect
5. #29391C2semidirect
6. #29315C2semidirect
7. #27948C2semidirect
8. #27648C2semidirect
9. #27031C2semidirect
10. #26555C2semidirect
11. #26091C2semidirect
12. #24681C2semidirect
13. #23678C2semidirect
15. #22962C2semidirect
16. C4⋊C4×SD16C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#342072
#300772
#296872
#293912
#293152
#279482
#276482
#270312
#265552
#260912
#246812
#236782
#235702
#229622
C4⋊C4×SD162
#226392