Finite Group Databasebeta

← #6565916

Subgroups — #6565916

Order 512.

1661 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34331C211yesmaximal, characteristic, complemented
3#30045C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29552C212yesmaximal, complemented
6#29393C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#28156C211yesmaximal, characteristic
9#27563C214yesmaximal, complemented
10#27075C214yesmaximal, complemented
11#24993C212yesmaximal, complemented
12#24186C212yesmaximal, complemented
13#23817C212yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23093C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22767C212yesmaximal, complemented
18C2×#559C2211yescharacteristic
19C2×#556—21no—
20C2×#459C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2858C2212yes—
24#2547—21no—
25#2545—22no—
26#2538—21no—
27#2538C2212yes—
28#2512—22no—
29#2512C2214yes—
30#2367—21no—
31#2367C2212yes—
32#2366C2211yescharacteristic
33#2359—22no—
34#2355C2211yescharacteristic
35#2354—21no—
36#2354—21no—
37#2354C2212yes—
38Q8×C4⋊C4C2211yescharacteristic
39#2349C2212yes—
40#2335—22no—
41#2326—21no—
42C4×C4⋊Q8C2212yes—
43C4×#127C2211yescharacteristic
44C4×#127C2211yescharacteristic
45C4×#124C2212yes—
46#2295C2211yescharacteristic
47#2293C2212yes—
48#2275—21no—
49C42×Q8C2211yescharacteristic
50#2101C2214yes—
51#2100C2212yes—
52#2095C2212yes—
53#2084C2211yescharacteristic
54#2073C2214yes—
55#2072C2211yescharacteristic
56#2071C2212yes—
57#2065C2212yes—
58#2057C2212yes—
59#2055C2212yes—
60#2042C2214yes—
61#2041C2211yescharacteristic
62#2024C2212yes—
63#2023C2212yes—
64#2023C2212yes—
65#2014C2211yescharacteristic
66#2012C2214yes—
67#2011C2212yes—
68#1984C2212yes—
69#1970C2214yes—
70#1957C2212yes—
71#1947C2212yes—
72#1934C2214yes—
73#1925C2214yes—
74#1925C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1916C2214yes—
78#1880C2214yes—
79#1879C2214yes—
80#1879C2214yes—
81#1879C2214yes—
82#1864C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1831C2211yescharacteristic
86#1803C2212yes—
87#1802C2212yes—
88#1802C2214yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2214yes—
92C4×#102C2214yes—
93C4×#102C2212yes—
94#1744C2212yes—
95C22×SD16C2312yes—
96C22×D8C2314yes—
97#559—24no—
98#559—24no—
99#556—48no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#126—24no—
106C2×#124—21no—
107C2×#124—22no—
108C2×#124—22no—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117—22no—
129C2×#117—22no—
130#459—28no—
131C4×SD16—28no—
132C2×#107C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#397—24no—
148#396—24no—
149#396—24no—
150#391—21no—
151#391—22no—
152#391C2311yescharacteristic
153#391C2311yescharacteristic
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390—22no—
160#390—22no—
161#390C2312yes—
162#390C2312yes—
163#390C2314yes—
164#390C2314yes—
165#390C2312yes—
166#389—22no—
167#389—22no—
168#389C2314yes—
169#389C2311yescharacteristic
170#389C2311yescharacteristic
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—24no—
179#388—24no—
180#387—22no—
181#387—21no—
182#387—21no—
183#386—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382C2312yes—
195#380—22no—
196#380—22no—
197#380—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215#376—21no—
216C2×#95—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2411yescharacteristic
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C2×SD16—24no—
247C2×SD16—42no—
248C2×SD16—42no—
249C2×D8—28no—
250C2×D8—44no—
251C2×D8—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—82no—
258#127—22no—
259#127—42no—
260#127#4114yes—
261#126—416no—
262#124—28no—
263#124—24no—
264#124—24no—
265#124—48no—
266#124—84no—
267#123—216no—
268C22⋊Q8—48no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8—48no—
275C4×Q8—44no—
276C4×Q8#4114yes—
277C4×D4—28no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—82no—
289#117—48no—
290#117—84no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—44no—
353#106—24no—
354C4⋊C8—24no—
355C4⋊C8—42no—
356#103—24no—
357#103—44no—
358#102—28no—
359#102—24no—
360#102—44no—
361#102—42no—
362#102—44no—
363#102—42no—
364#102—44no—
365C4×C8—24no—
366C4×C8—44no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—21no—
377#95—24no—
378#95—22no—
379#95—21no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8C2×#4112yes—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4C2×#4114yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447SD16—48no—
448D8—88no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—42no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4AGL(1,ℤ/8)14yes—
487C4⋊C4C2×#4112yes—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—41no—
517C42—48no—
518C42—42no—
519C42—82no—
520C42—82no—
521C42#13514yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—44no—
541Q8—42no—
542Q8—88no—
543D4—28no—
544D4—48no—
545D4—44no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—84no—
552D4—84no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4—164no—
599C2×C4—164no—
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4C2×#13512yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×AGL(1,ℤ/8)14yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4#57612yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4#58512yes—
611C2×C4#58512yes—
612C8—42no—
613C8—42no—
614C8—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22C2×#52511yescharacteristic
630C22#348711yescharacteristic
631C22#350011yescharacteristic
632C22C22×#13511yescharacteristic
633C22C22×AGL(1,ℤ/8)11yescharacteristic
634C22C2×#57611yescharacteristic
635C22#358611yescharacteristic
636C4—22no—
637C4—24no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—82no—
648C4—82no—
649C4—81no—
650C4—81no—
651C4—84no—
652C4#310614yes—
653C4#349312yes—
654C4#352112yes—
655C2—84no—
656C2—162no—
657C2#3601911yescharacteristic
658C2C2×#310611yescharacteristic
659C2#6059611yescharacteristic
660C2#6188211yescharacteristic
661C2#6190711yescharacteristic
662C2#6252711yescharacteristic
663C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28156C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #34331C2semidirect
3. #30045C2semidirect
4. #29984C2semidirect
5. #29552C2semidirect
6. #29393C2semidirect
7. #29139C2semidirect
9. #27563C2semidirect
10. #27075C2semidirect
11. #24993C2semidirect
12. #24186C2semidirect
13. #23817C2semidirect
14. #23678C2semidirect
15. #23093C2semidirect
16. #22828C2semidirect
17. #22767C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#343312
#300452
#299842
#295522
#293932
#291392
#281562
#275632
#270752
#249932
#241862
#238172
#236782
#230932
#228282
#227672