Finite Group Databasebeta

← #6993604

Subgroups — #6993604

Order 512.

1495 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic, complemented
3#32824C212yesmaximal, complemented
4#32302C214yesmaximal, complemented
5#31954C214yesmaximal
6#30309C212yesmaximal, complemented
7#29980C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic, complemented
9#29560C212yesmaximal, complemented
10#28539C212yesmaximal, complemented
11#27738C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic, complemented
13#25455C212yesmaximal, complemented
14#24815C212yesmaximal, complemented
15#24299C211yesmaximal, characteristic
16#23568C211yesmaximal, characteristic
17#22636C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#500C2212yes—
21C2×#477C2212yes—
22C2×#476C2212yes—
23C2×C4×SD16C2212yes—
24#2868C2212yes—
25#2864C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2660—22no—
29#2571C2211yescharacteristic
30#2567C2212yes—
31#2567C2212yes—
32#2563C2211yescharacteristic
33#2546—21no—
34#2546—21no—
35#2527—22no—
36#2502C2214yes—
37#2494—21no—
38#2494C2212yes—
39#2485—21no—
40#2477—22no—
41#2434—22no—
42#2420C2212yes—
43#2369C2211yescharacteristic
44#2352—21no—
45D4×C4⋊C4C2211yescharacteristic
46#2344—21no—
47#2317—21no—
48#2317—21no—
49C4×#121C2211yescharacteristic
50#2293C2212yes—
51C4×#117C2211yescharacteristic
52#2099C2214yes—
53#2097C2214yes—
54#2093C2214yes—
55#2084C2214yes—
56#2083C2214yes—
57#2074C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2058C2211yescharacteristic
61#2058C2212yes—
62#2057C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2036C2214yes—
68#2035C2212yes—
69#2023C2211yescharacteristic
70#2023C2212yes—
71#2013C2211yescharacteristic
72#2000C2212yes—
73#1990C2212yes—
74#1987C2212yes—
75#1973C2214yes—
76#1947C2214yes—
77#1939C2212yes—
78#1935C2214yes—
79#1928C2214yes—
80#1924C2214yes—
81#1877C2214yes—
82#1874C2212yes—
83#1853C2212yes—
84#1852C2214yes—
85#1847C2214yes—
86#1836C2214yes—
87#1831C2212yes—
88#1803C2212yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C23×Q8—21no—
96C22×SD16C2312yes—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—22no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C42C2311yescharacteristic
126#500—44no—
127#477—44no—
128#476—44no—
129C4×SD16—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#401—22no—
149#400—22no—
150#398—24no—
151#398—24no—
152#397—22no—
153#397—22no—
154#395—24no—
155#395—24no—
156#394—22no—
157#393—22no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2314yes—
163#391C2314yes—
164#390C2312yes—
165#390C2311yescharacteristic
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—21no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—21no—
188#386—22no—
189#385—24no—
190#385—24no—
191#385—22no—
192#384—22no—
193#384—22no—
194#384C2314yes—
195#384C2314yes—
196#384C2311yescharacteristic
197#382—21no—
198#382—22no—
199#382—22no—
200#382C2312yes—
201#382C2312yes—
202#381—22no—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207#380C2312yes—
208#380C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—82no—
240C22×Q8—82no—
241C22×Q8C2412yes—
242C22×D4—21no—
243C22×D4C2411yescharacteristic
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—42no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—24no—
259C4⋊Q8—44no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—84no—
265C4⋊Q8—84no—
266#125—44no—
267C22⋊Q8—42no—
268C22⋊Q8—42no—
269C22⋊Q8—44no—
270C22⋊Q8—82no—
271#121—22no—
272#121—41no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—44no—
276C4×D4—24no—
277C4×D4—24no—
278C4×D4—42no—
279C4×D4—44no—
280#117—24no—
281#117—24no—
282#117—44no—
283#117—84no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
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—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344#107—44no—
345#107—44no—
346#107—44no—
347#106—42no—
348#106—44no—
349C4⋊C8—24no—
350C4⋊C8—44no—
351#103—24no—
352#103—44no—
353#103—42no—
354#103—42no—
355#103—44no—
356#102—42no—
357#102—44no—
358C22⋊C8—24no—
359C22⋊C8—42no—
360#97—28no—
361C4×C8—24no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×Q8—816no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C2×D42−1+412yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4#13714yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448SD16—48no—
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—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4#13714yes—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—42no—
515C42—44no—
516C42—42no—
517C42—82no—
518C42H(5,𝔽2)14yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×#13712yes—
533C23C23×D411yescharacteristic
534Q8—42no—
535Q8—42no—
536Q8—42no—
537Q8—42no—
538Q8—84no—
539Q8—88no—
540Q8—88no—
541Q8—162no—
542D4—24no—
543D4—42no—
544D4—84no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
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—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4—88no—
595C2×C4—84no—
596C2×C4#53512yes—
597C2×C4#53512yes—
598C2×C4C2×#13712yes—
599C2×C4C2×#13714yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4#57612yes—
603C2×C4C2×H(5,𝔽2)12yes—
604C2×C4C2×2−1+411yescharacteristic
605C2×C4C2×2−1+412yes—
606C2×C4#58511yescharacteristic
607C8—24no—
608C8—42no—
609C8—44no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22C2×#53511yescharacteristic
625C22#348911yescharacteristic
626C22#349911yescharacteristic
627C22C22×#13711yescharacteristic
628C22C22×#13711yescharacteristic
629C22C2×#57611yescharacteristic
630C22#358511yescharacteristic
631C4—22no—
632C4—22no—
633C4—22no—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4#321814yes—
648C4#353812yes—
649C4#354712yes—
650C2—42no—
651C2—82no—
652C2#4299411yescharacteristic
653C2C2×#321811yescharacteristic
654C2#6081111yescharacteristic
655C2#6210811yescharacteristic
656C2#6212711yescharacteristic
657C2#6248011yescharacteristic
658C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. #35281C2semidirect
3. #32824C2semidirect
4. #32302C2semidirect
6. #30309C2semidirect
7. #29980C2semidirect
8. #29691C2semidirect
9. #29560C2semidirect
10. #28539C2semidirect
11. #27738C2semidirect
12. #25813C2semidirect
13. #25455C2semidirect
14. #24815C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#352812
#328242
#323022
#319542
#303092
#299802
#296912
#295602
#285392
#277382
#258132
#254552
#248152
#242992
#235682
#226362