Finite Group Databasebeta

← #6490815

Subgroups — #6490815

Order 512.

1711 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#35303C211yesmaximal, characteristic
2C2×#2330C211yesmaximal, characteristic, complemented
3#33111C212yesmaximal, complemented
4#33106C212yesmaximal, complemented
5#30072C212yesmaximal, complemented
6#30071C212yesmaximal, complemented
7#29268C212yesmaximal, complemented
8#29265C212yesmaximal, complemented
9#29216C212yesmaximal, complemented
10#28761C212yesmaximal, complemented
11#27914C211yesmaximal, characteristic, complemented
12#27909C211yesmaximal, characteristic, complemented
13#25913C211yesmaximal, characteristic, complemented
14#25907C211yesmaximal, characteristic, complemented
15#24066C212yesmaximal, complemented
16#24060C212yesmaximal, complemented
17#22232C212yesmaximal, complemented
18#22228C212yesmaximal, complemented
19#20779C211yesmaximal, characteristic, complemented
20C2×#532C2211yescharacteristic
21C2×#529C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#490C2212yes—
24C2×#488C2212yes—
25C2×#483C2212yes—
26C2×#480C2212yes—
27C2×#480C2212yes—
28C2×#480C2212yes—
29#2856C2211yescharacteristic
30#2856C2211yescharacteristic
31#2850C2211yescharacteristic
32#2850C2211yescharacteristic
33#2660C2211yescharacteristic
34#2657C2211yescharacteristic
35#2641C2211yescharacteristic
36#2640C2211yescharacteristic
37#2610—22no—
38#2603—22no—
39#2543—22no—
40#2430—22no—
41#2330—22no—
42#2330—22no—
43#2330C2212yes—
44#2330C2212yes—
45#2330C2212yes—
46#2330C2212yes—
47#2328C2211yescharacteristic
48#2328C2211yescharacteristic
49#2326C2211yescharacteristic
50#2326C2211yescharacteristic
51C2×#389—22no—
52C2×#389C2212yes—
53C2×#387—22no—
54C2×#384C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57#2095C2212yes—
58#2094C2212yes—
59#2093C2212yes—
60#2093C2212yes—
61#2059C2214yes—
62#2057C2214yes—
63#2042C2214yes—
64#2041C2214yes—
65#2026C2212yes—
66#2025C2212yes—
67#2025C2212yes—
68#2025C2212yes—
69#1992C2212yes—
70#1991C2212yes—
71#1983C2214yes—
72#1982C2214yes—
73#1959C2212yes—
74#1957C2212yes—
75#1949C2214yes—
76#1947C2214yes—
77#1928C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1874C2212yes—
82#1874C2212yes—
83#1874C2212yes—
84#1872C2212yes—
85#1864C2214yes—
86#1863C2214yes—
87#1862C2214yes—
88#1862C2214yes—
89#1847C2212yes—
90#1845C2212yes—
91#1840C2212yes—
92#1832C2212yes—
93#1817C2212yes—
94#1797C2212yes—
95#1768C2212yes—
96#1749C2211yescharacteristic
97C4×C22⋊C8C2211yescharacteristic
98#1746C2212yes—
99C22×#41—21no—
100#532—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#127C2311yescharacteristic
103C2×#127C2311yescharacteristic
104C2×#127C2311yescharacteristic
105#529—44no—
106C2×#126—21no—
107C2×#125—28no—
108C2×#125—22no—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×#125C2311yescharacteristic
112C2×#124—21no—
113C2×#124C2311yescharacteristic
114C2×#124C2311yescharacteristic
115C2×#124C2311yescharacteristic
116C2×#123—21no—
117C2×C22⋊Q8C2311yescharacteristic
118C2×#121—21no—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121C2311yescharacteristic
122C2×#121C2311yescharacteristic
123C2×#121C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C42—22no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#490—28no—
141#488—28no—
142#483—44no—
143#480—44no—
144#480—44no—
145#480—44no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#398—22no—
164#397—22no—
165#395—22no—
166#394—22no—
167#394—22no—
168#393—22no—
169#392—22no—
170#392—22no—
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#390—22no—
174#390—22no—
175#390C2311yescharacteristic
176#390C2311yescharacteristic
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389—24no—
184#389—24no—
185#389—24no—
186#389—24no—
187#389C2314yes—
188#389C2314yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—24no—
194#387—24no—
195#387—24no—
196#387—24no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—22no—
205#384—22no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2312yes—
209#384C2312yes—
210#382C2311yescharacteristic
211#382C2311yescharacteristic
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216#376D412yes—
217#376D412yes—
218#376C2311yescharacteristic
219#376C2311yescharacteristic
220#376C2311yescharacteristic
221#376C2311yescharacteristic
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234#336C2312yes—
235C2×#41—44no—
236C2×#41—84no—
237C22×Q8—21no—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—21no—
246C22×D4C2411yescharacteristic
247C22×D4C2411yescharacteristic
248C22×D4C2411yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—42no—
262#127—24no—
263#127—24no—
264#127—42no—
265#126—44no—
266#125—216no—
267#125—28no—
268#125—24no—
269#125—42no—
270#125#4118yescomplemented
271#124—24no—
272#124—24no—
273#124—42no—
274#124—44no—
275#123—44no—
276C22⋊Q8—24no—
277#121—24no—
278#121—24no—
279#121—24no—
280#121—44no—
281#121—84no—
282#121—84no—
283C4×Q8—42no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287C4×D4—44no—
288C4×D4—44no—
289#117—44no—
290C2×C4⋊C4—216no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—28no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—84no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—84no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#107—44no—
352#107—44no—
353#107—44no—
354#106—44no—
355C4⋊C8—28no—
356#103—28no—
357#103—44no—
358#102—28no—
359#102—28no—
360#102—28no—
361#102—44no—
362#102—44no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366#97—24no—
367C4×C8—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—21no—
385C24—21no—
386C24—21no—
387C24C22×D411yescharacteristic
388#41—816no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—42no—
393C2×Q8—82no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4H(5,𝔽2)14yes—
460C22×C4C2511yescharacteristic
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×#4118yes—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—84no—
517C42—84no—
518C42C2×#4114yes—
519C422−1+414yes—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23#53512yes—
539C23#57614yes—
540C23C23×D411yescharacteristic
541Q8—44no—
542Q8—84no—
543D4—44no—
544D4—44no—
545D4—44no—
546D4—84no—
547D4—84no—
548D4—84no—
549D4—84no—
550D4—84no—
551D4—164no—
552D4—164no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—416no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—82no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4#52514yes—
601C2×C4#53512yes—
602C2×C4#53614yes—
603C2×C4C22×#4112yes—
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×2−1+412yes—
606C8—44no—
607C8—44no—
608C22—22no—
609C22—24no—
610C22—22no—
611C22—24no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22#322014yes—
628C22C2×#52511yescharacteristic
629C22C2×#53511yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#57612yes—
632C22#358612yes—
633C4—24no—
634C4—22no—
635C4—24no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—42no—
641C4—48no—
642C4—42no—
643C4—84no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4#352214yes—
650C2—24no—
651C2—161no—
652C2—161no—
653C2—161no—
654C2#3524911yescharacteristic
655C2#6058412yes—
656C2C2×#322011yescharacteristic
657C2#6080911yescharacteristic
658C2#6087112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35303C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2330C2semidirect
3. #33111C2semidirect
4. #33106C2semidirect
5. #30072C2semidirect
6. #30071C2semidirect
7. #29268C2semidirect
8. #29265C2semidirect
9. #29216C2semidirect
10. #28761C2semidirect
11. #27914C2semidirect
12. #27909C2semidirect
13. #25913C2semidirect
14. #25907C2semidirect
15. #24066C2semidirect
16. #24060C2semidirect
17. #22232C2semidirect
18. #22228C2semidirect
19. #20779C2semidirect
270. #125#41semidirect

Maximal subgroups

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

subgroupindex
#353032
C2×#23302
#331112
#331062
#300722
#300712
#292682
#292652
#292162
#287612
#279142
#279092
#259132
#259072
#240662
#240602
#222322
#222282
#207792