Finite Group Databasebeta

← #6489400

Subgroups — #6489400

Order 512.

1626 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic, complemented
2#34276C211yesmaximal, characteristic
3#32964C212yesmaximal, complemented
4#32963C212yesmaximal
5#30082C212yesmaximal, complemented
6#30079C212yesmaximal, complemented
7#28723C211yesmaximal, characteristic, complemented
8#28720C211yesmaximal, characteristic, complemented
9#28520C211yesmaximal, characteristic, complemented
10#28511C211yesmaximal, characteristic, complemented
11#28213C212yesmaximal, complemented
12#28211C212yesmaximal, complemented
13#27958C212yesmaximal, complemented
14#27586C212yesmaximal, complemented
15#27584C212yesmaximal
16#25868C212yesmaximal
17#22231C212yesmaximal, complemented
18#22228C212yesmaximal, complemented
19#20781C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#520C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#491C2212yes—
24C2×#488C2212yes—
25C2×#477C2212yes—
26C2×#477C2212yes—
27C2×#475C2212yes—
28C2×#474C2212yes—
29#2868C2211yescharacteristic
30#2859C2212yes—
31#2859C2211yescharacteristic
32#2629C2212yes—
33#2628C2212yes—
34#2570—22no—
35#2550—22no—
36#2441—22no—
37#2439—22no—
38#2362—22no—
39#2360—22no—
40#2359—22no—
41#2359—22no—
42#2331C2212yes—
43#2330C2212yes—
44#2327C2211yescharacteristic
45#2327C2211yescharacteristic
46#2326C2211yescharacteristic
47#2326C2211yescharacteristic
48#2317C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53#2297C2211yescharacteristic
54C2×#391C2211yescharacteristic
55#2285C2212yes—
56C4×#117C2211yescharacteristic
57#2095C2212yes—
58#2095C2212yes—
59#2095C2212yes—
60#2094C2212yes—
61#2085C2214yes—
62#2083C2214yes—
63#2072C2212yes—
64#2072C2212yes—
65#2072C2212yes—
66#2071C2212yes—
67#2043C2214yes—
68#2042C2214yes—
69#1992C2212yes—
70#1991C2212yes—
71#1959C2212yes—
72#1956C2212yes—
73#1928C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1925C2214yes—
77#1865C2214yes—
78#1863C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1854C2214yes—
82#1852C2214yes—
83#1847C2212yes—
84#1845C2212yes—
85#1839C2212yes—
86#1832C2212yes—
87#1817C2212yes—
88#1805C2212yes—
89#1805C2212yes—
90#1805C2212yes—
91#1804C2212yes—
92#1798C2212yes—
93#1788C2214yes—
94#1787C2214yes—
95#1767C2211yescharacteristic
96C4×C4⋊C8C2211yescharacteristic
97#1750C2212yes—
98#1746C2212yes—
99C22×#41C2311yescharacteristic
100C23×Q8C2311yescharacteristic
101#536—24no—
102#536—24no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×#125C2312yes—
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113#520—44no—
114#519—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—21no—
129C2×#117—21no—
130C2×#117—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#491—28no—
138#488—28no—
139#477—44no—
140#477—44no—
141#475—44no—
142#474—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#400—22no—
161#399—22no—
162#398—24no—
163#396—24no—
164#395—22no—
165#393—22no—
166#391—22no—
167#391—24no—
168#391C2311yescharacteristic
169#391C2311yescharacteristic
170#391C2311yescharacteristic
171#391C2311yescharacteristic
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#390C2311yescharacteristic
175#390C2311yescharacteristic
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—24no—
184#389C2314yes—
185#389C2314yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—24no—
191#387—24no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—22no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#384C2312yes—
204#384C2312yes—
205#384C2312yes—
206#382—22no—
207#382—22no—
208#382C2312yes—
209#382C2311yescharacteristic
210#382C2311yescharacteristic
211#380—22no—
212#380C2314yes—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376C2311yescharacteristic
225#376C2311yescharacteristic
226C2×#95—22no—
227C43C2311yescharacteristic
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C2×#41—22no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41—42no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8C2411yescharacteristic
249C22×Q8C2411yescharacteristic
250C22×Q8C2411yescharacteristic
251C22×D4—21no—
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—84no—
268C4⋊Q8#4114yes—
269#125—24no—
270#125—24no—
271#125—84no—
272#124—22no—
273#124—22no—
274#124—22no—
275#124#4114yes—
276#123—216no—
277C22⋊Q8—216no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290#117—22no—
291#117—22no—
292#117—24no—
293#117—44no—
294#117—44no—
295#117—42no—
296#117—44no—
297#117—84no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2414yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D411yescharacteristic
338C2×C22⋊C4C2×D411yescharacteristic
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—44no—
361#107—44no—
362#107—44no—
363#106—44no—
364C4⋊C8—24no—
365C4⋊C8—24no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—44no—
370#103—44no—
371#103—44no—
372#102—28no—
373#102—44no—
374C22⋊C8—28no—
375#97—24no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—21no—
396C24C22×D411yescharacteristic
397#41—48no—
398#41—48no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—42no—
414C2×Q8—84no—
415C2×Q8C2×#4112yes—
416C2×Q8C2×#4112yes—
417C2×Q8C2×#4112yes—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×#4112yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C42−1+414yes—
462C22×C4C2511yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C42—24no—
512C42—24no—
513C42—24no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—84no—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)14yes—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23#53612yes—
542C23C23×D411yescharacteristic
543Q8—24no—
544Q8—24no—
545Q8—24no—
546Q8—48no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—164no—
551D4—24no—
552D4—48no—
553D4—42no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—162no—
601C2×C4—162no—
602C2×C4—162no—
603C2×C4#52514yes—
604C2×C4#53512yes—
605C2×C4#53512yes—
606C2×C4#53612yes—
607C2×C4#57814yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4C2×2−1+412yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—24no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—81no—
622C22—81no—
623C22—82no—
624C22C2×#52511yescharacteristic
625C22C2×#53511yescharacteristic
626C22C2×#53611yescharacteristic
627C22C2×#57812yes—
628C22#358512yes—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—24no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—84no—
646C4—84no—
647C4#352012yes—
648C4#352012yes—
649C2—42no—
650C2—82no—
651C2#3524011yescharacteristic
652C2#6060012yes—
653C2#6082111yescharacteristic
654C2#6082511yescharacteristic
655C2#6087412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #35294C2semidirect
3. #32964C2semidirect
5. #30082C2semidirect
6. #30079C2semidirect
7. #28723C2semidirect
8. #28720C2semidirect
9. #28520C2semidirect
10. #28511C2semidirect
11. #28213C2semidirect
12. #28211C2semidirect
13. #27958C2semidirect
14. #27586C2semidirect
17. #22231C2semidirect
18. #22228C2semidirect
19. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#342762
#329642
#329632
#300822
#300792
#287232
#287202
#285202
#285112
#282132
#282112
#279582
#275862
#275842
#258682
#222312
#222282
#207812