Finite Group Databasebeta

← #7598714

Subgroups — #7598714

Order 512.

1946 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic, complemented
2#59815C211yesmaximal, characteristic
3#31674C214yesmaximal, complemented
4#28283C211yesmaximal, characteristic, complemented
5#28279C211yesmaximal, characteristic, complemented
6#28113C214yesmaximal, complemented
7#25887C212yesmaximal, complemented
8#25882C212yesmaximal, complemented
9#25348C212yesmaximal, complemented
10#25347C212yesmaximal, complemented
11#24971C212yesmaximal, complemented
12#24967C212yesmaximal, complemented
13#22432C212yesmaximal, complemented
14#22431C212yesmaximal, complemented
15#21681C211yesmaximal, characteristic, complemented
16C4×#487C211yesmaximal, characteristic, complemented
17C4×#486C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#550C2212yes—
20C2×#547C2212yes—
21C2×#487C2211yescharacteristic
22C2×#486C2211yescharacteristic
23C2×#475C2212yes—
24C2×#474C2212yes—
25C2×#436C2212yes—
26C2×#436C2212yes—
27C2×#436C2212yes—
28C2×#417C2214yes—
29#2724—22no—
30#2682—22no—
31#2582—21no—
32#2547C2212yes—
33#2547C2212yes—
34#2541—21no—
35#2538C2212yes—
36#2537C2212yes—
37#2486C2214yes—
38#2447—22no—
39#2447—22no—
40#2445—22no—
41#2416—22no—
42#2407C2214yes—
43D4×C4⋊C4—21no—
44D4×C4⋊C4C2212yes—
45D4×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#127C2211yescharacteristic
51#2077C2214yes—
52#2076C2214yes—
53#2067C2214yes—
54#2066C2214yes—
55#2036C2214yes—
56#2036C2214yes—
57#2034C2214yes—
58#2034C2214yes—
59#1990C2211yescharacteristic
60#1990C2211yescharacteristic
61#1982C2212yes—
62#1981C2212yes—
63#1957C2211yescharacteristic
64#1956C2211yescharacteristic
65#1954C2212yes—
66#1954C2212yes—
67#1954C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1934C2212yes—
71#1934C2212yes—
72#1933C2212yes—
73#1933C2212yes—
74#1895C2214yes—
75#1894C2214yes—
76#1874C2214yes—
77#1872C2214yes—
78#1842C2211yescharacteristic
79#1834C2211yescharacteristic
80#1834C2211yescharacteristic
81#1824C2214yes—
82#1809C2214yes—
83#1803C2212yes—
84#1802C2212yes—
85#1788C2214yes—
86#1788C2214yes—
87#1788C2214yes—
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#102C2212yes—
91C4×#102C2212yes—
92C42×C8C2211yescharacteristic
93C22×#41C2314yes—
94C22×M16C2314yes—
95#559—48no—
96#550—28no—
97#550—44no—
98#547—28no—
99#547—44no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#127—22no—
104C2×#127D411yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#124—22no—
107C2×#124—22no—
108C2×#123—24no—
109C2×#123—24no—
110C2×C22⋊Q8C2314yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—24no—
114C2×#121C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2314yes—
129#487—24no—
130#486—24no—
131#475—44no—
132#474—44no—
133#436—44no—
134#436—44no—
135#436—44no—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141#417—48no—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#398—24no—
164#397—24no—
165#397—24no—
166#397—24no—
167#396—24no—
168#394—24no—
169#394—24no—
170#392—24no—
171#391—22no—
172#391—22no—
173#391—21no—
174#391—21no—
175#391C2312yes—
176#390—22no—
177#390—21no—
178#390—21no—
179#390D412yes—
180#390D412yes—
181#390D412yes—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#386—24no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384D411yescharacteristic
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#381—24no—
202#380—22no—
203#380C2314yes—
204#380C2314yes—
205#380C2314yes—
206#380C2314yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224C2×#41—28no—
225C2×#41—28no—
226C2×#41—44no—
227C2×#41—44no—
228C2×#41—44no—
229C2×#41—44no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8C2412yes—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×D412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2412yes—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—24no—
244C23×C4C2414yes—
245C2×M16—28no—
246C2×M16—44no—
247C2×M16—44no—
248C22×C8C2414yes—
249C22×C8C2411yescharacteristic
250C22×C8C2411yescharacteristic
251C22×C8C2411yescharacteristic
252C22×C8C2411yescharacteristic
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8#4114yes—
259#127—24no—
260#127—22no—
261#127—48no—
262#127#4114yes—
263#124—48no—
264#124—48no—
265#123—416no—
266#123—416no—
267C22⋊Q8—28no—
268C22⋊Q8—44no—
269#121—28no—
270#121—416no—
271#121—44no—
272#121—48no—
273#121—84no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4—48no—
289C4×D4—44no—
290C4×D4—82no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#106—44no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346C4⋊C8—42no—
347#103—24no—
348#103—24no—
349#103—24no—
350#103—42no—
351#103—44no—
352#103—44no—
353#102—24no—
354#102—24no—
355#102—24no—
356#102—42no—
357#102—44no—
358#102—44no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361C22⋊C8—44no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—22no—
365C4×C8—22no—
366C4×C8—22no—
367C4×C8—41no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378C24—22no—
379C24C22×D414yes—
380#41—416no—
381#41—88no—
382#41—88no—
383#41—88no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8C2×#4114yes—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—84no—
422C2×D4C2×#4114yes—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D414yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453M16—88no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8H(5,𝔽2)12yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—84no—
500C4⋊C4—81no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—84no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—41no—
518C42—44no—
519C42—44no—
520C42—44no—
521C42AGL(1,ℤ/8)14yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42H(5,𝔽2)12yes—
525C42H(5,𝔽2)12yes—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545Q8—84no—
546Q8—84no—
547D4—24no—
548D4—24no—
549D4—48no—
550D4—48no—
551D4—48no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—84no—
556D4—816no—
557D4—168no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—24no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—816no—
596C2×C4—88no—
597C2×C4—168no—
598C2×C4#54612yes—
599C2×C4#54612yes—
600C2×C4#54612yes—
601C2×C4#54612yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4C2×AGL(1,ℤ/8)12yes—
604C2×C4C2×AGL(1,ℤ/8)12yes—
605C2×C4#57712yes—
606C2×C4#57812yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4#58511yescharacteristic
613C8—22no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×#54611yescharacteristic
633C22C2×#54611yescharacteristic
634C22#349611yescharacteristic
635C22C22×AGL(1,ℤ/8)11yescharacteristic
636C22C2×#57711yescharacteristic
637C22C2×#57811yescharacteristic
638C22#358411yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—44no—
644C4—41no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#333014yes—
652C4#333214yes—
653C4#348111yescharacteristic
654C4#351411yescharacteristic
655C4#351411yescharacteristic
656C4#354311yescharacteristic
657C2—84no—
658C2—84no—
659C2#5942011yescharacteristic
660C2C2×#333011yescharacteristic
661C2C2×#333211yescharacteristic
662C2#6121211yescharacteristic
663C2#6121411yescharacteristic
664C2#6225911yescharacteristic
665C2#6234311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59827C2semidirect
3. #31674C2semidirect
4. #28283C2semidirect
5. #28279C2semidirect
6. #28113C2semidirect
7. #25887C2semidirect
8. #25882C2semidirect
9. #25348C2semidirect
10. #25347C2semidirect
11. #24971C2semidirect
12. #24967C2semidirect
13. #22432C2semidirect
14. #22431C2semidirect
15. #21681C2semidirect
16. C4×#487C2semidirect
17. C4×#486C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#598152
#316742
#282832
#282792
#281132
#258872
#258822
#253482
#253472
#249712
#249672
#224322
#224312
#216812
C4×#4872
C4×#4862