Finite Group Databasebeta

← #6500768

Subgroups — #6500768

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35659C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27595C212yesmaximal, complemented
4#27592C212yesmaximal, complemented
5#24884C214yesmaximal, complemented
6#24883C214yesmaximal, complemented
7#24867C214yesmaximal, complemented
8#24866C214yesmaximal, complemented
9#24006C212yesmaximal, complemented
10#24004C212yesmaximal, complemented
11#21947C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic, complemented
13#20603C212yesmaximal, complemented
14C22×2−1+4C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C2×#484C2214yes—
19C2×#483C2212yes—
20C2×#480C2212yes—
21C2×#417C2212yescomplemented
22C2×#419C2212yescomplemented
23C2×#416C2212yescomplemented
24C22×#103C2212yescomplemented
25C2×#408C2212yes—
26#2409—22no—
27#2409C2214yes—
28#2407—22no—
29#2407C2214yes—
30#2406—22no—
31#2404—22no—
32#2376—22no—
33#2358—22no—
34#2352C2214yes—
35D4×C4⋊C4C2214yes—
36#2329—22no—
37#2325—22no—
38#2314C2212yes—
39#2311C2211yescharacteristic
40#2311C2212yes—
41#2311C2211yescharacteristic
42#2310C2211yescharacteristic
43C2×#380C2212yes—
44#2097C2214yescomplemented
45#2095C2212yescomplemented
46#2093C2212yescomplemented
47#2074C2212yescomplemented
48#2073C2212yescomplemented
49#2072C2214yescomplemented
50#2036C2214yes—
51#2036C2214yes—
52#2036C2214yes—
53#2034C2214yes—
54#1928C2214yescomplemented
55#1928C2214yescomplemented
56#1927C2214yescomplemented
57#1925C2214yescomplemented
58#1874C2212yes—
59#1872C2214yes—
60#1872C2212yes—
61#1870C2214yes—
62#1869C2214yes—
63#1854C2214yes—
64#1854C2214yescomplemented
65#1852C2214yes—
66#1852C2214yescomplemented
67#1818C2212yes—
68#1809C2214yescomplemented
69#1808C2214yescomplemented
70#1788C2214yes—
71#1788C2214yescomplemented
72#1787C2214yes—
73#1787C2214yescomplemented
74#1779C2212yes—
75C2×2−1+4—28no—
76C2×2−1+4—44no—
77C2×2−1+4—42no—
78C2×2−1+4—42no—
79C22×#41—22no—
80C22×#41—21no—
81C22×#41C2311yescharacteristic
82C22×#41C2312yes—
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C22×M16C2312yes—
89C23×C8C2312yes—
90#525—28no—
91#525—42no—
92#525—42no—
93C2×#123—21no—
94C2×#123D412yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8D412yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×#121D412yes—
101C2×#121C2312yes—
102C2×#120—21no—
103C2×#120D412yes—
104C2×C4×D4—22no—
105C2×C4×D4—24no—
106C2×C4×D4C2314yes—
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—21no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—22no—
117#484—48no—
118#483—44no—
119#480—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124#419—44no—
125#417—44no—
126#416—44no—
127C2×#103—44no—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136#408—28no—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2314yes—
142#395—24no—
143#394—24no—
144#393—24no—
145#392—24no—
146#391—22no—
147#391—21no—
148#391—22no—
149#391C2314yes—
150#391C2312yes—
151#390—22no—
152#390—21no—
153#390C2312yes—
154#389—24no—
155#388—24no—
156#388—24no—
157#387—24no—
158#387—24no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—24no—
163#386—24no—
164#386—22no—
165#386—24no—
166#385—24no—
167#385—24no—
168#385—24no—
169#385—24no—
170#384C2312yes—
171#382C2312yes—
172#381—22no—
173#380—22no—
174#380—22no—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—24no—
182C2×#95—22no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C25C2×D411yescharacteristic
1912−1+4—432no—
1922−1+4—432no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—42no—
201C2×#41—41no—
202C2×#41—41no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—48no—
207C2×#41—44no—
208C2×#41—48no—
209C2×#41—44no—
210C2×#41—48no—
211C2×#41—44no—
212C2×#41#4118yes—
213C22×Q8—24no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—41no—
224C22×Q8—41no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8C2×D411yescharacteristic
229C22×Q8C2412yes—
230C22×Q8C2411yescharacteristic
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4C2×D414yes—
235C22×D4C2×D412yes—
236C22×D4C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—42no—
248C23×C4—41no—
249C23×C4—41no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C2×M16—24no—
256C2×M16—42no—
257C2×M16—42no—
258C22×C8—24no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263#123—28no—
264#123—44no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—24no—
268C22⋊Q8—44no—
269C22⋊Q8—42no—
270C22⋊Q8—48no—
271#121—28no—
272#121—24no—
273#121—42no—
274#120—28no—
275#120—44no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—48no—
279C4×D4—88no—
280#117—24no—
281#117—41no—
282#117—41no—
283#117—42no—
284#117—42no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—84no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42C2412yes—
324#107—44no—
325#107—44no—
326#107—44no—
327#106—44no—
328#103—28no—
329#103—28no—
330#103—28no—
331#103—44no—
332#103—44no—
333#103—44no—
334#102—28no—
335#102—44no—
336C22⋊C8—28no—
337C22⋊C8—28no—
338C22⋊C8—42no—
339C22⋊C8—42no—
340C22⋊C8—44no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350C24—21no—
351C24—22no—
352C24—24no—
353C24—41no—
354C24—42no—
355C24—41no—
356C24C22×D411yescharacteristic
357C24C22×D412yes—
358C24C22×D412yes—
359#41—216no—
360#41—48no—
361#41—48no—
362#41—48no—
363#41—48no—
364#41—48no—
365#41—48no—
366#41—48no—
367#41—432no—
368#41—464no—
369C2×Q8—24no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—216no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—48no—
389C2×Q8—432no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8#12314yes—
393C2×Q8H(5,𝔽2)12yes—
394C2×Q8H(5,𝔽2)14yes—
395C2×D4—24no—
396C2×D4—28no—
397C2×D4—28no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—44no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4#12318yes—
415C2×D4#12314yes—
416C2×D4H(5,𝔽2)12yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—41no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2511yescharacteristic
472M16—84no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—84no—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—42no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—82no—
508C22⋊C4—84no—
509C42—24no—
510C42—42no—
511C42—82no—
512C42—84no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—48no—
529C23—44no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—82no—
535C23—84no—
536C23—82no—
537C23C2×#12312yes—
538C23C2×#12314yes—
539C23#57814yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—24no—
544Q8—44no—
545Q8—48no—
546Q8—416no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550D4—24no—
551D4—44no—
552D4—48no—
553D4—416no—
554D4—48no—
555D4—416no—
556D4—42no—
557D4—88no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—48no—
577C2×C4—416no—
578C2×C4—48no—
579C2×C4—416no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—82no—
599C2×C4—84no—
600C2×C4—164no—
601C2×C4—164no—
602C2×C4—164no—
603C2×C4C2×#12314yes—
604C2×C4C2×#12312yes—
605C2×C4C22×#4112yes—
606C2×C4C2×H(5,𝔽2)12yes—
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C8—44no—
610C8—44no—
611C22—22no—
612C22—24no—
613C22—24no—
614C22—28no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—44no—
630C22—88no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22C22×#12311yescharacteristic
635C22#348411yescharacteristic
636C22#348411yescharacteristic
637C22C2×#57812yes—
638C22#358512yes—
639C4—22no—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4#349012yes—
650C4#349012yes—
651C2—44no—
652C2—42no—
653C2—44no—
654C2—82no—
655C2#3535011yescharacteristic
656C2#6052612yes—
657C2#6161912yes—
658C2#6163411yescharacteristic
659C2#6165111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35659C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27595C2semidirect
4. #27592C2semidirect
5. #24884C2semidirect
6. #24883C2semidirect
7. #24867C2semidirect
8. #24866C2semidirect
9. #24006C2semidirect
10. #24004C2semidirect
11. #21947C2semidirect
12. #21744C2semidirect
13. #20603C2semidirect
21. C2×#417C22semidirect
22. C2×#419C22semidirect
23. C2×#416C22semidirect
24. C22×#103C22semidirect
44. #2097C22semidirect
45. #2095C22semidirect
46. #2093C22semidirect
47. #2074C22semidirect
48. #2073C22semidirect
49. #2072C22semidirect
54. #1928C22semidirect
55. #1928C22semidirect
56. #1927C22semidirect
57. #1925C22semidirect
64. #1854C22semidirect
66. #1852C22semidirect
68. #1809C22semidirect
69. #1808C22semidirect
71. #1788C22semidirect
73. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#356592
#343602
#275952
#275922
#248842
#248832
#248672
#248662
#240062
#240042
#219472
#217442
#206032