Finite Group Databasebeta

← #6640545

Subgroups — #6640545

Order 512.

1799 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35102C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic, complemented
3#32983C211yesmaximal, characteristic, complemented
4#32979C211yesmaximal, characteristic, complemented
5#32265C214yesmaximal, complemented
6#32033C214yesmaximal, complemented
7#32008C214yesmaximal, complemented
8#32004C214yesmaximal, complemented
9#29893C211yesmaximal, characteristic, complemented
10#29890C211yesmaximal, characteristic, complemented
11#27617C212yesmaximal, complemented
12#24553C212yesmaximal, complemented
13#24552C212yesmaximal, complemented
14#24067C212yesmaximal, complemented
15#21738C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#481C2212yes—
21C22×C22⋊C8C2212yes—
22#2848C2211yescharacteristic
23#2848C2211yescharacteristic
24#2847C2211yescharacteristic
25#2847C2211yescharacteristic
26#2623—21no—
27#2613—21no—
28#2603—22no—
29#2598—22no—
30#2562C2212yes—
31#2540C2212yes—
32#2536C2212yes—
33#2535C2212yes—
34#2487—22no—
35#2487—22no—
36#2487C2214yes—
37#2487C2214yes—
38C2×#400—21no—
39C2×#399—21no—
40#2321—21no—
41#2321—21no—
42C2×#389C2212yes—
43C2×#386—21no—
44C2×#385—21no—
45#2283C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48#2101C2214yescomplemented
49#2085C2214yescomplemented
50#2077C2214yescomplemented
51#2077C2214yescomplemented
52#2076C2214yescomplemented
53#2075C2214yescomplemented
54#2059C2214yescomplemented
55#2042C2214yescomplemented
56#2036C2214yescomplemented
57#2035C2214yescomplemented
58#2034C2214yescomplemented
59#2033C2214yescomplemented
60#1919C2214yescomplemented
61#1918C2214yescomplemented
62#1893C2214yescomplemented
63#1892C2214yescomplemented
64#1874C2212yes—
65#1874C2214yescomplemented
66#1872C2212yes—
67#1872C2214yescomplemented
68#1869C2214yes—
69#1869C2214yes—
70#1865C2214yescomplemented
71#1862C2214yescomplemented
72#1853C2212yes—
73#1853C2214yescomplemented
74#1852C2212yes—
75#1852C2214yescomplemented
76#1824C2212yes—
77#1824C2212yes—
78#1817C2212yes—
79#1808C2214yescomplemented
80#1807C2214yescomplemented
81#1779C2211yescharacteristic
82#1779C2211yescharacteristic
83C23×D4D411yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C23×C8C2314yes—
88#529—44no—
89C2×#126—21no—
90C2×#125C2311yescharacteristic
91C2×#124C2311yescharacteristic
92C2×#123—21no—
93C2×C22⋊Q8C2311yescharacteristic
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×#121C2311yescharacteristic
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#120—21no—
100C2×#120—21no—
101C2×C4×Q8C2311yescharacteristic
102C2×C4×D4—21no—
103C2×C4×D4—44no—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—44no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42C2311yescharacteristic
123#484—44no—
124#481—44no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8—44no—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139#400—22no—
140#400—24no—
141#400—24no—
142#399—22no—
143#399—24no—
144#399—24no—
145#398—22no—
146#397—22no—
147#394—22no—
148#394—22no—
149#392—22no—
150#392—22no—
151#390—22no—
152#390—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#386—24no—
162#386—24no—
163#386—21no—
164#386—21no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#385—21no—
170#385—21no—
171#385—22no—
172#385—22no—
173#385—24no—
174#385—24no—
175#383—21no—
176#382C2312yes—
177#381—21no—
178#380—22no—
179#380—22no—
180#380—22no—
181#380C2312yes—
182#380C2312yes—
183#380C2312yes—
184#380C2312yes—
185#380C2314yes—
186#380C2314yes—
187#380C2314yes—
188#380C2311yescharacteristic
189#380C2311yescharacteristic
190#379—21no—
191#379—21no—
192C4×C4⋊C4C2312yes—
193C4×C22⋊C4—21no—
194#376—21no—
195C2×#95—21no—
196C2×#95—22no—
197C2×#95—22no—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208C25C2×D411yescharacteristic
209C25C2×D411yescharacteristic
210C22×Q8C2411yescharacteristic
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—42no—
217C22×D4C2×D412yes—
218C22×D4C2×D412yes—
219C22×D4C2411yescharacteristic
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—21no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—84no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22×C8—44no—
240C22×C8—44no—
241C22×C8—44no—
242C22×C8C2414yes—
243C22×C8C2414yes—
244#126—44no—
245#125—42no—
246#124—42no—
247#123—44no—
248C22⋊Q8—24no—
249C22⋊Q8—44no—
250C22⋊Q8—44no—
251#121—24no—
252#121—44no—
253#121—44no—
254#120—44no—
255#120—44no—
256C4×Q8—42no—
257C4×D4—42no—
258C4×D4—44no—
259C4×D4—44no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—44no—
265#117—44no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—84no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2411yescharacteristic
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—84no—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—44no—
328#107—44no—
329#106—44no—
330#106—44no—
331#103—44no—
332#103—44no—
333#102—44no—
334#102—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—44no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—21no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24#13514yes—
364C24#13714yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C22×D411yescharacteristic
368C2×Q8—22no—
369C2×Q8—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—416no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—82no—
387C2×D4—82no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—416no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—88no—
429C22×C4—82no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—816no—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4H(5,𝔽2)14yes—
441C22×C4C2511yescharacteristic
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—88no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—84no—
467C4⋊C4—84no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—42no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—42no—
497C42—82no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—28no—
514C23—48no—
515C23—48no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23—42no—
520C23—41no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—82no—
525C23—84no—
526C23—82no—
527C23#53412yes—
528C23C2×#13514yes—
529C23C2×#13512yes—
530C23C2×#13714yes—
531C23C2×#13712yes—
532C23C23×D411yescharacteristic
533Q8—44no—
534D4—48no—
535D4—48no—
536D4—44no—
537D4—84no—
538D4—164no—
539D4—164no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—84no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—84no—
588C2×C4—88no—
589C2×C4—88no—
590C2×C4—816no—
591C2×C4#53412yes—
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4#58514yes—
594C8—44no—
595C8—44no—
596C22—24no—
597C22—24no—
598C22—22no—
599C22—22no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—416no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—88no—
616C22—82no—
617C22—82no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#319014yes—
622C22C2×#53411yescharacteristic
623C22#348512yes—
624C22C22×#13511yescharacteristic
625C22C22×#13711yescharacteristic
626C22#358512yes—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4—81no—
639C4—84no—
640C4—81no—
641C4#353314yes—
642C2—24no—
643C2—44no—
644C2—44no—
645C2—161no—
646C2#3695511yescharacteristic
647C2C2×#319011yescharacteristic
648C2#6075311yescharacteristic
649C2#6169912yes—
650C2#6170912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35102C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2487C2semidirect
3. #32983C2semidirect
4. #32979C2semidirect
5. #32265C2semidirect
6. #32033C2semidirect
7. #32008C2semidirect
8. #32004C2semidirect
9. #29893C2semidirect
10. #29890C2semidirect
11. #27617C2semidirect
12. #24553C2semidirect
13. #24552C2semidirect
14. #24067C2semidirect
15. #21738C2semidirect
48. #2101C22semidirect
49. #2085C22semidirect
50. #2077C22semidirect
51. #2077C22semidirect
52. #2076C22semidirect
53. #2075C22semidirect
54. #2059C22semidirect
55. #2042C22semidirect
56. #2036C22semidirect
57. #2035C22semidirect
58. #2034C22semidirect
59. #2033C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect
62. #1893C22semidirect
63. #1892C22semidirect
65. #1874C22semidirect
67. #1872C22semidirect
70. #1865C22semidirect
71. #1862C22semidirect
73. #1853C22semidirect
75. #1852C22semidirect
79. #1808C22semidirect
80. #1807C22semidirect

Maximal subgroups

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

subgroupindex
#351022
C2×#24872
#329832
#329792
#322652
#320332
#320082
#320042
#298932
#298902
#276172
#245532
#245522
#240672
#217382