Finite Group Databasebeta

← #6503180

Subgroups — #6503180

Order 512.

2046 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#27855C214yesmaximal, complemented
4#26200C214yesmaximal, complemented
5#26128C214yesmaximal, complemented
6#24935C214yesmaximal, complemented
7#24863C214yesmaximal, complemented
8#24551C214yesmaximal, complemented
9#21938C211yesmaximal, characteristic, complemented
10#21759C211yesmaximal, characteristic, complemented
11C2×#1892C211yesmaximal, characteristic, complemented
12C2×#1869C211yesmaximal, characteristic
13C2×#1807C211yesmaximal, characteristic, complemented
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#518—21no—
17C22×C4×D4C2211yescharacteristic
18C2×#483C2214yescomplemented
19C22×#106C2212yescomplemented
20C22×#103C2212yescomplemented
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2212yes—
23#2855C2212yes—
24#2848C2212yes—
25#2627—21no—
26#2626—21no—
27#2579—22no—
28#2564—22no—
29#2487C2212yes—
30#2417C2214yes—
31#2409C2212yes—
32#2349—22no—
33#2349—22no—
34D4×C4⋊C4C2214yes—
35D4×C4⋊C4C2214yes—
36#2316—21no—
37#2315—21no—
38#2315—21no—
39#2315—21no—
40#2308—21no—
41#2285C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46#2099C2214yescomplemented
47#2096C2214yescomplemented
48#2084C2214yescomplemented
49#2077C2214yescomplemented
50#2077C2214yescomplemented
51#2072C2214yescomplemented
52#2035C2214yescomplemented
53#2035C2214yescomplemented
54#1927C2214yescomplemented
55#1927C2214yescomplemented
56#1920C2214yescomplemented
57#1920C2214yescomplemented
58#1892C2214yes—
59#1892C2214yes—
60#1892C2212yescomplemented
61#1892C2212yescomplemented
62#1872C2214yescomplemented
63#1872C2214yescomplemented
64#1869C2212yes—
65#1869C2214yes—
66#1869C2212yes—
67#1853C2214yescomplemented
68#1853C2214yescomplemented
69#1853C2214yescomplemented
70#1824C2212yescomplemented
71#1823C2214yescomplemented
72#1807C2214yes—
73#1807C2214yes—
74#1807C2214yescomplemented
75#1787C2212yescomplemented
76C2×#340C2212yescomplemented
77C2×#340C2212yescomplemented
78C2×#336C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8—24no—
84C23×C8—44no—
85C23×C8C2311yescharacteristic
86C23×C8C2312yes—
87C23×C8C2312yes—
88C23×C8C2312yes—
89C23×C8C2311yescharacteristic
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2314yes—
97#518—48no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—22no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4D411yescharacteristic
120C22×C22⋊C4D411yescharacteristic
121C22×C42C2311yescharacteristic
122#483—48no—
123C2×#106—44no—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#103—44no—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×C22⋊C8—24no—
134C2×C22⋊C8—44no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139#400—22no—
140#399—22no—
141#398—24no—
142#396—24no—
143#395—22no—
144#393—22no—
145#391C2314yes—
146#389—24no—
147#389—24no—
148#389C2314yes—
149#387—24no—
150#386—24no—
151#386—22no—
152#386—22no—
153#385—24no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#385—24no—
159#385—24no—
160#385—24no—
161#385—24no—
162#384C2314yes—
163#382—22no—
164#381—22no—
165#380—22no—
166#380—22no—
167#380—21no—
168#380—21no—
169#380—21no—
170#380—22no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—24no—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#380C2314yes—
181#380C2314yes—
182#380C2312yes—
183#379—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C2×#95—21no—
187C2×#95—22no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C25C2×D411yescharacteristic
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8C2412yes—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4SD1618yes—
211C22×D4C2×D412yes—
212C22×D4C2×D412yes—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4#4114yes—
233C23×C4#4114yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C22×C8—22no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—22no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—42no—
250C22×C8—42no—
251C22×C8—416no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—416no—
270C4×D4—416no—
271#117—48no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
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—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42C2×D412yes—
327C2×C42C2412yes—
328#106—28no—
329#106—44no—
330#106—44no—
331#106—44no—
332#103—28no—
333#103—44no—
334#103—44no—
335#103—44no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338C22⋊C8—28no—
339C22⋊C8—44no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352C24—22no—
353C24—22no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24#12314yes—
359C24C2×SD1614yes—
360C24C2×SD1614yes—
361C24#13714yes—
362C24C22×D412yes—
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C24C22×D411yescharacteristic
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—48no—
371C2×Q8—42no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—232no—
380C2×D4C2×SD1618yes—
381C2×D4C2×SD1618yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—84no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—82no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4—84no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—84no—
430C22×C4#12314yes—
431C22×C4#12314yes—
432C22×C4#12314yes—
433C22×C4C2×SD1614yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2511yescharacteristic
447C2×C8—28no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—48no—
457C2×C8—48no—
458C2×C8—44no—
459C2×C8—84no—
460C2×C8H(5,𝔽2)14yes—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—48no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—28no—
512C23—44no—
513C23—44no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—82no—
518C23—84no—
519C23—82no—
520C23#483116yescomplemented
521C23C2×#12312yes—
522C23C2×#12312yes—
523C23C2×#12312yes—
524C23C22×SD1614yes—
525C23C22×SD1612yes—
526C23C22×SD1612yes—
527C23C22×SD1612yes—
528C23C2×#13712yes—
529C23C2×#13712yes—
530C23C2×#13712yes—
531C23C23×D411yescharacteristic
532Q8—42no—
533Q8—42no—
534Q8—44no—
535Q8—84no—
536D4—216no—
537D4—216no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—164no—
582C2×C4—164no—
583C2×C4—168no—
584C2×C4—164no—
585C2×C4C2×#12312yes—
586C2×C4C2×#12312yes—
587C2×C4C2×#12312yes—
588C2×C4C22×SD1614yes—
589C2×C4C22×#4111yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4#58512yes—
593C8—24no—
594C8—42no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—28no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—44no—
613C22—42no—
614C22—88no—
615C22C2×#48314yes—
616C22C2×#48314yes—
617C22C2×#48314yes—
618C22C22×#12311yescharacteristic
619C22#349612yes—
620C22C23×SD1611yescharacteristic
621C22C22×#13711yescharacteristic
622C22#358412yes—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—82no—
630C4—84no—
631C4—84no—
632C4—84no—
633C4—82no—
634C4#349011yescharacteristic
635C4#349111yescharacteristic
636C4#350512yes—
637C2—24no—
638C2—24no—
639C2—24no—
640C2—44no—
641C2#3536011yescharacteristic
642C2C22×#48311yescharacteristic
643C2#6051611yescharacteristic
644C2#6224312yes—
645C2#6225412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
3. #27855C2semidirect
4. #26200C2semidirect
5. #26128C2semidirect
6. #24935C2semidirect
7. #24863C2semidirect
8. #24551C2semidirect
9. #21938C2semidirect
10. #21759C2semidirect
11. C2×#1892C2semidirect
13. C2×#1807C2semidirect
18. C2×#483C22semidirect
19. C22×#106C22semidirect
20. C22×#103C22semidirect
46. #2099C22semidirect
47. #2096C22semidirect
48. #2084C22semidirect
49. #2077C22semidirect
50. #2077C22semidirect
51. #2072C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
54. #1927C22semidirect
55. #1927C22semidirect
56. #1920C22semidirect
57. #1920C22semidirect
60. #1892C22semidirect
61. #1892C22semidirect
62. #1872C22semidirect
63. #1872C22semidirect
67. #1853C22semidirect
68. #1853C22semidirect
69. #1853C22semidirect
70. #1824C22semidirect
71. #1823C22semidirect
74. #1807C22semidirect
75. #1787C22semidirect
76. C2×#340C22semidirect
77. C2×#340C22semidirect
520. C23#483semidirect

Maximal subgroups

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

subgroupindex
#350102
#341812
#278552
#262002
#261282
#249352
#248632
#245512
#219382
#217592
C2×#18922
C2×#18692
C2×#18072