Finite Group Databasebeta

← #6644707

Subgroups — #6644707

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3#33007C212yesmaximal
4#32455C214yesmaximal, complemented
5#32373C214yesmaximal, complemented
6#31663C214yesmaximal, complemented
7#31549C214yesmaximal, complemented
8#29642C212yesmaximal, complemented
9#29109C211yesmaximal, characteristic
10#28311C211yesmaximal, characteristic
11#28123C212yesmaximal
12#27969C211yesmaximal, characteristic, complemented
13#27919C211yesmaximal, characteristic, complemented
14#27653C212yesmaximal, complemented
15#21916C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#450C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2646—21no—
26#2627—21no—
27#2590—22no—
28#2588—22no—
29#2563—22no—
30#2563—22no—
31#2563C2214yes—
32#2563C2214yes—
33#2511C2212yescomplemented
34#2507C2212yescomplemented
35#2451C2212yescomplemented
36#2446C2212yescomplemented
37C2×#393—21no—
38C2×#393—21no—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#380—21no—
47C2×#380—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2101C2214yescomplemented
50#2097C2214yescomplemented
51#2093C2212yes—
52#2085C2214yescomplemented
53#2074C2214yescomplemented
54#2071C2212yes—
55#2067C2214yes—
56#2057C2214yescomplemented
57#2057C2214yescomplemented
58#2055C2214yes—
59#2024C2214yescomplemented
60#2024C2214yescomplemented
61#2023C2212yes—
62#2013C2212yes—
63#1993C2212yescomplemented
64#1973C2214yes—
65#1960C2212yescomplemented
66#1949C2214yescomplemented
67#1936C2214yes—
68#1925C2214yes—
69#1917C2214yescomplemented
70#1917C2214yescomplemented
71#1880C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1847C2212yescomplemented
77#1836C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1805C2214yescomplemented
83#1797C2212yes—
84#1786C2212yes—
85#1749C2211yescharacteristic
86#1749C2211yescharacteristic
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4—44no—
101C2×C4×D4C2314yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#484—44no—
117#484—44no—
118#483—44no—
119#450—44no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×#97C2312yes—
133C2×#97C2312yes—
134#400—22no—
135#400—22no—
136#400—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#395—22no—
142#393—24no—
143#393—24no—
144#393—24no—
145#393—24no—
146#391—21no—
147#391—22no—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390—22no—
153#390—21no—
154#390—21no—
155#390C2314yes—
156#390C2312yes—
157#390C2312yes—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—22no—
180#382—21no—
181#382—21no—
182#382C2314yes—
183#382C2312yes—
184#382C2312yes—
185#381—22no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—24no—
190#380—24no—
191#380—24no—
192#380—24no—
193#380C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C25—21no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8—44no—
222C22×Q8—82no—
223C22×Q8—82no—
224C22×Q8C2412yes—
225C22×D4—24no—
226C22×D4—21no—
227C22×D4—24no—
228C22×D4—42no—
229C22×D4C2412yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—24no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×SD16—28no—
245C2×SD16—44no—
246C2×SD16—44no—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—48no—
250C4⋊Q8—42no—
251C4⋊Q8—48no—
252C4⋊Q8—42no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—44no—
256C4×Q8—44no—
257C4×D4—48no—
258C4×D4—48no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—82no—
303C2×C22⋊C4—82no—
304C2×C22⋊C4—82no—
305C2×C22⋊C4—82no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#107—44no—
320#107—44no—
321#106—44no—
322C4⋊C8—44no—
323#103—44no—
324#103—44no—
325#103—44no—
326#102—28no—
327#102—44no—
328#102—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331#97—42no—
332#97—42no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356#95C2×D412yes—
357C24—24no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24C22×D411yescharacteristic
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—48no—
371C2×Q8—42no—
372C2×Q8—84no—
373C2×Q8—84no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×Q8—816no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—416no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D42−1+414yes—
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—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—48no—
423C22×C4—816no—
424C22×C4#13714yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4C2511yescharacteristic
436SD16—88no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C42—24no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—42no—
486C42—41no—
487C42—41no—
488C42—82no—
489C42—82no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—24no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23#53512yes—
509C23C2×#13712yes—
510C23#57512yes—
511C23C23×D411yescharacteristic
512Q8—44no—
513Q8—44no—
514Q8—816no—
515Q8—162no—
516D4—28no—
517D4—48no—
518D4—44no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—816no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4#53512yes—
577C2×C4C2×#13714yes—
578C2×C4C2×H(5,𝔽2)12yes—
579C2×C4C2×2−1+412yes—
580C2×C4#58512yes—
581C8—44no—
582C8—44no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—816no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#321514yes—
598C22C2×#53511yescharacteristic
599C22#348511yescharacteristic
600C22#348611yescharacteristic
601C22C22×#13711yescharacteristic
602C22C2×#57511yescharacteristic
603C22#358511yescharacteristic
604C22#358511yescharacteristic
605C4—24no—
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—42no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#353514yes—
622C2—24no—
623C2—84no—
624C2#3711811yescharacteristic
625C2C2×#321511yescharacteristic
626C2#6080611yescharacteristic
627C2#6171011yescharacteristic
628C2#6172111yescharacteristic
629C2#6182611yescharacteristic
630C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35015C2semidirect
4. #32455C2semidirect
5. #32373C2semidirect
6. #31663C2semidirect
7. #31549C2semidirect
8. #29642C2semidirect
12. #27969C2semidirect
13. #27919C2semidirect
14. #27653C2semidirect
15. #21916C2semidirect
33. #2511C22semidirect
34. #2507C22semidirect
35. #2451C22semidirect
36. #2446C22semidirect
49. #2101C22semidirect
50. #2097C22semidirect
52. #2085C22semidirect
53. #2074C22semidirect
56. #2057C22semidirect
57. #2057C22semidirect
59. #2024C22semidirect
60. #2024C22semidirect
63. #1993C22semidirect
65. #1960C22semidirect
66. #1949C22semidirect
69. #1917C22semidirect
70. #1917C22semidirect
76. #1847C22semidirect
77. #1836C22semidirect
82. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#25632
#330072
#324552
#323732
#316632
#315492
#296422
#291092
#283112
#281232
#279692
#279192
#276532
#219162