Finite Group Databasebeta

← #6507912

Subgroups — #6507912

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2421C211yesmaximal, characteristic
3#29111C211yesmaximal, characteristic, complemented
4#28986C214yesmaximal, complemented
5#28852C214yesmaximal, complemented
6#28313C211yesmaximal, characteristic, complemented
7#27983C211yesmaximal, characteristic, complemented
8#27956C211yesmaximal, characteristic, complemented
9#26981C214yesmaximal, complemented
10#25043C214yesmaximal, complemented
11#24551C212yesmaximal
12#24380C212yesmaximal
13#22084C212yesmaximal, complemented
14#21455C212yesmaximal, complemented
15#20526C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#483C2212yes—
19C2×#467C2214yes—
20C2×#451C2212yes—
21C2×#441C2214yescomplemented
22C22×C22⋊C8C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2848C2212yes—
26#2646—21no—
27#2626—21no—
28#2589—22no—
29#2587—22no—
30#2497C2212yes—
31#2492C2212yes—
32#2429C2212yes—
33#2423C2212yes—
34#2421—22no—
35#2421—22no—
36#2421C2214yes—
37#2421C2214yes—
38C2×#400—21no—
39C2×#397—21no—
40#2329—21no—
41#2312—21no—
42C4×C22⋊Q8C2212yes—
43C2×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#388—21no—
46C2×#387—21no—
47C2×#382C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2096C2212yescomplemented
51#2085C2214yescomplemented
52#2077C2214yes—
53#2074C2214yescomplemented
54#2072C2212yescomplemented
55#2036C2214yes—
56#2035C2214yes—
57#2024C2212yescomplemented
58#2016C2212yescomplemented
59#2001C2214yes—
60#1993C2212yescomplemented
61#1971C2214yescomplemented
62#1960C2212yescomplemented
63#1944C2214yescomplemented
64#1935C2214yescomplemented
65#1935C2214yescomplemented
66#1920C2214yescomplemented
67#1892C2214yes—
68#1892C2214yes—
69#1878C2214yescomplemented
70#1872C2212yes—
71#1872C2212yes—
72#1865C2214yescomplemented
73#1863C2214yescomplemented
74#1853C2212yes—
75#1852C2212yes—
76#1847C2212yescomplemented
77#1836C2212yescomplemented
78#1823C2212yes—
79#1807C2214yes—
80#1787C2212yes—
81#1763C2214yescomplemented
82#1752C2214yes—
83#1752C2214yes—
84#1749C2211yescharacteristic
85#1749C2212yes—
86#1749C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×Q16C2314yes—
90C23×C8C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—22no—
101C2×C4×Q8—44no—
102C2×C4×Q8C2314yes—
103C2×C4×Q8C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4D411yescharacteristic
113C22×C22⋊C4D411yescharacteristic
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#483—28no—
118#467—48no—
119#451—28no—
120#441—48no—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×C22⋊C8—28no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2312yes—
135C2×#97C2312yes—
136C2×#97C2314yes—
137#400—22no—
138#400—22no—
139#400—24no—
140#400—24no—
141#399—22no—
142#398—22no—
143#398—22no—
144#397—24no—
145#397—24no—
146#396—22no—
147#396—22no—
148#393—22no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2314yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—24no—
170#388—24no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#382—21no—
183#382—21no—
184#382C2312yes—
185#382C2312yes—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380C2312yes—
191#380C2314yes—
192#380C2312yes—
193#380C2312yes—
194#379—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4C2312yes—
201C4×C22⋊C4C2312yes—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C22×Q8—24no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×Q8C2412yes—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—44no—
245C22×C8—28no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C22⋊Q8—28no—
253C22⋊Q8—24no—
254C22⋊Q8—24no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257C22⋊Q8—48no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—28no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—48no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—44no—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#106—28no—
325#106—44no—
326#103—28no—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332C22⋊C8—24no—
333C22⋊C8—28no—
334C22⋊C8#4118yes—
335#97—42no—
336#97—42no—
337#97—44no—
338#95—24no—
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—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24—22no—
363C24—22no—
364C24—41no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24#13714yes—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—416no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8H(5,𝔽2)14yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
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—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—88no—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C2×#4114yes—
440C22×C4C2511yescharacteristic
441Q16—416no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—88no—
448C2×C8C2×#4118yes—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—81no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42—48no—
494C42—82no—
495C42—82no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—44no—
508C23—44no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23#53212yes—
514C23C2×#13714yes—
515C23C2×#13712yes—
516C23#57512yes—
517C23C23×D411yescharacteristic
518Q8—28no—
519Q8—48no—
520Q8—44no—
521Q8—44no—
522Q8—44no—
523Q8—162no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
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—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—84no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4#53212yes—
585C2×C4C22×#4112yes—
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4#58512yes—
588C8—28no—
589C8—44no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—88no—
604C22#318114yes—
605C22C2×#53211yescharacteristic
606C22#348411yescharacteristic
607C22#348511yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57511yescharacteristic
610C22#358611yescharacteristic
611C22#358611yescharacteristic
612C4—22no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4#351314yes—
629C2—24no—
630C2—44no—
631C2#3539411yescharacteristic
632C2C2×#318111yescharacteristic
633C2#6070511yescharacteristic
634C2#6157311yescharacteristic
635C2#6157811yescharacteristic
636C2#6170511yescharacteristic
637C2#6171211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29111C2semidirect
4. #28986C2semidirect
5. #28852C2semidirect
6. #28313C2semidirect
7. #27983C2semidirect
8. #27956C2semidirect
9. #26981C2semidirect
10. #25043C2semidirect
13. #22084C2semidirect
14. #21455C2semidirect
21. C2×#441C22semidirect
50. #2096C22semidirect
51. #2085C22semidirect
53. #2074C22semidirect
54. #2072C22semidirect
57. #2024C22semidirect
58. #2016C22semidirect
60. #1993C22semidirect
61. #1971C22semidirect
62. #1960C22semidirect
63. #1944C22semidirect
64. #1935C22semidirect
65. #1935C22semidirect
66. #1920C22semidirect
69. #1878C22semidirect
72. #1865C22semidirect
73. #1863C22semidirect
76. #1847C22semidirect
77. #1836C22semidirect
81. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#24212
#291112
#289862
#288522
#283132
#279832
#279562
#269812
#250432
#245512
#243802
#220842
#214552
#205262