Finite Group Databasebeta

← #6508306

Subgroups — #6508306

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2448C211yesmaximal, characteristic
3#29335C211yesmaximal, characteristic, complemented
4#29302C211yesmaximal, characteristic, complemented
5#29029C214yesmaximal, complemented
6#28895C214yesmaximal, complemented
7#26830C214yesmaximal, complemented
8#26259C214yesmaximal, complemented
9#25864C211yesmaximal, characteristic
10#25730C211yesmaximal, characteristic
11#24508C212yesmaximal, complemented
12#24507C212yesmaximal
13#22043C212yesmaximal
14C4×#451C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C2×#497C2212yes—
19C2×#489C2212yes—
20C2×#451C2212yes—
21C2×#450C2212yes—
22C2×C4×Q16C2214yescomplemented
23C2×#432C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yescomplemented
32#2507C2212yescomplemented
33#2451C2212yescomplemented
34#2448—22no—
35#2448—22no—
36#2448C2214yes—
37#2448C2214yes—
38#2446C2212yescomplemented
39C2×#399—21no—
40C2×#397—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#391C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2099C2212yescomplemented
52#2084C2212yescomplemented
53#2067C2214yes—
54#2066C2214yescomplemented
55#2057C2212yescomplemented
56#2056C2214yescomplemented
57#2055C2214yes—
58#2043C2212yescomplemented
59#2042C2214yes—
60#2015C2214yes—
61#1973C2214yes—
62#1971C2214yescomplemented
63#1968C2214yes—
64#1959C2212yes—
65#1959C2212yes—
66#1944C2214yescomplemented
67#1936C2214yes—
68#1935C2214yescomplemented
69#1935C2214yescomplemented
70#1934C2214yes—
71#1926C2214yes—
72#1880C2214yes—
73#1878C2214yescomplemented
74#1872C2212yes—
75#1872C2212yes—
76#1863C2214yescomplemented
77#1853C2212yes—
78#1852C2212yes—
79#1787C2212yes—
80#1786C2212yes—
81#1771C2214yescomplemented
82#1771C2214yescomplemented
83C4×#103C2214yescomplemented
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C22×Q16C2314yes—
90C22×SD16C2314yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#124—21no—
94C2×#124—21no—
95C2×#124—44no—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×C22⋊Q8—21no—
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⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#497—44no—
121#489—44no—
122#451—28no—
123#450—28no—
124C4×Q16—48no—
125#432—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×#97C2314yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#400—22no—
145#400—22no—
146#399—24no—
147#399—24no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#397—24no—
153#397—24no—
154#395—22no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#391C2314yes—
161#391C2312yes—
162#391C2312yes—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—24no—
180#388—24no—
181#388—24no—
182#388—24no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#381—22no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—22no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×Q8C2412yes—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×SD16—28no—
247C2×SD16—44no—
248C2×SD16—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253#124—24no—
254#124—24no—
255#124—42no—
256#124—42no—
257#124—44no—
258#124—44no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262C4×Q8—28no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—48no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4—82no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2412yes—
330#107—44no—
331#106—44no—
332C4⋊C8—28no—
333#103—28no—
334#103—28no—
335#103—44no—
336#103—44no—
337#103—44no—
338#102—28no—
339#102—44no—
340#102—44no—
341C22⋊C8—24no—
342C22⋊C8#4118yes—
343#97—28no—
344#97—42no—
345#97—42no—
346C4×C8—44no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24C22×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—416no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8H(5,𝔽2)14yes—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—88no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4#13714yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4C2511yescharacteristic
448Q16—416no—
449SD16—88no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8C2×#4118yes—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—41no—
503C42—41no—
504C42—48no—
505C42—82no—
506C42—82no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23#53212yes—
526C23C2×#13712yes—
527C23#57512yes—
528C23C23×D411yescharacteristic
529Q8—28no—
530Q8—48no—
531Q8—44no—
532Q8—44no—
533Q8—44no—
534Q8—162no—
535D4—44no—
536D4—44no—
537D4—84no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—44no—
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—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4#53212yes—
595C2×C4C2×#13714yes—
596C2×C4C22×#4112yes—
597C2×C4C2×H(5,𝔽2)12yes—
598C2×C4#58512yes—
599C8—28no—
600C8—44no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—168no—
615C22#317214yes—
616C22C2×#53211yescharacteristic
617C22#348411yescharacteristic
618C22#348711yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57511yescharacteristic
621C22#358511yescharacteristic
622C22#358611yescharacteristic
623C4—22no—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—41no—
632C4—41no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4#351314yes—
640C2—24no—
641C2—162no—
642C2#3539611yescharacteristic
643C2C2×#317211yescharacteristic
644C2#6070311yescharacteristic
645C2#6157311yescharacteristic
646C2#6157711yescharacteristic
647C2#6189211yescharacteristic
648C2#6189411yescharacteristic
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
3. #29335C2semidirect
4. #29302C2semidirect
5. #29029C2semidirect
6. #28895C2semidirect
7. #26830C2semidirect
8. #26259C2semidirect
11. #24508C2semidirect
14. C4×#451C2semidirect
15. #20534C2semidirect
22. C2×C4×Q16C22semidirect
31. #2511C22semidirect
32. #2507C22semidirect
33. #2451C22semidirect
38. #2446C22semidirect
51. #2099C22semidirect
52. #2084C22semidirect
54. #2066C22semidirect
55. #2057C22semidirect
56. #2056C22semidirect
58. #2043C22semidirect
62. #1971C22semidirect
66. #1944C22semidirect
68. #1935C22semidirect
69. #1935C22semidirect
73. #1878C22semidirect
76. #1863C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect
83. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24482
#293352
#293022
#290292
#288952
#268302
#262592
#258642
#257302
#245082
#245072
#220432
C4×#4512
#205342