Finite Group Databasebeta

← #6508303

Subgroups — #6508303

Order 512.

1720 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#35015C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic
3#29335C211yesmaximal, characteristic, complemented
4#29301C211yesmaximal, characteristic, complemented
5#29023C214yesmaximal, complemented
6#28889C214yesmaximal, complemented
7#26824C214yesmaximal, complemented
8#26256C214yesmaximal, complemented
9#25864C211yesmaximal, characteristic
10#25730C211yesmaximal, characteristic
11#24508C212yesmaximal, complemented
12#24507C212yesmaximal
13#22044C212yesmaximal, complemented
14C4×#450C212yesmaximal
15#20534C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#497C2212yes—
19C2×#489C2212yes—
20C2×#451C2212yes—
21C2×#450C2212yes—
22C2×C4×SD16C2214yes—
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#2447—22no—
35#2447—22no—
36#2447C2214yes—
37#2447C2214yes—
38#2446C2212yescomplemented
39C2×#397—21no—
40C2×#397—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#385—21no—
49C2×#381—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2099C2212yescomplemented
52#2083C2212yescomplemented
53#2067C2214yes—
54#2066C2214yescomplemented
55#2058C2212yescomplemented
56#2056C2214yescomplemented
57#2055C2214yes—
58#2043C2212yescomplemented
59#2042C2214yes—
60#2015C2214yes—
61#1973C2214yes—
62#1971C2214yescomplemented
63#1969C2214yescomplemented
64#1959C2212yes—
65#1959C2212yes—
66#1943C2214yescomplemented
67#1936C2214yes—
68#1935C2214yescomplemented
69#1935C2214yescomplemented
70#1933C2214yescomplemented
71#1928C2214yescomplemented
72#1878C2214yescomplemented
73#1877C2214yes—
74#1872C2212yes—
75#1872C2212yes—
76#1864C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1787C2212yes—
80#1786C2212yes—
81#1771C2214yescomplemented
82#1771C2214yescomplemented
83C4×#102C2214yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×Q16C2314yes—
91C22×SD16C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#124—21no—
95C2×#124—21no—
96C2×#124—44no—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2312yes—
101C2×C4×Q8C2312yes—
102C2×C4×D4—22no—
103C2×C4×D4—44no—
104C2×C4×D4C2314yes—
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×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#497—44no—
119#489—44no—
120#451—28no—
121#450—28no—
122C4×SD16—48no—
123#432—28no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#397—24no—
149#397—24no—
150#397—24no—
151#397—24no—
152#395—22no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#391C2312yes—
162#391C2314yes—
163#390—22no—
164#390C2314yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#381—22no—
193#381—24no—
194#381—24no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C2×#95—21no—
207C2×#95—21no—
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—21no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×Q8C2412yes—
228C22×D4—24no—
229C22×D4—21no—
230C22×D4—24no—
231C22×D4—42no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—28no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×SD16—28no—
251C2×SD16—28no—
252C2×SD16—44no—
253C22×C8C2414yes—
254C22×C8C2414yes—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257#124—24no—
258#124—24no—
259#124—42no—
260#124—42no—
261#124—44no—
262#124—44no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—44no—
266C4×Q8—44no—
267C4×D4—28no—
268C4×D4—44no—
269C4×D4—48no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—82no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2412yes—
328#107—44no—
329#106—44no—
330C4⋊C8—28no—
331#103—28no—
332#103—44no—
333#103—44no—
334#103—44no—
335#103—44no—
336#102—28no—
337#102—28no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8#4118yes—
341#97—28no—
342#97—42no—
343#97—42no—
344C4×C8—44no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
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—24no—
370C24—22no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24C22×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—88no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—24no—
397C2×D4—416no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4H(5,𝔽2)14yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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—41no—
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—48no—
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—42no—
437C22×C4—48no—
438C22×C4—84no—
439C22×C4—816no—
440C22×C4#13714yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C2×#4114yes—
451C22×C4C2511yescharacteristic
452Q16—88no—
453SD16—416no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8C2×#4118yes—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—84no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—22no—
502C42—22no—
503C42—42no—
504C42—41no—
505C42—41no—
506C42—82no—
507C42—82no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—24no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23#53212yes—
527C23C2×#13712yes—
528C23#57512yes—
529C23C23×D411yescharacteristic
530Q8—44no—
531Q8—44no—
532Q8—44no—
533Q8—44no—
534Q8—84no—
535Q8—162no—
536D4—28no—
537D4—48no—
538D4—44no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
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—44no—
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—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—816no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4#53212yes—
597C2×C4C2×#13714yes—
598C2×C4C22×#4112yes—
599C2×C4C2×H(5,𝔽2)12yes—
600C2×C4#58512yes—
601C8—28no—
602C8—44no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—816no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#317214yes—
618C22C2×#53211yescharacteristic
619C22#348411yescharacteristic
620C22#348711yescharacteristic
621C22C22×#13711yescharacteristic
622C22C2×#57511yescharacteristic
623C22#358511yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—24no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#351314yes—
642C2—24no—
643C2—84no—
644C2#3539611yescharacteristic
645C2C2×#317211yescharacteristic
646C2#6070311yescharacteristic
647C2#6157011yescharacteristic
648C2#6157511yescharacteristic
649C2#6188911yescharacteristic
650C2#6189511yescharacteristic
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. #29301C2semidirect
5. #29023C2semidirect
6. #28889C2semidirect
7. #26824C2semidirect
8. #26256C2semidirect
11. #24508C2semidirect
13. #22044C2semidirect
15. #20534C2semidirect
31. #2511C22semidirect
32. #2507C22semidirect
33. #2451C22semidirect
38. #2446C22semidirect
51. #2099C22semidirect
52. #2083C22semidirect
54. #2066C22semidirect
55. #2058C22semidirect
56. #2056C22semidirect
58. #2043C22semidirect
62. #1971C22semidirect
63. #1969C22semidirect
66. #1943C22semidirect
68. #1935C22semidirect
69. #1935C22semidirect
70. #1933C22semidirect
71. #1928C22semidirect
72. #1878C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24472
#293352
#293012
#290232
#288892
#268242
#262562
#258642
#257302
#245082
#245072
#220442
C4×#4502
#205342