Finite Group Databasebeta

← #6459155

Subgroups — #6459155

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic, complemented
2#34946C211yesmaximal, characteristic
3#27814C212yesmaximal, complemented
4#27768C212yesmaximal, complemented
5#27748C212yesmaximal, complemented
6#27728C212yesmaximal, complemented
7#27699C214yesmaximal, complemented
8#27256C214yesmaximal
9#23818C214yesmaximal, complemented
10C4×#488C214yesmaximal, complemented
11#21400C211yesmaximal, characteristic, complemented
12#21385C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic
14C8×#117C211yesmaximal, characteristic, complemented
15C2×C4×#102C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#488C2214yes—
19C2×#431C2211yescharacteristic
20C22×#102C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2571C2212yescomplemented
23#2570—22no—
24#2569—22no—
25#2563C2212yes—
26#2562—22no—
27#2561—22no—
28#2552C2212yescomplemented
29#2547C2212yes—
30#2503C2212yes—
31#2442C2214yes—
32#2435C2212yes—
33#2402—22no—
34#2398—22no—
35C4×#125C2214yescomplemented
36C4×#124C2214yes—
37#2296—21no—
38#2285—21no—
39#2285C2211yescharacteristic
40#2280—21no—
41#2280—21no—
42C4×#117C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44#2101C2214yescomplemented
45#2101C2214yescomplemented
46#2074C2214yescomplemented
47#2073C2214yes—
48#2042C2214yes—
49#2042C2214yes—
50#2026C2214yescomplemented
51#2025C2214yes—
52#1991C2214yescomplemented
53#1990C2212yescomplemented
54#1987C2211yescharacteristic, complemented
55#1987C2211yescharacteristic, complemented
56#1981C2211yescharacteristic, complemented
57#1981C2211yescharacteristic, complemented
58#1958C2214yes—
59#1956C2212yescomplemented
60#1934C2214yes—
61#1934C2214yes—
62#1933C2214yescomplemented
63#1933C2214yescomplemented
64#1892C2214yes—
65#1872C2214yes—
66#1835C2214yescomplemented
67#1823C2212yes—
68#1802C2212yes—
69#1802C2212yes—
70#1786C2212yes—
71C8×C4⋊C4C2212yescomplemented
72C8×C4⋊C4C2212yescomplemented
73C4×#103C2212yescomplemented
74C4×#103C2214yescomplemented
75C4×#103C2212yescomplemented
76C4×#102C2212yes—
77C4×#102C2214yes—
78C4×#102C2212yes—
79C4×#102C2212yes—
80C4×#102C2212yes—
81C8×C22⋊C4C2214yes—
82#1744C2212yescomplemented
83C2×#340C2212yes—
84C42×C8C2212yescomplemented
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87C23×C8C2312yes—
88#533—24no—
89#533—24no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—21no—
93C2×#125C2314yes—
94C2×#124C2314yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8—21no—
98C2×C4×Q8C2312yes—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×#117—21no—
106C2×#117—21no—
107C2×#117D411yescharacteristic
108C2×#117D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C42—21no—
113C22×C42C2311yescharacteristic
114#488—216no—
115#431—24no—
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102—28no—
123C2×#102C2314yes—
124C2×#102C2312yes—
125C2×#102C2312yes—
126C2×#102C2312yes—
127C2×#102C2314yes—
128C2×#102C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C4×C8—24no—
131C2×C4×C8C2314yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#400—24no—
139#400—24no—
140#398—24no—
141#398—24no—
142#397—24no—
143#394—24no—
144#393—24no—
145#391C2314yes—
146#390C2314yes—
147#389—22no—
148#389—22no—
149#389C2314yes—
150#389C2314yes—
151#388—24no—
152#388—24no—
153#387—24no—
154#387—24no—
155#385—24no—
156#384—22no—
157#384C2312yes—
158#382—22no—
159#382—21no—
160#382C2314yes—
161#382C2314yes—
162#381—22no—
163#381—22no—
164#380—21no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—24no—
170#380C2312yes—
171#379—22no—
172#379—22no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2311yescharacteristic
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—24no—
185C4×C22⋊C4—24no—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188#376D412yes—
189C2×#95—21no—
190C43C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2312yes—
196#340C2312yes—
197#340C2314yes—
198#340C2312yes—
199#340C2314yes—
200#340C2312yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C22×Q8—21no—
205C22×Q8C2412yes—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—42no—
225C22×C8C2412yes—
226C22×C8C2412yes—
227C22×C8C2412yes—
228C22×C8C2412yes—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233#125—28no—
234#125—44no—
235#125—44no—
236#124—28no—
237#124—44no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240C22⋊Q8—44no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—44no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249C4×D4—44no—
250#117—24no—
251#117—24no—
252#117—44no—
253#117—44no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—44no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D414yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42C2414yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—22no—
306C4⋊C8—22no—
307#103—28no—
308#103—28no—
309#103—44no—
310#103—44no—
311#102—24no—
312#102—28no—
313#102—24no—
314#102—24no—
315#102—44no—
316#102#4118yes—
317C22⋊C8—24no—
318C4×C8—28no—
319C4×C8—22no—
320C4×C8—22no—
321C4×C8—24no—
322C4×C8—24no—
323C4×C8—44no—
324#95—22no—
325#95—21no—
326#95—21no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95C2×D414yes—
338#95C2×D414yes—
339C24—22no—
340C24—44no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24C22×D411yescharacteristic
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—42no—
351C2×Q8—82no—
352C2×D4—28no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4C2×#4114yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—82no—
384C22×C4—816no—
385C22×C4#12414yes—
386C22×C4#12414yes—
387C22×C4#13514yes—
388C22×C4#13514yes—
389C22×C4C22×D414yes—
390C22×C4C22×D414yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4C2511yescharacteristic
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—48no—
409C2×C8C2×#4114yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—44no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4C2×#4114yes—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C42—24no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—44no—
450C42—44no—
451C42—42no—
452C42—44no—
453C42—42no—
454C42—44no—
455C42#12414yes—
456C42#12414yes—
457C42C2×#4112yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C42C2×#4112yes—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—48no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—88no—
472C23—88no—
473C23—84no—
474C23—84no—
475C23C2×#12412yes—
476C23C2×#13512yes—
477C23C2×#13512yes—
478C23C23×D411yescharacteristic
479Q8—42no—
480Q8—42no—
481Q8—44no—
482Q8—84no—
483D4—24no—
484D4—24no—
485D4—48no—
486D4—42no—
487D4—42no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—88no—
518C2×C4—84no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4#488116yescomplemented
522C2×C4C2×#12412yes—
523C2×C4C2×#12412yes—
524C2×C4C2×#12412yes—
525C2×C4C2×#12412yes—
526C2×C4C2×#12412yes—
527C2×C4C2×#13512yes—
528C2×C4C2×#13512yes—
529C2×C4C2×#13512yes—
530C2×C4C2×#13512yes—
531C2×C4C22×#4111yescharacteristic
532C2×C4C22×#4112yes—
533C2×C4C22×#4111yescharacteristic
534C2×C4#58512yes—
535C8—24no—
536C8—42no—
537C8—42no—
538C8—42no—
539C22—22no—
540C22—24no—
541C22—24no—
542C22—22no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—816no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C2×#48814yes—
552C22C22×#12411yescharacteristic
553C22D4×#4112yes—
554C22C22×#13511yescharacteristic
555C22C22×#13511yescharacteristic
556C22#358412yes—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—44no—
561C4—44no—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—84no—
567C4—82no—
568C4C2×#48814yes—
569C4C2×#48814yes—
570C4#344411yescharacteristic
571C4#345511yescharacteristic
572C4#347512yes—
573C2—24no—
574C2—84no—
575C2#3481311yescharacteristic
576C2C22×#48811yescharacteristic
577C2#6061211yescharacteristic
578C2#6142712yes—
579C2#6142812yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34948C2semidirect
3. #27814C2semidirect
4. #27768C2semidirect
5. #27748C2semidirect
6. #27728C2semidirect
7. #27699C2semidirect
9. #23818C2semidirect
10. C4×#488C2semidirect
11. #21400C2semidirect
12. #21385C2semidirect
14. C8×#117C2semidirect
22. #2571C22semidirect
28. #2552C22semidirect
35. C4×#125C22semidirect
44. #2101C22semidirect
45. #2101C22semidirect
46. #2074C22semidirect
50. #2026C22semidirect
52. #1991C22semidirect
53. #1990C22semidirect
54. #1987C22semidirect
55. #1987C22semidirect
56. #1981C22semidirect
57. #1981C22semidirect
59. #1956C22semidirect
62. #1933C22semidirect
63. #1933C22semidirect
66. #1835C22semidirect
71. C8×C4⋊C4C22semidirect
72. C8×C4⋊C4C22semidirect
73. C4×#103C22semidirect
74. C4×#103C22semidirect
75. C4×#103C22semidirect
82. #1744C22semidirect
84. C42×C8C22semidirect
521. C2×C4#488semidirect

Maximal subgroups

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

subgroupindex
#349482
#349462
#278142
#277682
#277482
#277282
#276992
#272562
#238182
C4×#4882
#214002
#213852
#207322
C8×#1172
C2×C4×#1022