Finite Group Databasebeta

← #6489542

Subgroups — #6489542

Order 512.

4169 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#3239C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#32983C212yesmaximal, complemented
5#32979C212yesmaximal
6#29642C212yesmaximal, complemented
7#29639C212yesmaximal, complemented
8#25908C214yesmaximal, complemented
9#25816C214yesmaximal, complemented
10#24301C214yesmaximal, complemented
11#24298C214yesmaximal, complemented
12#21841C212yesmaximal, complemented
13#20715C211yesmaximal, characteristic, complemented
14C24×Q8—21no—
15C22×C4⋊Q8—22no—
16C22×C4⋊Q8—21no—
17C22×C4⋊Q8—22no—
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8—216no—
20C22×C4⋊Q8C2214yes—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2214yes—
23C22×C4⋊Q8C2212yes—
24C2×#529C2211yescharacteristic
25C23×C4⋊C4—21no—
26C23×C4⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#3239—216no—
29C2×#501C2214yescomplemented
30C2×#489C2214yescomplemented
31C2×#484C2214yes—
32C2×#481C2214yes—
33C2×#475C2214yescomplemented
34C2×#474C2214yescomplemented
35C2×#432C2212yes—
36#2855C2211yescharacteristic
37#2854C2211yescharacteristic
38#2853—21no—
39#2852—22no—
40#2851C2212yes—
41#2850C2212yes—
42#2849—21no—
43#2848C2211yescharacteristic
44#2847C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C4×#123—21no—
47C4×#121C2212yes—
48C4×#120—21no—
49#2290—22no—
50#2283C2211yescharacteristic
51#2058C2218yescomplemented
52#2057C2218yescomplemented
53#2024C2218yescomplemented
54#2023C2218yescomplemented
55#1993C2218yescomplemented
56#1988C2218yescomplemented
57#1874C2218yes—
58#1872C2218yes—
59#1853C2218yes—
60#1852C2218yes—
61#1824C2214yes—
62#1824C2214yes—
63#1817C2212yes—
64#1797C2214yes—
65#1779C2212yes—
66#1749C2214yes—
67C23×Q8—216no—
68C23×Q8—24no—
69C23×Q8—22no—
70C23×Q8—22no—
71C23×Q8—24no—
72C23×Q8—416no—
73C24×C4—22no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4C2311yescharacteristic
77C2×C4⋊Q8—216no—
78C2×C4⋊Q8—216no—
79C2×C4⋊Q8—232no—
80C2×C4⋊Q8—232no—
81C2×C4⋊Q8—232no—
82C2×C4⋊Q8—232no—
83C2×C4⋊Q8—216no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8D4116yescomplemented
93C2×C4⋊Q8C2318yes—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2314yes—
96#529—28no—
97C2×#126—21no—
98C2×#125C2311yescharacteristic
99C2×#124C2311yescharacteristic
100C2×#123—21no—
101C2×#123—21no—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C22⋊Q8C2312yes—
104C2×#121C2311yescharacteristic
105C2×#121C2312yes—
106C2×#120—21no—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—22no—
110C2×C4×D4C2311yescharacteristic
111C2×#117—21no—
112C22×C4⋊C4—216no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—216no—
120C22×C4⋊C4C2318yes—
121C22×C4⋊C4C2314yes—
122C22×C42—28no—
123C22×C42D412yes—
124C22×C42D411yescharacteristic
125C22×C42D412yes—
126C22×C42D411yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#501—216no—
132#489—216no—
133#484—216no—
134#481—216no—
135#475—216no—
136#474—216no—
137#432—28no—
138C2×#107C2318yes—
139C2×#106C2318yes—
140C2×C4⋊C8C2314yes—
141C2×#103C2318yes—
142C2×#102C2318yes—
143C2×C22⋊C8C2318yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2314yes—
146#391C2312yes—
147#390C2312yes—
148#389C2312yes—
149#389C2312yes—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#386—22no—
159#386—21no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—21no—
164#385—22no—
165#385—21no—
166#385—22no—
167#385—22no—
168#385—22no—
169#385—21no—
170#383—21no—
171#382C2312yes—
172#381—21no—
173#381—21no—
174#380C2311yescharacteristic
175#380C2312yes—
176#380C2311yescharacteristic
177#380C2312yes—
178#379—21no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2312yes—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184#376—21no—
185#340C23116yes—
186#340C23116yes—
187#336C2318yes—
188C25C2×D411yescharacteristic
189C22×Q8—216no—
190C22×Q8—232no—
191C22×Q8—232no—
192C22×Q8—216no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—24no—
196C22×Q8—4128no—
197C22×Q8—416no—
198C22×Q8—416no—
199C22×Q8—416no—
200C22×Q8—416no—
201C22×Q8—416no—
202C22×Q8—416no—
203C22×Q8C2411yescharacteristic
204C22×D4—21no—
205C22×D4—22no—
206C22×D4C2411yescharacteristic
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—216no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—21no—
218C23×C4—416no—
219C23×C4C2×D412yes—
220C23×C4C2×D412yes—
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D412yes—
228C23×C4C2×D411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C22×C8C2418yes—
232C4⋊Q8—232no—
233C4⋊Q8—232no—
234C4⋊Q8—232no—
235C4⋊Q8—232no—
236C4⋊Q8—232no—
237C4⋊Q8—232no—
238C4⋊Q8—216no—
239C4⋊Q8—216no—
240C4⋊Q8—28no—
241C4⋊Q8—28no—
242C4⋊Q8C2×D4116yes—
243C4⋊Q8C2×D4116yes—
244#126—44no—
245#125—24no—
246#124—24no—
247#123—44no—
248#123—44no—
249C22⋊Q8—24no—
250C22⋊Q8—28no—
251#121—24no—
252#121—28no—
253#120—44no—
254C4×Q8—24no—
255C4×D4—24no—
256C4×D4—44no—
257C4×D4—48no—
258#117—44no—
259C2×C4⋊C4—216no—
260C2×C4⋊C4—216no—
261C2×C4⋊C4—232no—
262C2×C4⋊C4—232no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—232no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4C2×D4116yes—
280C2×C4⋊C4C2411yescharacteristic
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2418yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—21no—
306C2×C42—22no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D418yes—
315C2×C42C2412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318#107—216no—
319#106—216no—
320C4⋊C8—28no—
321#103—216no—
322#102—216no—
323C22⋊C8—28no—
324C22⋊C8—216no—
325#97—28no—
326#95—21no—
327#95—21no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—21no—
342C24#13714yes—
343C24C22×D411yescharacteristic
344C24C22×D412yes—
345C2×Q8—232no—
346C2×Q8—216no—
347C2×Q8—216no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—4128no—
351C2×Q8—464no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×Q8—464no—
355C2×Q8—4128no—
356C2×Q8—416no—
357C2×Q8—416no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—42no—
365C2×D4—42no—
366C22×C4—232no—
367C22×C4—232no—
368C22×C4—232no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—232no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—464no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4#12014yes—
410C22×C4#12014yes—
411C22×C4AGL(1,ℤ/8)18yes—
412C22×C4#13714yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C42−1+418yes—
423C22×C4C2511yescharacteristic
424C2×C8—28no—
425C2×C8—216no—
426C4⋊C4—232no—
427C4⋊C4—232no—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—28no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4C22×D4116yes—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C42—24no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—24no—
463C42—42no—
464C42—44no—
465C42#12014yes—
466C42#12014yes—
467C42C22×D414yes—
468C42C22×D414yes—
469C23—28no—
470C23—28no—
471C23—216no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—216no—
476C23—28no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—41no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23C2×#12012yes—
486C23#53614yes—
487C23C2×AGL(1,ℤ/8)14yes—
488C23C2×#13714yes—
489C23C2×#13712yes—
490C23C2×#13712yes—
491C23C23×D411yescharacteristic
492Q8—432no—
493Q8—464no—
494Q8—432no—
495Q8—44no—
496D4—44no—
497D4—88no—
498D4—84no—
499C2×C4—216no—
500C2×C4—28no—
501C2×C4—216no—
502C2×C4—216no—
503C2×C4—216no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—216no—
521C2×C4—28no—
522C2×C4—216no—
523C2×C4—216no—
524C2×C4—432no—
525C2×C4—464no—
526C2×C4—416no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4C2×#12012yes—
542C2×C4C2×#12012yes—
543C2×C4C2×#12012yes—
544C2×C4C2×#12012yes—
545C2×C4C2×#12012yes—
546C2×C4#53614yes—
547C2×C4C2×AGL(1,ℤ/8)14yes—
548C2×C4C2×AGL(1,ℤ/8)14yes—
549C2×C4C2×#13714yes—
550C2×C4C23×D412yes—
551C2×C4C2×2−1+414yes—
552C8—48no—
553C22—232no—
554C22—22no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—24no—
563C22—24no—
564C22—81no—
565C22—82no—
566C22—82no—
567C22—81no—
568C22—81no—
569C22#300414yes—
570C22#323918yescomplemented
571C22C22×#12011yescharacteristic
572C22C2×#53612yes—
573C22C22×AGL(1,ℤ/8)12yes—
574C22C22×#13711yescharacteristic
575C22C22×#13711yescharacteristic
576C4—22no—
577C4—24no—
578C4—22no—
579C4—24no—
580C4—22no—
581C4—22no—
582C4—24no—
583C4—416no—
584C4—81no—
585C4—82no—
586C4—82no—
587C4—81no—
588C4—81no—
589C4#300314yes—
590C4#348614yes—
591C2—28no—
592C2—24no—
593C2—161no—
594C2#3524211yescharacteristic
595C2C2×#300311yescharacteristic
596C2C2×#300411yescharacteristic
597C2C2×#323912yes—
598C2#6086012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4⋊Q8C2semidirect
2. C2×#3239C2semidirect
4. #32983C2semidirect
6. #29642C2semidirect
7. #29639C2semidirect
8. #25908C2semidirect
9. #25816C2semidirect
10. #24301C2semidirect
11. #24298C2semidirect
12. #21841C2semidirect
13. #20715C2semidirect
29. C2×#501C22semidirect
30. C2×#489C22semidirect
33. C2×#475C22semidirect
34. C2×#474C22semidirect
51. #2058C22semidirect
52. #2057C22semidirect
53. #2024C22semidirect
54. #2023C22semidirect
55. #1993C22semidirect
56. #1988C22semidirect
92. C2×C4⋊Q8D4semidirect
570. C22#3239semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#32392
#366212
#329832
#329792
#296422
#296392
#259082
#258162
#243012
#242982
#218412
#207152