Finite Group Databasebeta

← #6487431

Subgroups — #6487431

Order 512.

4345 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#3218C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#32882C212yesmaximal, complemented
5#32879C212yesmaximal
6#29627C212yesmaximal, complemented
7#29624C212yesmaximal, complemented
8#25861C214yesmaximal, complemented
9#25726C214yesmaximal, complemented
10#24212C214yesmaximal, complemented
11#24209C214yesmaximal, complemented
12#21811C212yesmaximal, complemented
13#20715C211yesmaximal, characteristic, complemented
14C24×Q8C2212yes—
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⋊Q8C2214yes—
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2212yes—
24C2×#529C2211yescharacteristic
25C23×C4⋊C4—21no—
26C23×C4⋊C4—21no—
27C23×C42C2211yescharacteristic
28#3218—216no—
29C2×#497C2214yescomplemented
30C2×#491C2214yescomplemented
31C2×#462C2214yescomplemented
32C2×#461C2214yescomplemented
33C2×#451C2214yes—
34C2×#451C2218yes—
35C2×#448C2214yes—
36C2×#448C2218yes—
37C2×#432C2212yes—
38#2855C2211yescharacteristic
39#2854C2211yescharacteristic
40#2853—21no—
41#2852—22no—
42#2851C2212yes—
43#2850C2212yes—
44#2849—21no—
45#2848C2211yescharacteristic
46#2847C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C4×#123—21no—
49C4×#121C2212yes—
50C4×#120—21no—
51#2290—22no—
52#2283C2211yescharacteristic
53#2043C2218yescomplemented
54#2043C2218yescomplemented
55#2016C2218yescomplemented
56#2013C2218yescomplemented
57#1981C2218yescomplemented
58#1960C2218yescomplemented
59#1817C2212yes—
60#1797C2214yes—
61#1787C2218yes—
62#1787C2218yes—
63#1787C2214yes—
64#1787C2214yes—
65#1779C2212yes—
66#1749C2214yes—
67C23×Q8—216no—
68C23×Q8—24no—
69C23×Q8—22no—
70C23×Q8—416no—
71C23×Q8C2318yes—
72C23×Q8C2314yes—
73C24×C4—22no—
74C24×C4D411yescharacteristic
75C24×C4D411yescharacteristic
76C24×C4C2311yescharacteristic
77C22×Q16C2318yes—
78C22×SD16C2318yes—
79C2×C4⋊Q8—216no—
80C2×C4⋊Q8—216no—
81C2×C4⋊Q8—232no—
82C2×C4⋊Q8—232no—
83C2×C4⋊Q8—216no—
84C2×C4⋊Q8—232no—
85C2×C4⋊Q8—232no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8D4116yescomplemented
95C2×C4⋊Q8C2318yes—
96C2×C4⋊Q8C2314yes—
97C2×C4⋊Q8C2314yes—
98#529—28no—
99C2×#126—21no—
100C2×#125C2311yescharacteristic
101C2×#124C2311yescharacteristic
102C2×#123—21no—
103C2×#123—21no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121C2312yes—
107C2×#121C2311yescharacteristic
108C2×#120—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4C2311yescharacteristic
113C2×#117—21no—
114C22×C4⋊C4—216no—
115C22×C4⋊C4—216no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C42—28no—
125C22×C42D412yes—
126C22×C42D411yescharacteristic
127C22×C42D412yes—
128C22×C42D411yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#497—216no—
134#491—216no—
135#462—216no—
136#461—216no—
137#451—216no—
138#451—416no—
139#448—216no—
140#448—416no—
141#432—28no—
142C2×C4⋊C8C2314yes—
143C2×#103C2318yes—
144C2×#103C2318yes—
145C2×#103C23116yes—
146C2×#103C23116yes—
147C2×C22⋊C8C2318yes—
148C2×C22⋊C8C2314yes—
149C2×#97C2314yes—
150#391C2312yes—
151#390C2312yes—
152#389C2312yes—
153#389C2312yes—
154#388—22no—
155#388—22no—
156#388—22no—
157#388—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#386—22no—
163#386—21no—
164#386—22no—
165#386—22no—
166#386—22no—
167#386—21no—
168#385—22no—
169#385—21no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—21no—
174#383—21no—
175#382C2312yes—
176#381—21no—
177#381—21no—
178#380C2312yes—
179#380C2311yescharacteristic
180#380C2312yes—
181#380C2311yescharacteristic
182#379—21no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188#376—21no—
189#336C2318yes—
190C25C2×D411yescharacteristic
191C22×Q8—216no—
192C22×Q8—232no—
193C22×Q8—232no—
194C22×Q8—28no—
195C22×Q8—24no—
196C22×Q8—28no—
197C22×Q8—216no—
198C22×Q8—216no—
199C22×Q8—416no—
200C22×Q8—416no—
201C22×Q8—4128no—
202C22×Q8—416no—
203C22×Q8—416no—
204C22×Q8—416no—
205C22×Q8C2×D4116yes—
206C22×Q8C2411yescharacteristic
207C22×Q8C2418yes—
208C22×D4—21no—
209C22×D4—22no—
210C22×D4C2411yescharacteristic
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—216no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—21no—
222C23×C4—416no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C2×Q16—216no—
236C2×Q16—416no—
237C2×SD16—216no—
238C2×SD16—416no—
239C22×C8C2418yes—
240C4⋊Q8—232no—
241C4⋊Q8—232no—
242C4⋊Q8—232no—
243C4⋊Q8—232no—
244C4⋊Q8—232no—
245C4⋊Q8—232no—
246C4⋊Q8—216no—
247C4⋊Q8—216no—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8C2×D4116yes—
251C4⋊Q8C2×D4116yes—
252#126—44no—
253#125—24no—
254#124—24no—
255#123—44no—
256#123—44no—
257C22⋊Q8—24no—
258C22⋊Q8—28no—
259#121—24no—
260#121—28no—
261#120—44no—
262C4×Q8—24no—
263C4×D4—24no—
264C4×D4—44no—
265C4×D4—48no—
266#117—44no—
267C2×C4⋊C4—216no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—232no—
270C2×C4⋊C4—232no—
271C2×C4⋊C4—232no—
272C2×C4⋊C4—232no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D418yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C4⋊C8—28no—
326#103—216no—
327#103—216no—
328#103—416no—
329#103—416no—
330C22⋊C8—28no—
331C22⋊C8—216no—
332#97—28no—
333#95—21no—
334#95—21no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—21no—
349C24#13714yes—
350C24C22×D411yescharacteristic
351C24C22×D412yes—
352C2×Q8—232no—
353C2×Q8—216no—
354C2×Q8—28no—
355C2×Q8—22no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—24no—
359C2×Q8—432no—
360C2×Q8—464no—
361C2×Q8—464no—
362C2×Q8—4128no—
363C2×Q8—464no—
364C2×Q8—416no—
365C2×Q8—416no—
366C2×Q8—432no—
367C2×Q8—416no—
368C2×Q8C22×D4116yes—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—42no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—42no—
376C2×D4—42no—
377C22×C4—232no—
378C22×C4—232no—
379C22×C4—232no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—232no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—464no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4#12014yes—
421C22×C4#12014yes—
422C22×C4#13718yes—
423C22×C4#13714yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4H(5,𝔽2)18yes—
434C22×C4C2511yescharacteristic
435Q16—816no—
436SD16—816no—
437C2×C8—28no—
438C2×C8—216no—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42—24no—
473C42—24no—
474C42—24no—
475C42—42no—
476C42—44no—
477C42#12014yes—
478C42#12014yes—
479C42C22×D414yes—
480C42C22×D414yes—
481C23—28no—
482C23—28no—
483C23—216no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—216no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23C2×#12012yes—
498C23#53514yes—
499C23C2×#13714yes—
500C23C2×#13714yes—
501C23C2×#13712yes—
502C23C2×#13712yes—
503C23C23×D411yescharacteristic
504Q8—216no—
505Q8—216no—
506Q8—432no—
507Q8—464no—
508Q8—416no—
509Q8—44no—
510D4—44no—
511D4—88no—
512D4—84no—
513C2×C4—216no—
514C2×C4—216no—
515C2×C4—216no—
516C2×C4—28no—
517C2×C4—216no—
518C2×C4—28no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—28no—
534C2×C4—216no—
535C2×C4—28no—
536C2×C4—216no—
537C2×C4—216no—
538C2×C4—432no—
539C2×C4—464no—
540C2×C4—416no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4C2×#12012yes—
556C2×C4C2×#12012yes—
557C2×C4C2×#12012yes—
558C2×C4C2×#12012yes—
559C2×C4C2×#12012yes—
560C2×C4#53514yes—
561C2×C4C2×#13714yes—
562C2×C4C2×#13714yes—
563C2×C4C2×#13714yes—
564C2×C4C23×D412yes—
565C2×C4C2×H(5,𝔽2)14yes—
566C8—48no—
567C22—232no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—24no—
575C22—22no—
576C22—24no—
577C22—24no—
578C22—82no—
579C22—82no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22#300414yes—
584C22#321818yescomplemented
585C22C22×#12011yescharacteristic
586C22C2×#53512yes—
587C22C22×#13711yescharacteristic
588C22C22×#13712yes—
589C22C22×#13711yescharacteristic
590C4—22no—
591C4—24no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—24no—
596C4—24no—
597C4—416no—
598C4—82no—
599C4—82no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4#300414yes—
604C4#348414yes—
605C2—28no—
606C2—24no—
607C2—161no—
608C2#3522211yescharacteristic
609C2C2×#300411yescharacteristic
610C2C2×#300411yescharacteristic
611C2C2×#321812yes—
612C2#6080012yes—
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×#3218C2semidirect
4. #32882C2semidirect
6. #29627C2semidirect
7. #29624C2semidirect
8. #25861C2semidirect
9. #25726C2semidirect
10. #24212C2semidirect
11. #24209C2semidirect
12. #21811C2semidirect
13. #20715C2semidirect
29. C2×#497C22semidirect
30. C2×#491C22semidirect
31. C2×#462C22semidirect
32. C2×#461C22semidirect
53. #2043C22semidirect
54. #2043C22semidirect
55. #2016C22semidirect
56. #2013C22semidirect
57. #1981C22semidirect
58. #1960C22semidirect
94. C2×C4⋊Q8D4semidirect
584. C22#3218semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#32182
#366212
#328822
#328792
#296272
#296242
#258612
#257262
#242122
#242092
#218112
#207152