Finite Group Databasebeta

← #6489541

Subgroups — #6489541

Order 512.

4169 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#3238C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#32982C212yesmaximal, complemented
5#32978C212yesmaximal
6#29641C212yesmaximal, complemented
7#29638C212yesmaximal, complemented
8#25908C214yesmaximal, complemented
9#25816C214yesmaximal, complemented
10#24302C214yesmaximal, complemented
11#24299C214yesmaximal, complemented
12#21840C212yesmaximal, 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#3238—216no—
29C2×#501C2214yescomplemented
30C2×#489C2214yescomplemented
31C2×#483C2214yes—
32C2×#480C2214yes—
33C2×#477C2214yescomplemented
34C2×#476C2214yescomplemented
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#1987C2218yescomplemented
57#1874C2218yes—
58#1872C2218yes—
59#1853C2218yes—
60#1852C2218yes—
61#1823C2214yes—
62#1823C2214yes—
63#1817C2212yes—
64#1797C2214yes—
65#1779C2212yes—
66#1749C2214yes—
67C23×Q8—216no—
68C23×Q8—24no—
69C23×Q8—24no—
70C23×Q8—22no—
71C23×Q8—22no—
72C23×Q8—48no—
73C23×Q8—48no—
74C24×C4—22no—
75C24×C4D411yescharacteristic
76C24×C4D411yescharacteristic
77C24×C4C2311yescharacteristic
78C2×C4⋊Q8—216no—
79C2×C4⋊Q8—216no—
80C2×C4⋊Q8—232no—
81C2×C4⋊Q8—232no—
82C2×C4⋊Q8—232no—
83C2×C4⋊Q8—232no—
84C2×C4⋊Q8—216no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—22no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8D4116yescomplemented
94C2×C4⋊Q8C2318yes—
95C2×C4⋊Q8C2314yes—
96C2×C4⋊Q8C2314yes—
97#529—28no—
98C2×#126—21no—
99C2×#125C2311yescharacteristic
100C2×#124C2311yescharacteristic
101C2×#123—21no—
102C2×#123—21no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C2×#121C2311yescharacteristic
106C2×#121C2312yes—
107C2×#120—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4C2311yescharacteristic
112C2×#117—21no—
113C22×C4⋊C4—216no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—216no—
121C22×C4⋊C4C2318yes—
122C22×C4⋊C4C2314yes—
123C22×C42—28no—
124C22×C42D412yes—
125C22×C42D411yescharacteristic
126C22×C42D412yes—
127C22×C42D411yescharacteristic
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#501—216no—
133#489—216no—
134#483—216no—
135#480—216no—
136#477—216no—
137#476—216no—
138#432—28no—
139C2×#107C2318yes—
140C2×#106C2318yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2318yes—
143C2×#102C2318yes—
144C2×C22⋊C8C2318yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147#391C2312yes—
148#390C2312yes—
149#389C2312yes—
150#389C2312yes—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#386—22no—
160#386—21no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—21no—
165#385—22no—
166#385—21no—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—21no—
171#383—21no—
172#382C2312yes—
173#381—21no—
174#381—21no—
175#380C2311yescharacteristic
176#380C2311yescharacteristic
177#380C2312yes—
178#380C2312yes—
179#379—21no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2312yes—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185#376—21no—
186#340C23116yes—
187#340C23116yes—
188#336C2318yes—
189C25C2×D411yescharacteristic
190C22×Q8—216no—
191C22×Q8—232no—
192C22×Q8—232no—
193C22×Q8—216no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—28no—
197C22×Q8—4128no—
198C22×Q8—416no—
199C22×Q8—416no—
200C22×Q8—48no—
201C22×Q8—48no—
202C22×Q8—48no—
203C22×Q8—416no—
204C22×Q8—416no—
205C22×Q8—48no—
206C22×Q8C2411yescharacteristic
207C22×D4—21no—
208C22×D4—22no—
209C22×D4C2411yescharacteristic
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—216no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—21no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C22×C8C2418yes—
236C4⋊Q8—232no—
237C4⋊Q8—232no—
238C4⋊Q8—232no—
239C4⋊Q8—232no—
240C4⋊Q8—232no—
241C4⋊Q8—232no—
242C4⋊Q8—216no—
243C4⋊Q8—216no—
244C4⋊Q8—28no—
245C4⋊Q8—28no—
246C4⋊Q8C2×D4116yes—
247C4⋊Q8C2×D4116yes—
248#126—44no—
249#125—24no—
250#124—24no—
251#123—44no—
252#123—44no—
253C22⋊Q8—24no—
254C22⋊Q8—28no—
255#121—24no—
256#121—28no—
257#120—44no—
258C4×Q8—24no—
259C4×D4—24no—
260C4×D4—44no—
261C4×D4—48no—
262#117—44no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—216no—
265C2×C4⋊C4—232no—
266C2×C4⋊C4—232no—
267C2×C4⋊C4—232no—
268C2×C4⋊C4—216no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—232no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4C2×D4116yes—
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2411yescharacteristic
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2418yes—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—28no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D418yes—
319C2×C42C2412yes—
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—216no—
323#106—216no—
324C4⋊C8—28no—
325#103—216no—
326#102—216no—
327C22⋊C8—28no—
328C22⋊C8—216no—
329#97—28no—
330#95—21no—
331#95—21no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—24no—
345C24—21no—
346C24AGL(1,ℤ/8)14yes—
347C24C22×D411yescharacteristic
348C24C22×D412yes—
349C2×Q8—232no—
350C2×Q8—216no—
351C2×Q8—216no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—4128no—
355C2×Q8—4128no—
356C2×Q8—464no—
357C2×Q8—464no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—416no—
363C2×Q8—416no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—44no—
372C22×C4—232no—
373C22×C4—232no—
374C22×C4—232no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
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—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—232no—
402C22×C4—216no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—464no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4#12014yes—
417C22×C4#12014yes—
418C22×C4AGL(1,ℤ/8)14yes—
419C22×C4#13718yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D414yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C42−1+418yes—
430C22×C4C2511yescharacteristic
431C2×C8—28no—
432C2×C8—216no—
433C4⋊C4—232no—
434C4⋊C4—232no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—216no—
438C4⋊C4—28no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4C22×D4116yes—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C42—24no—
465C42—24no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—42no—
471C42—44no—
472C42#12014yes—
473C42#12014yes—
474C42C22×D414yes—
475C42C22×D414yes—
476C23—28no—
477C23—28no—
478C23—216no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—216no—
483C23—28no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23C2×#12012yes—
493C23#53614yes—
494C23C2×AGL(1,ℤ/8)14yes—
495C23C2×AGL(1,ℤ/8)12yes—
496C23C2×AGL(1,ℤ/8)12yes—
497C23C2×#13714yes—
498C23C23×D411yescharacteristic
499Q8—432no—
500Q8—432no—
501Q8—464no—
502Q8—44no—
503D4—44no—
504D4—88no—
505D4—84no—
506C2×C4—216no—
507C2×C4—28no—
508C2×C4—216no—
509C2×C4—28no—
510C2×C4—216no—
511C2×C4—216no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—216no—
528C2×C4—28no—
529C2×C4—216no—
530C2×C4—216no—
531C2×C4—464no—
532C2×C4—432no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4C2×#12012yes—
550C2×C4C2×#12012yes—
551C2×C4C2×#12012yes—
552C2×C4C2×#12012yes—
553C2×C4C2×#12012yes—
554C2×C4#53614yes—
555C2×C4C2×AGL(1,ℤ/8)14yes—
556C2×C4C2×#13714yes—
557C2×C4C2×#13714yes—
558C2×C4C23×D412yes—
559C2×C4C2×2−1+414yes—
560C8—48no—
561C22—232no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—24no—
572C22—81no—
573C22—82no—
574C22—82no—
575C22—81no—
576C22—81no—
577C22#300314yes—
578C22#323818yescomplemented
579C22C22×#12011yescharacteristic
580C22C2×#53612yes—
581C22C22×AGL(1,ℤ/8)11yescharacteristic
582C22C22×AGL(1,ℤ/8)11yescharacteristic
583C22C22×#13712yes—
584C4—22no—
585C4—24no—
586C4—22no—
587C4—22no—
588C4—24no—
589C4—22no—
590C4—24no—
591C4—416no—
592C4—81no—
593C4—82no—
594C4—82no—
595C4—81no—
596C4—81no—
597C4#300414yes—
598C4#348614yes—
599C2—28no—
600C2—24no—
601C2—161no—
602C2#3524211yescharacteristic
603C2C2×#300311yescharacteristic
604C2C2×#300411yescharacteristic
605C2C2×#323812yes—
606C2#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×#3238C2semidirect
4. #32982C2semidirect
6. #29641C2semidirect
7. #29638C2semidirect
8. #25908C2semidirect
9. #25816C2semidirect
10. #24302C2semidirect
11. #24299C2semidirect
12. #21840C2semidirect
13. #20715C2semidirect
29. C2×#501C22semidirect
30. C2×#489C22semidirect
33. C2×#477C22semidirect
34. C2×#476C22semidirect
51. #2058C22semidirect
52. #2057C22semidirect
53. #2024C22semidirect
54. #2023C22semidirect
55. #1993C22semidirect
56. #1987C22semidirect
93. C2×C4⋊Q8D4semidirect
578. C22#3238semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#32382
#366212
#329822
#329782
#296412
#296382
#259082
#258162
#243022
#242992
#218402
#207152