Finite Group Databasebeta

← #6489537

Subgroups — #6489537

Order 512.

3241 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
1C23×#125C211yesmaximal, characteristic, complemented
2C2×#3232C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#32983C211yesmaximal, characteristic, complemented
5#32982C211yesmaximal, characteristic, complemented
6#32979C211yesmaximal, characteristic
7#32978C211yesmaximal, characteristic
8#29643C212yesmaximal, complemented
9#29640C212yesmaximal, complemented
10#25907C214yesmaximal, complemented
11#25815C214yesmaximal, complemented
12#24307C214yesmaximal, complemented
13#24305C214yesmaximal, complemented
14#21836C212yesmaximal, complemented
15#20714C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C22×#125—22no—
18C22×#125—21no—
19C22×#125—22no—
20C22×#125—21no—
21C22×#125—216no—
22C22×#125C2214yes—
23C22×#125C2212yes—
24C22×#125C2214yes—
25C22×#125C2212yes—
26C23×C4⋊C4—21no—
27C23×C4⋊C4—21no—
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C42C2211yescharacteristic
31#3232—216no—
32C2×#499C2214yescomplemented
33C2×#488C2214yescomplemented
34C2×#484C2212yes—
35C2×#481C2212yes—
36C2×#483C2212yes—
37C2×#480C2212yes—
38C2×#479C2214yescomplemented
39C2×#478C2214yescomplemented
40C2×#431C2212yes—
41#2855C2211yescharacteristic
42#2854C2211yescharacteristic
43#2853—21no—
44#2852—22no—
45#2851C2212yes—
46#2850C2212yes—
47#2849—21no—
48#2848C2211yescharacteristic
49#2847C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C4×#123—21no—
52C4×#121C2212yes—
53C4×#120—21no—
54#2290—22no—
55#2283C2211yescharacteristic
56#2059C2218yescomplemented
57#2059C2218yescomplemented
58#2026C2218yescomplemented
59#2025C2218yescomplemented
60#1991C2218yescomplemented
61#1989C2218yescomplemented
62#1874C2214yes—
63#1874C2214yes—
64#1872C2214yes—
65#1872C2214yes—
66#1853C2214yes—
67#1853C2214yes—
68#1852C2214yes—
69#1852C2214yes—
70#1824C2212yes—
71#1824C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1817C2212yes—
75#1797C2214yes—
76#1779C2211yescharacteristic
77#1779C2211yescharacteristic
78C4×C22⋊C8C2214yes—
79C24×C4—21no—
80C24×C4—21no—
81C24×C4D411yescharacteristic
82C24×C4D411yescharacteristic
83C24×C4C2311yescharacteristic
84#529—28no—
85C2×#126—21no—
86C2×#125—216no—
87C2×#125—216no—
88C2×#125—232no—
89C2×#125—232no—
90C2×#125—232no—
91C2×#125—232no—
92C2×#125—28no—
93C2×#125—28no—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×#125—24no—
99C2×#125—22no—
100C2×#125—22no—
101C2×#125—28no—
102C2×#125D4116yescomplemented
103C2×#125C2318yes—
104C2×#125C2311yescharacteristic
105C2×#125C2314yes—
106C2×#125C2314yes—
107C2×#124C2311yescharacteristic
108C2×#123—21no—
109C2×#123—21no—
110C2×C22⋊Q8C2311yescharacteristic
111C2×C22⋊Q8C2312yes—
112C2×#121C2311yescharacteristic
113C2×#121C2312yes—
114C2×#120—21no—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—22no—
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C22×C4⋊C4—216no—
121C22×C4⋊C4—216no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—24no—
124C22×C4⋊C4—24no—
125C22×C4⋊C4—24no—
126C22×C4⋊C4—24no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4—28no—
135C22×C4⋊C4—28no—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C42—28no—
141C22×C42D412yes—
142C22×C42D411yescharacteristic
143C22×C42D412yes—
144C22×C42D411yescharacteristic
145C22×C42C2312yes—
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149#499—216no—
150#488—216no—
151#484—28no—
152#483—28no—
153#481—28no—
154#480—28no—
155#479—216no—
156#478—216no—
157#431—28no—
158C2×#107C2314yes—
159C2×#107C2314yes—
160C2×#106C2314yes—
161C2×#106C2314yes—
162C2×C4⋊C8C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2314yes—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2312yes—
170C2×C22⋊C8C2312yes—
171C2×C4×C8C2314yes—
172#391C2312yes—
173#390C2312yes—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—21no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—21no—
190#385—22no—
191#385—21no—
192#385—22no—
193#385—22no—
194#385—22no—
195#385—21no—
196#383—21no—
197#382C2312yes—
198#381—21no—
199#381—21no—
200#380C2311yescharacteristic
201#380C2311yescharacteristic
202#380C2312yes—
203#380C2312yes—
204#379—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210#376—21no—
211#340C2318yes—
212#340C2318yes—
213#340C2318yes—
214#340C2318yes—
215#336C2318yes—
216C25C2×D411yescharacteristic
217C22×Q8C2411yescharacteristic
218C22×D4—21no—
219C22×D4—22no—
220C22×D4C2411yescharacteristic
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—21no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—48no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C22×C8C2414yes—
251C22×C8C2414yes—
252#126—44no—
253#125—232no—
254#125—232no—
255#125—232no—
256#125—232no—
257#125—232no—
258#125—216no—
259#125—216no—
260#125—216no—
261#125—216no—
262#125—28no—
263#125—28no—
264#125—24no—
265#125C2×D4116yes—
266#125C2×D4116yes—
267#124—24no—
268#123—44no—
269#123—44no—
270C22⋊Q8—24no—
271C22⋊Q8—28no—
272#121—24no—
273#121—28no—
274#120—44no—
275C4×Q8—24no—
276C4×D4—24no—
277C4×D4—48no—
278C4×D4—44no—
279#117—44no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—216no—
283C2×C4⋊C4—216no—
284C2×C4⋊C4—232no—
285C2×C4⋊C4—232no—
286C2×C4⋊C4—232no—
287C2×C4⋊C4—232no—
288C2×C4⋊C4—216no—
289C2×C4⋊C4—216no—
290C2×C4⋊C4—216no—
291C2×C4⋊C4—216no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—216no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4C2×D418yes—
314C2×C4⋊C4C2×D418yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2414yes—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—28no—
338C2×C42—28no—
339C2×C42—28no—
340C2×C42—28no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D418yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#107—28no—
356#106—28no—
357#106—28no—
358C4⋊C8—28no—
359#103—28no—
360#103—28no—
361#102—28no—
362#102—28no—
363C22⋊C8—24no—
364C22⋊C8—24no—
365C22⋊C8—28no—
366C22⋊C8—28no—
367C4×C8—28no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379C24—22no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—21no—
384C24#13514yes—
385C24C22×D411yescharacteristic
386C24C22×D412yes—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—42no—
395C2×D4—44no—
396C2×D4—42no—
397C22×C4—232no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—432no—
441C22×C4—432no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4#12014yes—
451C22×C4#12014yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C42−1+418yes—
462C22×C4C2511yescharacteristic
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—28no—
466C2×C8—28no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—232no—
470C4⋊C4—232no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4C22×D418yes—
491C4⋊C4C22×D418yes—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C42—24no—
506C42—24no—
507C42—24no—
508C42—24no—
509C42—24no—
510C42—24no—
511C42—44no—
512C42—42no—
513C42#12014yes—
514C42#12014yes—
515C42C22×D414yes—
516C42C22×D414yes—
517C23—28no—
518C23—28no—
519C23—216no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—216no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23C2×#12012yes—
534C23#53614yes—
535C23C2×#13514yes—
536C23C2×#13512yes—
537C23#57514yes—
538C23#57512yes—
539C23C23×D411yescharacteristic
540Q8—44no—
541D4—44no—
542D4—88no—
543D4—84no—
544C2×C4—216no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—216no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—432no—
572C2×C4—44no—
573C2×C4—416no—
574C2×C4—432no—
575C2×C4—44no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—84no—
594C2×C4C2×#12012yes—
595C2×C4C2×#12012yes—
596C2×C4C2×#12012yes—
597C2×C4C2×#12012yes—
598C2×C4C2×#12012yes—
599C2×C4#53614yes—
600C2×C4C23×D412yes—
601C2×C4C2×2−1+414yes—
602C8—44no—
603C8—44no—
604C22—232no—
605C22—24no—
606C22—24no—
607C22—22no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—22no—
614C22—24no—
615C22—82no—
616C22—81no—
617C22—82no—
618C22—81no—
619C22—81no—
620C22#300214yes—
621C22#323218yescomplemented
622C22C22×#12011yescharacteristic
623C22C2×#53612yes—
624C22C22×#13511yescharacteristic
625C22C2×#57512yes—
626C22C2×#57511yescharacteristic
627C4—22no—
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—48no—
635C4—42no—
636C4—48no—
637C4—82no—
638C4—81no—
639C4—82no—
640C4—81no—
641C4—81no—
642C4#348614yes—
643C2—28no—
644C2—24no—
645C2—161no—
646C2#3524211yescharacteristic
647C2C2×#300211yescharacteristic
648C2#6035911yescharacteristic
649C2C2×#323212yes—
650C2#6086112yes—
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×#125C2semidirect
2. C2×#3232C2semidirect
4. #32983C2semidirect
5. #32982C2semidirect
8. #29643C2semidirect
9. #29640C2semidirect
10. #25907C2semidirect
11. #25815C2semidirect
12. #24307C2semidirect
13. #24305C2semidirect
14. #21836C2semidirect
15. #20714C2semidirect
32. C2×#499C22semidirect
33. C2×#488C22semidirect
38. C2×#479C22semidirect
39. C2×#478C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #2026C22semidirect
59. #2025C22semidirect
60. #1991C22semidirect
61. #1989C22semidirect
102. C2×#125D4semidirect
621. C22#3232semidirect

Maximal subgroups

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

subgroupindex
C23×#1252
C2×#32322
#366212
#329832
#329822
#329792
#329782
#296432
#296402
#259072
#258152
#243072
#243052
#218362
#207142