Finite Group Databasebeta

← #6565918

Subgroups — #6565918

Order 512.

1713 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34313C211yesmaximal, characteristic
3#30039C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic, complemented
5#29554C212yesmaximal, complemented
6#29392C212yesmaximal, complemented
7#29139C211yesmaximal, characteristic, complemented
8#28160C211yesmaximal, characteristic, complemented
9#27552C214yesmaximal, complemented
10#27064C214yesmaximal, complemented
11#24994C212yesmaximal, complemented
12#24188C212yesmaximal
13#23815C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23095C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22770C212yesmaximal, complemented
18C2×#550C2211yescharacteristic
19C2×#520—21no—
20C2×#460C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2582C2212yes—
25#2571—21no—
26#2570—22no—
27#2568—21no—
28#2511C2214yes—
29#2446—22no—
30#2366C2211yescharacteristic
31#2366C2212yes—
32#2365—21no—
33#2361—21no—
34#2361—22no—
35#2355—21no—
36#2355C2212yes—
37Q8×C4⋊C4C2211yescharacteristic
38D4×C4⋊C4C2212yes—
39D4×C4⋊C4C2211yescharacteristic
40#2341—21no—
41#2324—22no—
42#2324—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#127C2212yes—
47#2293C2211yescharacteristic
48#2293C2212yes—
49C42×Q8C2211yescharacteristic
50#2100C2212yes—
51#2099C2214yes—
52#2094C2212yes—
53#2084C2211yescharacteristic
54#2072C2214yes—
55#2071C2212yes—
56#2071C2211yescharacteristic
57#2066C2212yes—
58#2058C2212yes—
59#2055C2212yes—
60#2043C2211yescharacteristic
61#2041C2214yes—
62#2023C2212yes—
63#2023C2212yes—
64#2023C2212yes—
65#2014C2214yes—
66#2013C2212yes—
67#2013C2211yescharacteristic
68#1984C2212yes—
69#1968C2214yes—
70#1957C2212yes—
71#1948C2212yes—
72#1933C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1916C2214yes—
78#1877C2214yes—
79#1877C2214yes—
80#1877C2214yes—
81#1877C2214yes—
82#1864C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1831C2211yescharacteristic
86#1803C2212yes—
87#1802C2212yes—
88#1802C2214yes—
89C8×C4⋊C4C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×#103C2212yes—
92C4×#102C2214yes—
93C4×#102C2214yes—
94#1744C2212yes—
95C22×#41—21no—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#550—24no—
99#550—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2312yes—
105C2×#127C2311yescharacteristic
106C2×#121—21no—
107C2×#121—22no—
108#520—48no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C2×#117—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130#460—28no—
131C4×SD16—28no—
132C2×#106C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#400—24no—
149#398—24no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2314yes—
156#391C2314yes—
157#390—21no—
158#390—22no—
159#390—22no—
160#390—21no—
161#390—21no—
162#390—22no—
163#390C2312yes—
164#390C2311yescharacteristic
165#390C2311yescharacteristic
166#390C2312yes—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—21no—
171#389—22no—
172#389—21no—
173#389C2314yes—
174#389C2311yescharacteristic
175#389C2311yescharacteristic
176#389C2312yes—
177#388—22no—
178#388—21no—
179#388—22no—
180#388—21no—
181#387—24no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#386—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#382—22no—
199#382—22no—
200#380—21no—
201#380—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2314yes—
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C2×#41—28no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C2×SD16—24no—
251C2×SD16—28no—
252C2×SD16—42no—
253C2×SD16—42no—
254C2×SD16—44no—
255C2×SD16—44no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—44no—
260C4⋊Q8—42no—
261C4⋊Q8—44no—
262C4⋊Q8—82no—
263#127—24no—
264#127—24no—
265#127—22no—
266#127#4114yes—
267#121—28no—
268#121—216no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—42no—
274C4×Q8—44no—
275C4×Q8—84no—
276C4×Q8SD1618yescomplemented
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—48no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4—84no—
288C4×D4—82no—
289C4×D4#4114yes—
290#117—48no—
291#117—44no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42C2×D412yes—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#106—24no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—42no—
358#103—24no—
359#103—42no—
360#103—44no—
361#103—42no—
362#102—24no—
363#102—28no—
364#102—44no—
365#102—44no—
366#102—44no—
367C4×C8—24no—
368C4×C8—44no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—22no—
378#95—21no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24—21no—
387#41—432no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8C2×SD1614yes—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—82no—
416C2×D4—81no—
417C2×D4—82no—
418C2×D4C2×#4114yes—
419C2×D4C2×#4112yes—
420C22×C4—28no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—82no—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453SD16—48no—
454SD16—88no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×SD1618yes—
495C4⋊C4C2×SD1614yes—
496C4⋊C4C2×#4112yes—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—82no—
506C22⋊C4—81no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—41no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—44no—
523C42—42no—
524C42—84no—
525C42C2×SD1618yes—
526C42C2×SD1614yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C23—24no—
530C23—22no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—41no—
543C23C23×D411yescharacteristic
544Q8—28no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—88no—
550D4—24no—
551D4—28no—
552D4—416no—
553D4—48no—
554D4—44no—
555D4—42no—
556D4—84no—
557D4—88no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—416no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—164no—
603C2×C4—164no—
604C2×C4—164no—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4C22×SD1614yes—
608C2×C4C22×SD1612yes—
609C2×C4C22×SD1614yes—
610C2×C4C22×SD1612yes—
611C2×C4C22×SD1612yes—
612C2×C4#57512yes—
613C2×C4#57612yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4#58512yes—
617C2×C4#58512yes—
618C8—42no—
619C8—42no—
620C8—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—82no—
628C22—81no—
629C22—82no—
630C22—81no—
631C22—82no—
632C22—81no—
633C22—82no—
634C22—81no—
635C22C2×#52511yescharacteristic
636C22#348711yescharacteristic
637C22#350011yescharacteristic
638C22C23×SD1611yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57611yescharacteristic
641C22#358611yescharacteristic
642C4—24no—
643C4—22no—
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—81no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—84no—
658C4#310114yes—
659C4#349312yes—
660C4#352112yes—
661C2—84no—
662C2—82no—
663C2#3601911yescharacteristic
664C2C2×#310111yescharacteristic
665C2#6059611yescharacteristic
666C2#6188311yescharacteristic
667C2#6189811yescharacteristic
668C2#6252511yescharacteristic
669C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
3. #30039C2semidirect
4. #29981C2semidirect
5. #29554C2semidirect
6. #29392C2semidirect
7. #29139C2semidirect
8. #28160C2semidirect
9. #27552C2semidirect
10. #27064C2semidirect
11. #24994C2semidirect
13. #23815C2semidirect
14. #23677C2semidirect
15. #23095C2semidirect
16. #22828C2semidirect
17. #22770C2semidirect
276. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365982
#343132
#300392
#299812
#295542
#293922
#291392
#281602
#275522
#270642
#249942
#241882
#238152
#236772
#230952
#228282
#227702