Finite Group Databasebeta

← #6623903

Subgroups — #6623903

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2C4×#559C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#33026C212yesmaximal, complemented
5#28509C211yesmaximal, characteristic
6#28384C212yesmaximal, complemented
7#28252C211yesmaximal, characteristic
8#28204C212yesmaximal, complemented
9#27516C214yesmaximal, complemented
10#27088C214yesmaximal, complemented
11#25061C212yesmaximal, complemented
12#24190C212yesmaximal
13#23101C212yesmaximal
14#22825C211yesmaximal, characteristic, complemented
15#22814C212yesmaximal, complemented
16#21614C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#477C2212yes—
20C2×#476C2212yes—
21C2×#476C2212yes—
22C2×#462C2211yescharacteristic
23C2×#460C2212yes—
24C2×#461C2212yes—
25C2×C4×Q16C2212yes—
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2582C2212yes—
29#2571—21no—
30#2570—22no—
31#2568—21no—
32#2511C2214yes—
33#2446—22no—
34#2374—22no—
35#2367—21no—
36#2367C2212yes—
37#2366C2211yescharacteristic
38#2365—21no—
39#2357—21no—
40#2357C2212yes—
41#2355—21no—
42#2355C2211yescharacteristic
43#2355C2212yes—
44#2355C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#124—21no—
49C4×#124C2212yes—
50C4×#123—22no—
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53C42×D4—21no—
54C42×D4C2212yes—
55#2097C2214yes—
56#2094C2214yes—
57#2085C2214yes—
58#2055C2214yes—
59#2042C2214yes—
60#2015C2214yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2211yescharacteristic
64#1973C2214yes—
65#1971C2214yes—
66#1957C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1947C2212yes—
70#1928C2212yes—
71#1927C2214yes—
72#1927C2214yes—
73#1927C2214yes—
74#1925C2214yes—
75#1919C2214yes—
76#1917C2214yes—
77#1878C2212yes—
78#1877C2214yes—
79#1877C2214yes—
80#1862C2214yes—
81#1844C2212yes—
82#1842C2211yescharacteristic
83#1831C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1762C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97#559—24no—
98#559—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#124—21no—
105C2×#124C2312yes—
106C2×#123—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C22×C42—22no—
123#477—44no—
124#476—28no—
125#476—44no—
126#462—24no—
127#461—44no—
128#460—28no—
129C4×Q16—28no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—24no—
151#400—24no—
152#398—24no—
153#391—22no—
154#391—21no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2314yes—
160#391C2314yes—
161#391C2312yes—
162#390—21no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—22no—
168#389—21no—
169#389—22no—
170#389—21no—
171#389C2314yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—24no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#386—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—21no—
193#382—22no—
194#382—22no—
195#382C2312yes—
196#381—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
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⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C43—21no—
226C43C2311yescharacteristic
227C43C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—22no—
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2412yes—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—42no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262#127—22no—
263#127#4114yes—
264#124—28no—
265#124—24no—
266#124—24no—
267#123—216no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—42no—
278C4×Q8—41no—
279C4×Q8—84no—
280C4×Q8#4114yes—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—82no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—24no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—84no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—24no—
353#107—44no—
354#106—28no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—22no—
359C4⋊C8—42no—
360C4⋊C8#4114yes—
361#103—24no—
362#103—24no—
363#103—24no—
364#103—24no—
365#103—44no—
366#103—44no—
367#102—28no—
368#102—28no—
369#102—44no—
370#97—44no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8C2×#4112yes—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4C2×#4114yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2511yescharacteristic
451Q16—48no—
452SD16—88no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8C2×#4114yes—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4C2×#4112yes—
495C4⋊C42−1+412yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—22no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42—41no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—81no—
525C42—82no—
526C42—82no—
527C42—84no—
528C42#13714yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42H(5,𝔽2)12yes—
533C23—24no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23C23×D411yescharacteristic
541Q8—22no—
542Q8—22no—
543Q8—44no—
544Q8—41no—
545Q8—41no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—88no—
551D4—24no—
552D4—24no—
553D4—48no—
554D4—42no—
555D4—42no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4#55012yes—
608C2×C4#55012yes—
609C2×C4C2×#13712yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4#57512yes—
613C2×C4#57712yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4#58511yescharacteristic
620C8—24no—
621C8—42no—
622C8—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×#52511yescharacteristic
631C22C2×#55011yescharacteristic
632C22#348911yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57511yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358511yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—81no—
652C4—84no—
653C4#311614yes—
654C4#340014yes—
655C4#347511yescharacteristic
656C4#348211yescharacteristic
657C4#351111yescharacteristic
658C4#351411yescharacteristic
659C2—84no—
660C2#3650411yescharacteristic
661C2C2×#311611yescharacteristic
662C2#6059711yescharacteristic
663C2C2×#340011yescharacteristic
664C2#6140511yescharacteristic
665C2#6211211yescharacteristic
666C2#6214411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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. #33198C2semidirect
4. #33026C2semidirect
6. #28384C2semidirect
8. #28204C2semidirect
9. #27516C2semidirect
10. #27088C2semidirect
11. #25061C2semidirect
14. #22825C2semidirect
15. #22814C2semidirect
16. #21614C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×#5592
#331982
#330262
#285092
#283842
#282522
#282042
#275162
#270882
#250612
#241902
#231012
#228252
#228142
#216142
C4×#4622