Finite Group Databasebeta

← #6623456

Subgroups — #6623456

Order 512.

1613 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2C4×#559C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#33066C212yesmaximal, complemented
5#29139C211yesmaximal, characteristic
6#29131C212yesmaximal, complemented
7#28252C211yesmaximal, characteristic
8#28159C212yesmaximal, complemented
9#27452C214yesmaximal, complemented
10#26915C214yesmaximal, complemented
11#24995C212yesmaximal, complemented
12#24190C212yesmaximal
13#23003C212yesmaximal
14#22825C211yesmaximal, characteristic, complemented
15#22763C212yesmaximal, complemented
16#21504C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#559C2211yescharacteristic
19C2×#477C2212yes—
20C2×#462C2211yescharacteristic
21C2×#460C2212yes—
22C2×#461C2212yes—
23C2×#440C2214yes—
24C2×C4×Q16C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2582C2212yes—
28#2567—21no—
29#2552—21no—
30#2550—22no—
31#2507—22no—
32#2451C2214yes—
33#2374—22no—
34#2369—21no—
35#2367—21no—
36#2367C2212yes—
37#2366C2211yescharacteristic
38#2357—21no—
39#2357C2212yes—
40#2355—21no—
41#2355C2211yescharacteristic
42#2355C2211yescharacteristic
43Q8×C4⋊C4C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#124—21no—
48C4×#124C2212yes—
49C4×#123—22no—
50C42×Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52C42×D4—21no—
53C42×D4C2212yes—
54#2101C2214yes—
55#2074C2214yes—
56#2055C2214yes—
57#2042C2214yes—
58#2023C2214yes—
59#2015C2214yes—
60#1987C2212yes—
61#1984C2212yes—
62#1981C2211yescharacteristic
63#1957C2212yes—
64#1947C2212yes—
65#1947C2211yescharacteristic
66#1947C2212yes—
67#1936C2214yes—
68#1935C2214yes—
69#1928C2214yes—
70#1928C2212yes—
71#1918C2214yes—
72#1918C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1878C2212yes—
76#1878C2214yes—
77#1877C2214yes—
78#1862C2214yes—
79#1862C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1831C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
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×#125—22no—
105C2×#124—21no—
106C2×#124C2312yes—
107C2×#123—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×C4×D4C2312yes—
123C22×C42—22no—
124#477—44no—
125#462—24no—
126#461—28no—
127#460—28no—
128#440—48no—
129C4×Q16—28no—
130C2×#107C2314yes—
131C2×#107C2312yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#398—24no—
149#398—24no—
150#395—24no—
151#391—22no—
152#391—21no—
153#391—24no—
154#391C2312yes—
155#391C2314yes—
156#391C2312yes—
157#390—21no—
158#390—21no—
159#390—21no—
160#390C2312yes—
161#390C2312yes—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—24no—
170#389C2314yes—
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—22no—
181#386—22no—
182#385—22no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382—21no—
191#382—22no—
192#382C2312yes—
193#381—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C43—21no—
223C43C2311yescharacteristic
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×Q8C2412yes—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2412yes—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C2×Q16—24no—
247C2×Q16—24no—
248C2×Q16—42no—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—42no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#127—22no—
262#127#4114yes—
263#125—84no—
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—44no—
280C4×Q8—44no—
281C4×Q8#4114yes—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4—82no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
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×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—84no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—28no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—22no—
360C4⋊C8—22no—
361C4⋊C8—42no—
362#103—24no—
363#103—28no—
364#103—24no—
365#103—24no—
366#103—24no—
367#103—44no—
368#102—28no—
369#102—44no—
370#102—44no—
371#97—44no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—21no—
405C2×Q8—44no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8C2×#4114yes—
409C2×Q8C2×#4112yes—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4C2×#4114yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—28no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450Q16—48no—
451SD16—416no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—81no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C2×#4112yes—
496C4⋊C4H(5,𝔽2)12yes—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—21no—
515C42—22no—
516C42—21no—
517C42—41no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—42no—
523C42—42no—
524C42—42no—
525C42—81no—
526C42—82no—
527C42—82no—
528C42—84no—
529C42#13714yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42C2×#4112yes—
533C422−1+412yes—
534C23—24no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23C23×D411yescharacteristic
542Q8—22no—
543Q8—24no—
544Q8—24no—
545Q8—22no—
546Q8—44no—
547Q8—41no—
548Q8—42no—
549Q8—41no—
550Q8—42no—
551Q8—164no—
552D4—24no—
553D4—24no—
554D4—48no—
555D4—42no—
556D4—42no—
557D4—84no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4#52512yes—
607C2×C4#52512yes—
608C2×C4#54812yes—
609C2×C4#54812yes—
610C2×C4C2×#13712yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4#57512yes—
614C2×C4#57612yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—42no—
622C8—44no—
623C8—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22C2×#52511yescharacteristic
632C22C2×#54811yescharacteristic
633C22#348811yescharacteristic
634C22C22×#13711yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57611yescharacteristic
637C22#358611yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—84no—
651C4—82no—
652C4—81no—
653C4—82no—
654C4#311614yes—
655C4#337514yes—
656C4#347511yescharacteristic
657C4#348111yescharacteristic
658C4#351111yescharacteristic
659C4#351511yescharacteristic
660C2—84no—
661C2#3649211yescharacteristic
662C2C2×#311611yescharacteristic
663C2#6059611yescharacteristic
664C2C2×#337511yescharacteristic
665C2#6132011yescharacteristic
666C2#6196411yescharacteristic
667C2#6199511yescharacteristic
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. #36563C2semidirect
3. #33198C2semidirect
4. #33066C2semidirect
6. #29131C2semidirect
8. #28159C2semidirect
9. #27452C2semidirect
10. #26915C2semidirect
11. #24995C2semidirect
14. #22825C2semidirect
15. #22763C2semidirect
16. #21504C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#365632
C4×#5592
#331982
#330662
#291392
#291312
#282522
#281592
#274522
#269152
#249952
#241902
#230032
#228252
#227632
#215042
C4×#4622