Finite Group Databasebeta

← #6593794

Subgroups — #6593794

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36176C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#33149C211yesmaximal, characteristic, complemented
4#33005C212yesmaximal
5#27589C212yesmaximal, complemented
6#27084C214yesmaximal, complemented
7#26772C214yesmaximal, complemented
8#26531C212yesmaximal, complemented
9#25856C211yesmaximal, characteristic, complemented
10#25319C212yesmaximal, complemented
11#24509C212yesmaximal
12#24209C211yesmaximal, characteristic
13#23221C212yesmaximal, complemented
14#23100C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic, complemented
16C4×#481C211yesmaximal, characteristic
17#20741C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#487C2212yes—
21C2×#481C2211yescharacteristic
22C2×#461C2212yes—
23C2×#448C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2854C2212yes—
27#2571C2211yescharacteristic
28#2563C2211yescharacteristic
29#2532—22no—
30#2513C2212yes—
31#2508C2212yes—
32#2495—21no—
33#2488—22no—
34#2486—21no—
35#2369C2211yescharacteristic
36#2365—21no—
37#2365—21no—
38#2365—22no—
39#2365C2212yes—
40#2365C2214yes—
41#2365C2212yes—
42#2354C2211yescharacteristic
43#2352—21no—
44#2346—21no—
45C4×#121C2211yescharacteristic
46C2×#391C2212yes—
47C2×#389—21no—
48C2×#387—22no—
49#2283C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2074C2214yes—
55#2068C2214yes—
56#2055C2214yes—
57#2043C2214yes—
58#2043C2212yes—
59#2013C2212yes—
60#1987C2212yes—
61#1981C2212yes—
62#1973C2214yes—
63#1956C2212yes—
64#1947C2212yes—
65#1939C2212yes—
66#1935C2214yes—
67#1928C2214yes—
68#1927C2214yes—
69#1926C2212yes—
70#1921C2214yes—
71#1919C2212yes—
72#1917C2214yes—
73#1880C2214yes—
74#1880C2214yes—
75#1874C2211yescharacteristic
76#1865C2214yes—
77#1865C2214yes—
78#1853C2211yescharacteristic
79#1852C2212yes—
80#1824C2211yescharacteristic
81#1824C2212yes—
82#1817C2212yes—
83#1805C2214yes—
84#1805C2214yes—
85#1797C2211yescharacteristic
86#1787C2212yes—
87#1787C2212yes—
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×SD16C2314yes—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#123—22no—
104C2×C22⋊Q8—21no—
105C2×#121C2311yescharacteristic
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—28no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2314yes—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C42—22no—
123C22×C42—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#487—28no—
127#481—24no—
128#461—28no—
129#448—44no—
130#432—44no—
131C2×#107C2314yes—
132C2×#106C2312yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#398—22no—
153#397—22no—
154#397—22no—
155#393—22no—
156#391—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391C2314yes—
161#391C2314yes—
162#390C2312yes—
163#390C2312yes—
164#390C2311yescharacteristic
165#389—21no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389—21no—
170#389—21no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2311yescharacteristic
175#388—22no—
176#388—22no—
177#388—22no—
178#388—21no—
179#388—22no—
180#388—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#386—22no—
187#386—21no—
188#384C2311yescharacteristic
189#383—22no—
190#382—21no—
191#382—22no—
192#382C2312yes—
193#382C2312yes—
194#381—22no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380C2312yes—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—24no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#376—22no—
221C2×#95—22no—
222C2×#95—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8—24no—
240C22×Q8—24no—
241C22×Q8—44no—
242C22×Q8C2414yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—28no—
261C4⋊Q8—28no—
262C4⋊Q8—28no—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8#4118yes—
266#123—84no—
267C22⋊Q8—82no—
268#121—22no—
269#121—22no—
270C4×Q8—28no—
271C4×Q8—28no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×Q8—48no—
276C4×D4—24no—
277C4×D4—44no—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4#4114yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—48no—
336C2×C42—44no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—44no—
346#106—24no—
347#106—42no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#103—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#103—44no—
355#103—44no—
356#103—44no—
357#102—24no—
358#102—42no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—24no—
362#97—44no—
363C4×C8—24no—
364#95—24no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24—21no—
388C24C22×D411yescharacteristic
389C2×Q8—28no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—416no—
396C2×Q8—416no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8C2×#4118yes—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—41no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4#13714yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4112yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4C2511yescharacteristic
456SD16—88no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4C2×#4114yes—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C42—22no—
508C42—22no—
509C42—24no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—48no—
517C42—48no—
518C42—44no—
519C42—44no—
520C42—42no—
521C42—44no—
522C42—82no—
523C42C2×#4114yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23#54612yes—
536C23C2×#13712yes—
537C23#57712yes—
538C23C23×D411yescharacteristic
539Q8—28no—
540Q8—28no—
541Q8—416no—
542Q8—44no—
543Q8—44no—
544Q8—162no—
545D4—42no—
546D4—42no—
547D4—84no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—416no—
577C2×C4—48no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4#52512yes—
604C2×C4#52512yes—
605C2×C4#54612yes—
606C2×C4C2×#13712yes—
607C2×C4C2×#13712yes—
608C2×C4#57512yes—
609C2×C4C22×#4111yescharacteristic
610C2×C4C22×#4112yes—
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4#58512yes—
613C8—42no—
614C8—42no—
615C8—44no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#332914yes—
630C22C2×#52511yescharacteristic
631C22C2×#54611yescharacteristic
632C22#348711yescharacteristic
633C22C22×#13711yescharacteristic
634C22C2×#57511yescharacteristic
635C22C2×#57711yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—48no—
641C4—41no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—82no—
652C4—81no—
653C4#311714yes—
654C4#347512yes—
655C4#351312yes—
656C2—24no—
657C2—161no—
658C2#3616211yescharacteristic
659C2C2×#311711yescharacteristic
660C2#6058211yescharacteristic
661C2C2×#332911yescharacteristic
662C2#6115911yescharacteristic
663C2#6189311yescharacteristic
664C2#6189911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2365C2semidirect
3. #33149C2semidirect
5. #27589C2semidirect
6. #27084C2semidirect
7. #26772C2semidirect
8. #26531C2semidirect
9. #25856C2semidirect
10. #25319C2semidirect
13. #23221C2semidirect
14. #23100C2semidirect
15. #22603C2semidirect
17. #20741C2semidirect

Maximal subgroups

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

subgroupindex
#361762
C2×#23652
#331492
#330052
#275892
#270842
#267722
#265312
#258562
#253192
#245092
#242092
#232212
#231002
#226032
C4×#4812
#207412