Finite Group Databasebeta

← #6792858

Subgroups — #6792858

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33106C211yesmaximal, characteristic
4#31579C214yesmaximal, complemented
5#31463C214yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#28792C212yesmaximal, complemented
8#27973C211yesmaximal, characteristic
9#27750C212yesmaximal, complemented
10#27681C211yesmaximal, characteristic, complemented
11#26529C212yesmaximal, complemented
12#26109C212yesmaximal, complemented
13#24600C212yesmaximal, complemented
14#24066C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21861C212yesmaximal, complemented
17C4×#483C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#483C2211yescharacteristic
20C2×#480C2212yes—
21C2×#431C2212yes—
22#2856C2211yescharacteristic
23#2854C2212yes—
24#2571—21no—
25#2571—22no—
26#2571—21no—
27#2571C2214yes—
28#2571C2212yes—
29#2571C2212yes—
30#2552C2211yescharacteristic
31#2548C2211yescharacteristic
32#2512—21no—
33#2501—21no—
34#2479—22no—
35#2454C2212yes—
36#2448C2212yes—
37#2425—22no—
38C2×#400—22no—
39#2366C2211yescharacteristic
40#2361—21no—
41#2354—21no—
42Q8×C4⋊C4C2211yescharacteristic
43C4×C22⋊Q8C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382—21no—
48C2×#382C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2101C2214yes—
52#2100C2212yes—
53#2095C2212yes—
54#2092C2214yes—
55#2086C2214yes—
56#2085C2214yes—
57#2073C2214yes—
58#2071C2214yes—
59#2067C2214yes—
60#2059C2214yes—
61#2042C2214yes—
62#1991C2214yes—
63#1982C2212yes—
64#1958C2214yes—
65#1957C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1942C2214yes—
69#1939C2212yes—
70#1934C2214yes—
71#1927C2212yes—
72#1925C2214yes—
73#1920C2212yes—
74#1916C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1874C2212yes—
78#1872C2212yes—
79#1872C2211yescharacteristic
80#1864C2214yes—
81#1863C2214yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1834C2212yes—
85#1823C2211yescharacteristic
86#1817C2212yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C4×#106C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×#127C2311yescharacteristic
96#529—28no—
97#529—44no—
98C2×#126—22no—
99C2×#125C2312yes—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×#123—22no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124#483—24no—
125#480—44no—
126#431—44no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145#400—24no—
146#400—24no—
147#400—24no—
148#400—24no—
149#399—22no—
150#399—22no—
151#398—22no—
152#397—22no—
153#396—22no—
154#394—22no—
155#391C2311yescharacteristic
156#390—21no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#389C2312yes—
174#389C2312yes—
175#389C2311yescharacteristic
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—21no—
181#386—22no—
182#386—22no—
183#385—21no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2311yescharacteristic
192#382—22no—
193#382—22no—
194#382—24no—
195#382—22no—
196#382—22no—
197#382C2314yes—
198#382C2314yes—
199#380—22no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C22⋊C4—21no—
215C2×#95—21no—
216C2×#95—22no—
217C2×#95—22no—
218C2×#95—22no—
219C2×#95D412yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2411yescharacteristic
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—22no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2411yescharacteristic
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250#127—24no—
251#126—48no—
252#125—24no—
253#125—42no—
254#124—24no—
255#124—42no—
256#123—48no—
257#123—84no—
258C22⋊Q8—22no—
259C22⋊Q8#4114yes—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—82no—
264C4×Q8—24no—
265C4×Q8—24no—
266C4×Q8—42no—
267C4×D4—24no—
268C4×D4—42no—
269C4×D4—48no—
270C4×D4—44no—
271#117—48no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—84no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
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×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—44no—
343#107—44no—
344#106—24no—
345#106—42no—
346C4⋊C8—24no—
347C4⋊C8—44no—
348#103—24no—
349#103—44no—
350#103—42no—
351#102—44no—
352#102—44no—
353#102—44no—
354#102—44no—
355C22⋊C8—22no—
356C22⋊C8—22no—
357C22⋊C8—24no—
358C4×C8—24no—
359C4×C8—44no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—22no—
380C24C22×D411yescharacteristic
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8C2×#4112yes—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—81no—
397C2×D4—81no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—82no—
431C22×C4#13514yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×#4112yes—
490C22⋊C4—24no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—44no—
512C42—42no—
513C42—44no—
514C42—44no—
515C42—42no—
516C422−1+414yes—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23#54612yes—
529C23C2×#13512yes—
530C23#57612yes—
531C23C23×D411yescharacteristic
532Q8—24no—
533Q8—42no—
534D4—44no—
535D4—44no—
536D4—88no—
537D4—84no—
538D4—162no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—84no—
590C2×C4—88no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4#53612yes—
595C2×C4#53612yes—
596C2×C4#54612yes—
597C2×C4C2×#13512yes—
598C2×C4C2×#13512yes—
599C2×C4#57512yes—
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×2−1+412yes—
603C2×C4#58512yes—
604C8—42no—
605C8—42no—
606C8—44no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22#330514yes—
621C22C2×#53611yescharacteristic
622C22C2×#54611yescharacteristic
623C22#348711yescharacteristic
624C22C22×#13511yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57611yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—84no—
644C4#323214yes—
645C4#351512yes—
646C4#355912yes—
647C2—24no—
648C2—162no—
649C2#3890711yescharacteristic
650C2C2×#323211yescharacteristic
651C2#6087111yescharacteristic
652C2C2×#330511yescharacteristic
653C2#6116511yescharacteristic
654C2#6188211yescharacteristic
655C2#6189811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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×#2571C2semidirect
4. #31579C2semidirect
5. #31463C2semidirect
6. #29912C2semidirect
7. #28792C2semidirect
9. #27750C2semidirect
10. #27681C2semidirect
11. #26529C2semidirect
12. #26109C2semidirect
13. #24600C2semidirect
14. #24066C2semidirect
15. #22599C2semidirect
16. #21861C2semidirect
17. C4×#483C2semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#25712
#331062
#315792
#314632
#299122
#287922
#279732
#277502
#276812
#265292
#261092
#246002
#240662
#225992
#218612
C4×#4832