Finite Group Databasebeta

← #6792849

Subgroups — #6792849

Order 512.

1487 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#31581C214yesmaximal, complemented
5#31465C214yesmaximal, complemented
6#29913C212yesmaximal, complemented
7#28790C212yesmaximal, complemented
8#27972C211yesmaximal, characteristic, complemented
9#27748C212yesmaximal, complemented
10#27678C211yesmaximal, characteristic
11#26527C212yesmaximal, complemented
12#26107C212yesmaximal, complemented
13#24601C212yesmaximal, complemented
14#24066C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21861C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#529C2212yes—
19C2×#483C2212yes—
20C2×#480C2211yescharacteristic
21C2×#431C2212yes—
22#2857C2211yescharacteristic
23#2855C2212yes—
24#2571—21no—
25#2571—22no—
26#2571—21no—
27#2571C2214yes—
28#2571C2212yes—
29#2571C2212yes—
30#2552C2211yescharacteristic
31#2547C2211yescharacteristic
32#2496—21no—
33#2486—21no—
34#2478—22no—
35#2453C2212yes—
36#2447C2212yes—
37#2419—22no—
38C2×#400—22no—
39#2365C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2211yescharacteristic
42#2344—21no—
43C4×#121C2211yescharacteristic
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#2097C2214yes—
54#2094C2212yes—
55#2088C2214yes—
56#2085C2214yes—
57#2074C2214yes—
58#2072C2214yes—
59#2066C2214yes—
60#2059C2214yes—
61#2042C2214yes—
62#1991C2214yes—
63#1981C2212yes—
64#1958C2214yes—
65#1956C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1941C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1927C2214yes—
72#1925C2212yes—
73#1918C2212yes—
74#1916C2214yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1864C2214yes—
81#1863C2214yes—
82#1853C2212yes—
83#1852C2211yescharacteristic
84#1834C2212yes—
85#1823C2211yescharacteristic
86#1817C2212yes—
87#1805C2214yes—
88#1804C2214yes—
89#1802C2212yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96#529—28no—
97#529—44no—
98C2×#126—22no—
99C2×#125C2312yes—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8C2312yes—
103C2×#121C2311yescharacteristic
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C2×#117—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#483—44no—
123#480—24no—
124#431—44no—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2311yescharacteristic
140C2×C22⋊C8C2311yescharacteristic
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—24no—
144#400—24no—
145#400—24no—
146#400—24no—
147#400—22no—
148#398—22no—
149#398—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#390C2311yescharacteristic
157#389—22no—
158#389—22no—
159#389—24no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389C2314yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#389C2311yescharacteristic
172#388—21no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#386—21no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—21no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#382—22no—
188#382—22no—
189#382—24no—
190#382—22no—
191#382—22no—
192#382C2314yes—
193#382C2314yes—
194#381—22no—
195#381—22no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380C2311yescharacteristic
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#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8C2412yes—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2411yescharacteristic
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250#126—48no—
251#125—24no—
252#125—42no—
253#124—24no—
254#124—42no—
255#123—48no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258#121—22no—
259#121#4114yes—
260C4×Q8—24no—
261C4×Q8—42no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—42no—
265C4×D4—48no—
266C4×D4—44no—
267#117—48no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—84no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—44no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—24no—
337#107—42no—
338#106—44no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—44no—
343#103—44no—
344#103—44no—
345#103—44no—
346#102—24no—
347#102—44no—
348#102—42no—
349C22⋊C8—22no—
350C22⋊C8—22no—
351C22⋊C8—24no—
352C4×C8—24no—
353C4×C8—44no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C24C22×D411yescharacteristic
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×D4—24no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4C2×#4112yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4#13514yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4112yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4C2×#4112yes—
486C22⋊C4—24no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—81no—
495C42—22no—
496C42—22no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—44no—
505C42—44no—
506C42—44no—
507C42—42no—
508C422−1+414yes—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23#54612yes—
521C23C2×#13512yes—
522C23#57712yes—
523C23C23×D411yescharacteristic
524Q8—44no—
525Q8—44no—
526D4—24no—
527D4—48no—
528D4—42no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—48no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4—84no—
582C2×C4—88no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#53612yes—
586C2×C4#53612yes—
587C2×C4#54612yes—
588C2×C4C2×#13512yes—
589C2×C4C2×#13512yes—
590C2×C4#57512yes—
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)11yescharacteristic
593C2×C4C2×2−1+412yes—
594C2×C4#58512yes—
595C8—42no—
596C8—42no—
597C8—44no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#330314yes—
612C22C2×#53611yescharacteristic
613C22C2×#54611yescharacteristic
614C22#348711yescharacteristic
615C22C22×#13511yescharacteristic
616C22C2×#57511yescharacteristic
617C22C2×#57711yescharacteristic
618C22#358611yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—82no—
635C4#323214yes—
636C4#351512yes—
637C4#355912yes—
638C2—24no—
639C2—82no—
640C2#3890711yescharacteristic
641C2C2×#323211yescharacteristic
642C2#6087211yescharacteristic
643C2C2×#330311yescharacteristic
644C2#6116511yescharacteristic
645C2#6188211yescharacteristic
646C2#6190411yescharacteristic
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×#2571C2semidirect
3. #33108C2semidirect
4. #31581C2semidirect
5. #31465C2semidirect
6. #29913C2semidirect
7. #28790C2semidirect
8. #27972C2semidirect
9. #27748C2semidirect
11. #26527C2semidirect
12. #26107C2semidirect
13. #24601C2semidirect
15. #22599C2semidirect

Maximal subgroups

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

subgroupindex
#361162
C2×#25712
#331082
#315812
#314652
#299132
#287902
#279722
#277482
#276782
#265272
#261072
#246012
#240662
#225992
#218612
C4×#4802