Finite Group Databasebeta

← #6565050

Subgroups — #6565050

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34332C211yesmaximal, characteristic
3#30116C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29335C212yesmaximal, complemented
6#29219C211yesmaximal, characteristic, complemented
7#28206C211yesmaximal, characteristic, complemented
8#27610C212yesmaximal, complemented
9#27345C214yesmaximal, complemented
10#26743C214yesmaximal, complemented
11#24969C212yesmaximal
12#24122C212yesmaximal
13#23763C212yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23069C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic
17#22763C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#477C2212yes—
20C2×C4×Q16C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563—21no—
25#2538C2212yes—
26#2497—22no—
27#2492C2214yes—
28#2439—22no—
29#2370—21no—
30#2369—21no—
31#2369C2211yescharacteristic
32#2360—22no—
33#2360C2212yes—
34#2355C2212yes—
35#2354—21no—
36Q8×C4⋊C4C2211yescharacteristic
37D4×C4⋊C4—21no—
38D4×C4⋊C4C2211yescharacteristic
39#2345C2212yes—
40#2330—21no—
41#2329—22no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2295C2211yescharacteristic
46C2×#389C2212yes—
47#2285C2212yes—
48#2275—21no—
49C42×D4C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2099C2212yes—
53#2096C2212yes—
54#2083C2212yes—
55#2083C2211yescharacteristic
56#2077C2214yes—
57#2072C2211yescharacteristic
58#2068C2212yes—
59#2058C2212yes—
60#2056C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2036C2214yes—
64#2035C2214yes—
65#2024C2212yes—
66#2016C2211yescharacteristic
67#2001C2214yes—
68#1990C2212yes—
69#1988C2214yes—
70#1954C2212yes—
71#1948C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1924C2214yes—
77#1918C2214yes—
78#1918C2214yes—
79#1878C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1852C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1823C2214yes—
86#1817C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×Q16C2312yes—
96#554—24no—
97#554—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C42C2312yes—
121#477—28no—
122C4×Q16—28no—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139#400—24no—
140#396—24no—
141#393—24no—
142#391—21no—
143#391—22no—
144#391—24no—
145#391C2312yes—
146#391C2311yescharacteristic
147#391C2311yescharacteristic
148#391C2314yes—
149#390—22no—
150#389—22no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2311yescharacteristic
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2312yes—
167#388—24no—
168#387—21no—
169#387—22no—
170#387—21no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2314yes—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376—21no—
216C2×#95—22no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C22×D4C2×D411yescharacteristic
237C23×C4—22no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C2×Q16—24no—
243C2×Q16—42no—
244C2×Q16—42no—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—44no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253C4⋊Q8#4114yes—
254#125—28no—
255C22⋊Q8—216no—
256C22⋊Q8—24no—
257C22⋊Q8—24no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260C4×Q8—22no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—44no—
264C4×Q8—82no—
265C4×Q8#4114yes—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—44no—
272#117—84no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—84no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—44no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—24no—
337#107—28no—
338#106—44no—
339C4⋊C8—24no—
340C4⋊C8—42no—
341#103—24no—
342#103—24no—
343#103—28no—
344#103—42no—
345#103—42no—
346#103—44no—
347#103—44no—
348C22⋊C8—44no—
349C22⋊C8—44no—
350#97—44no—
351C4×C8—24no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—24no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24C22×D412yes—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8C2×#4112yes—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4#13714yes—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429Q16—48no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—84no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4C2×#4112yes—
476C4⋊C4C2×#4114yes—
477C22⋊C4—28no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—42no—
491C42—42no—
492C42—41no—
493C42—44no—
494C42—42no—
495C42—82no—
496C42—84no—
497C42—82no—
498C42—82no—
499C42#13714yes—
500C42C2×#4112yes—
501C42C2×#4112yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23C2×#13714yes—
509C23C23×D411yescharacteristic
510Q8—24no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—42no—
516Q8—84no—
517Q8—164no—
518D4—28no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—84no—
570C2×C4—88no—
571C2×C4—164no—
572C2×C4#52512yes—
573C2×C4#52512yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4#57612yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4#58512yes—
582C2×C4#58512yes—
583C8—42no—
584C8—42no—
585C8—44no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C2×#52511yescharacteristic
594C22#348511yescharacteristic
595C22#350011yescharacteristic
596C22C22×#13711yescharacteristic
597C22C22×#13711yescharacteristic
598C22C2×#57611yescharacteristic
599C22#358611yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—44no—
611C4—82no—
612C4—81no—
613C4—84no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4#311314yes—
618C4#349312yes—
619C4#352412yes—
620C2—44no—
621C2#3601111yescharacteristic
622C2C2×#311311yescharacteristic
623C2#6059611yescharacteristic
624C2#6172311yescharacteristic
625C2#6172611yescharacteristic
626C2#6253711yescharacteristic
627C2#6257711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30116C2semidirect
4. #29983C2semidirect
5. #29335C2semidirect
6. #29219C2semidirect
7. #28206C2semidirect
8. #27610C2semidirect
9. #27345C2semidirect
10. #26743C2semidirect
13. #23763C2semidirect
14. #23679C2semidirect
15. #23069C2semidirect
17. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343322
#301162
#299832
#293352
#292192
#282062
#276102
#273452
#267432
#249692
#241222
#237632
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