Finite Group Databasebeta

← #6495327

Subgroups — #6495327

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33305C211yesmaximal, characteristic, complemented
4#33297C211yesmaximal, characteristic, complemented
5#33232C211yesmaximal, characteristic, complemented
6#33230C211yesmaximal, characteristic, complemented
7#29808C212yesmaximal, complemented
8#29807C212yesmaximal
9#28620C214yesmaximal, complemented
10#28619C214yesmaximal, complemented
11#25604C214yesmaximal, complemented
12#25603C214yesmaximal, complemented
13#22270C212yesmaximal, complemented
14#22268C212yesmaximal, complemented
15#20463C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#529C2212yes—
18C2×#497C2214yes—
19C2×#497C2214yes—
20C2×#494C2212yes—
21C2×#492C2212yes—
22C2×#491C2214yes—
23C2×#489C2214yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2862C2212yes—
28#2861C2212yes—
29#2860—21no—
30#2860—21no—
31#2859—21no—
32#2859—21no—
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857—21no—
36#2857—21no—
37#2857C2212yes—
38#2857C2211yescharacteristic
39#2857C2212yes—
40#2856C2211yescharacteristic
41#2854—21no—
42#2854—21no—
43#2850C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2212yes—
49C4×#127C2211yescharacteristic
50C4×#124C2212yes—
51C4×#121—21no—
52C4×#121—21no—
53#2293—21no—
54#2293—21no—
55#2285—21no—
56#2285—21no—
57C2×C43C2211yescharacteristic
58#2056C2218yes—
59#2055C2218yes—
60#2001C2218yes—
61#2000C2218yes—
62#1993C2214yes—
63#1992C2214yes—
64#1960C2214yes—
65#1959C2214yes—
66#1957C2212yes—
67#1956C2212yes—
68#1956C2212yes—
69#1956C2212yes—
70#1880C2218yes—
71#1878C2218yes—
72#1878C2218yes—
73#1877C2218yes—
74#1834C2212yes—
75#1834C2212yes—
76#1767C2212yes—
77#1767C2212yes—
78#1749C2212yes—
79#1749C2212yes—
80#1744C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C22×Q16C2318yes—
84C22×SD16C2318yes—
85#533—48no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#127C2311yescharacteristic
94#529—48no—
95C2×#126—22no—
96C2×#125—22no—
97C2×#125C2312yes—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2314yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—22no—
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C2×#117—22no—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#497—48no—
124#497—48no—
125#494—28no—
126#492—28no—
127#491—48no—
128#489—48no—
129#431—28no—
130C2×C4⋊C8C2314yes—
131C2×#103C2318yes—
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×#102C2318yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#391—22no—
143#391—22no—
144#391C2314yes—
145#391C2312yes—
146#391C2314yes—
147#391C2312yes—
148#390—21no—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390C2312yes—
157#390C2312yes—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#386—22no—
183#386—22no—
184#384—21no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382—22no—
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43D411yescharacteristic
214C43D411yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218C43C2311yescharacteristic
219#340C2318yes—
220#340C2318yes—
221#336C2314yes—
222#336C2314yes—
223C22×Q8—22no—
224C22×Q8C2412yes—
225C22×Q8C2414yes—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C23×C4—22no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×Q16—216no—
239C2×Q16—48no—
240C2×Q16—48no—
241C2×SD16—216no—
242C2×SD16—48no—
243C2×SD16—48no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—28no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254#127—24no—
255#126—48no—
256#125—28no—
257#125—48no—
258#124—28no—
259#124—28no—
260#123—48no—
261C22⋊Q8—44no—
262C22⋊Q8—48no—
263C22⋊Q8—48no—
264#121—44no—
265#121—44no—
266#121—44no—
267C4×Q8—44no—
268C4×Q8—48no—
269C4×Q8—48no—
270C4×D4—44no—
271C4×D4—44no—
272C4×D4—48no—
273C4×D4—44no—
274#117—48no—
275#117—48no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C4⋊C8—28no—
352#103—48no—
353#103—48no—
354#103—48no—
355#102—48no—
356C22⋊C8—28no—
357#97—24no—
358#97—24no—
359#97—24no—
360#97—24no—
361C4×C8—24no—
362C4×C8#4118yes—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373C24—22no—
374C24C22×D411yescharacteristic
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—28no—
380C2×Q8—24no—
381C2×Q8—28no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4#13714yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C2511yescharacteristic
440Q16—816no—
441SD16—816no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8C2×#4118yes—
448C4⋊C4—28no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—44no—
511C42—48no—
512C42#13714yes—
513C42C2×#4114yes—
514C42H(5,𝔽2)14yes—
515C42H(5,𝔽2)14yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23C2×#13712yes—
530C23C23×D411yescharacteristic
531Q8—48no—
532Q8—48no—
533Q8—48no—
534Q8—88no—
535D4—48no—
536D4—88no—
537D4—84no—
538D4—84no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—88no—
588C2×C4#53212yes—
589C2×C4#53212yes—
590C2×C4#53512yes—
591C2×C4#53512yes—
592C2×C4#53512yes—
593C2×C4#53512yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57512yes—
600C2×C4#57512yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4C2×H(5,𝔽2)12yes—
604C8—28no—
605C8—44no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×#53211yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×#53511yescharacteristic
623C22C22×#13711yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57511yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—84no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4#317214yes—
644C4#318114yes—
645C4#350814yes—
646C2—42no—
647C2—162no—
648C2#3529311yescharacteristic
649C2C2×#317211yescharacteristic
650C2C2×#318111yescharacteristic
651C2#6080011yescharacteristic
652C2#6080111yescharacteristic
653C2#6080411yescharacteristic
654C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33305C2semidirect
4. #33297C2semidirect
5. #33232C2semidirect
6. #33230C2semidirect
7. #29808C2semidirect
9. #28620C2semidirect
10. #28619C2semidirect
11. #25604C2semidirect
12. #25603C2semidirect
13. #22270C2semidirect
14. #22268C2semidirect
15. #20463C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366312
#333052
#332972
#332322
#332302
#298082
#298072
#286202
#286192
#256042
#256032
#222702
#222682
#204632