Finite Group Databasebeta

← #6461889

Subgroups — #6461889

Order 512.

1878 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34637C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic, complemented
3#29571C214yesmaximal, complemented
4#29569C214yesmaximal, complemented
5#29567C214yesmaximal, complemented
6#29565C214yesmaximal, complemented
7#23554C211yesmaximal, characteristic, complemented
8#23508C211yesmaximal, characteristic
9#23455C211yesmaximal, characteristic, complemented
10#23429C211yesmaximal, characteristic
11#21237C212yesmaximal, complemented
12#21093C212yesmaximal, complemented
13#21059C212yesmaximal, complemented
14#20926C212yesmaximal, complemented
15#20365C211yesmaximal, characteristic, complemented
16C22×#137C2212yes—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#417C2212yes—
24C2×#419C2212yes—
25C2×C4×M16C2212yes—
26#2642C2212yes—
27#2641C2212yes—
28#2640C2212yes—
29#2640C2212yes—
30#2639—21no—
31#2639—21no—
32#2639—21no—
33#2639—21no—
34#2398—22no—
35#2398—22no—
36#2320—22no—
37#2320—22no—
38#2320C2214yes—
39#2320C2214yes—
40#2297C2212yes—
41#2297C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47C2×#387—21no—
48C2×#383—21no—
49C2×#382—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#2059C2214yes—
53#2058C2212yes—
54#2057C2212yes—
55#2043C2212yes—
56#2042C2214yes—
57#2041C2212yes—
58#2026C2214yes—
59#2024C2212yes—
60#2024C2212yes—
61#2016C2212yes—
62#2015C2214yes—
63#2014C2212yes—
64#1973C2214yes—
65#1971C2214yes—
66#1969C2214yes—
67#1968C2214yes—
68#1936C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1933C2214yes—
72#1928C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1746C2214yes—
81#1746C2214yes—
82#1736C2212yes—
83#1736C2212yes—
84#1728C2211yescharacteristic
85#1728C2212yes—
86#1728C2211yescharacteristic
87C22×#41C2312yes—
88C23×Q8C2311yescharacteristic
89C2×#137—28no—
90C2×#137—28no—
91C2×#137—44no—
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C22×M16C2312yes—
96#533—44no—
97#532—44no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127C2311yescharacteristic
102C2×#125C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×#121—21no—
107C2×#121—21no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2314yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×#117C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#428—44no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131#419—44no—
132#417—28no—
133#417—28no—
134#417—44no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C4×M16—28no—
144C2×#97C2314yes—
145C2×C4×C8C2314yes—
146#400—22no—
147#400—22no—
148#400—22no—
149#400—22no—
150#397—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#394—22no—
155#394—22no—
156#394—22no—
157#394—22no—
158#393—22no—
159#393—22no—
160#393—22no—
161#393—22no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391C2314yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—21no—
177#389—21no—
178#389—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2314yes—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#383—24no—
191#383—24no—
192#382—24no—
193#382—24no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4C2314yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200#376D412yes—
201#376D412yes—
202#376C2312yes—
203#376C2312yes—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214C2×#95D411yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2314yes—
222#336C2314yes—
223#336C2312yes—
224#336C2312yes—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—24no—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8#4114yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256#137—432no—
257C2×Q16—28no—
258C2×Q16—28no—
259C2×Q16—44no—
260C2×SD16—28no—
261C2×SD16—28no—
262C2×SD16—44no—
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16#4118yes—
269C22×C8C2414yes—
270C22×C8C2414yes—
271C4⋊Q8—42no—
272C4⋊Q8—42no—
273C4⋊Q8—42no—
274#127—42no—
275#125—42no—
276#125—42no—
277#124—42no—
278#124—42no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—44no—
282#121—44no—
283C4×Q8—216no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—44no—
289C4×D4—44no—
290#117—24no—
291#117—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4—82no—
311C2×C4⋊C4#4114yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—82no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345#107—44no—
346#106—44no—
347#103—28no—
348#103—28no—
349#103—44no—
350#103—44no—
351#103—44no—
352#102—28no—
353#102—28no—
354#102—44no—
355#97—28no—
356C4×C8—28no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24C22×D411yescharacteristic
385#41—48no—
386#41—48no—
387#41—48no—
388#41—48no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—216no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—48no—
405C2×Q8—81no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8C2×#4114yes—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—81no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4C2511yescharacteristic
479Q16—416no—
480SD16—416no—
481M16—216no—
482M16—48no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8C2×#4118yes—
489C4⋊C4—216no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—84no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C4⋊C4C2×#4114yes—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C42—216no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—44no—
533C42—84no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23#52512yes—
546C23#53212yes—
547C23#53512yes—
548C23C23×D411yescharacteristic
549Q8—28no—
550Q8—48no—
551Q8—44no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—84no—
557Q8—84no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—84no—
563D4—84no—
564D4—88no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4#52512yes—
622C2×C4#53212yes—
623C2×C4#53512yes—
624C2×C4#55118yes—
625C2×C4#55518yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C22×#4112yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C8—28no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22#261014yes—
646C22C2×#52511yescharacteristic
647C22C2×#53211yescharacteristic
648C22C2×#53511yescharacteristic
649C22#358511yescharacteristic
650C22#358611yescharacteristic
651C22#358611yescharacteristic
652C22#358611yescharacteristic
653C4—42no—
654C4—42no—
655C4—44no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—44no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#261014yes—
669C4C2×#55112yes—
670C4C2×#55512yes—
671C2—24no—
672C2—162no—
673C2C2×#261011yescharacteristic
674C2#6056511yescharacteristic
675C2#6056611yescharacteristic
676C2#6070311yescharacteristic
677C2#6070511yescharacteristic
678C2#6080911yescharacteristic
679C2#6081011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2320C2semidirect
3. #29571C2semidirect
4. #29569C2semidirect
5. #29567C2semidirect
6. #29565C2semidirect
7. #23554C2semidirect
9. #23455C2semidirect
11. #21237C2semidirect
12. #21093C2semidirect
13. #21059C2semidirect
14. #20926C2semidirect
15. #20365C2semidirect

Maximal subgroups

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

subgroupindex
#346372
C2×#23202
#295712
#295692
#295672
#295652
#235542
#235082
#234552
#234292
#212372
#210932
#210592
#209262
#203652