Finite Group Databasebeta

← #6569355

Subgroups — #6569355

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30317C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#28979C214yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28571C212yesmaximal, complemented
8#28374C211yesmaximal, characteristic, complemented
9#27857C212yesmaximal, complemented
10#27787C214yesmaximal, complemented
11#25387C212yesmaximal
12#25341C212yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23755C212yesmaximal, complemented
15#23095C212yesmaximal, complemented
16#23062C212yesmaximal, complemented
17#20555C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#498C2212yes—
20C2×#475C2212yes—
21C2×#465C2214yes—
22C2×#460C2212yes—
23C2×C8×D4C2212yes—
24#2868C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2212yes—
27#2854C2212yes—
28#2602—22no—
29#2592—21no—
30#2571—21no—
31#2563—21no—
32#2557—22no—
33#2547C2211yescharacteristic
34#2544—21no—
35#2541C2211yescharacteristic
36#2538C2212yes—
37#2497—22no—
38#2492C2214yes—
39#2491C2212yes—
40#2439—22no—
41#2422C2212yes—
42#2369—21no—
43#2367—21no—
44#2366C2211yescharacteristic
45#2365C2211yescharacteristic
46#2355C2212yes—
47#2354—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#121C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2094C2212yes—
56#2084C2212yes—
57#2084C2212yes—
58#2077C2214yes—
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2043C2211yescharacteristic
63#2043C2211yescharacteristic
64#2035C2214yes—
65#2034C2214yes—
66#2024C2212yes—
67#2023C2214yes—
68#2013C2212yes—
69#2000C2214yes—
70#1990C2212yes—
71#1990C2212yes—
72#1968C2214yes—
73#1968C2214yes—
74#1957C2212yes—
75#1948C2212yes—
76#1948C2212yes—
77#1933C2214yes—
78#1927C2214yes—
79#1926C2214yes—
80#1919C2214yes—
81#1893C2214yes—
82#1892C2214yes—
83#1877C2214yes—
84#1807C2214yes—
85C4×#106C2214yes—
86C8×C4⋊C4C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#102C2214yes—
91C8×C22⋊C4C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C22×SD16C2314yes—
96C23×C8C2314yes—
97#559—48no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127—21no—
105C2×#127C2311yescharacteristic
106C2×#124—21no—
107C2×#124—22no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8C2314yes—
113C2×#121C2312yes—
114C2×#121C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C22×C42C2312yes—
126#498—44no—
127#475—28no—
128#465—48no—
129#460—28no—
130C8×D4—28no—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×C22⋊C8C2314yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#397—22no—
149#397—22no—
150#396—24no—
151#394—22no—
152#394—22no—
153#393—24no—
154#393—22no—
155#392—22no—
156#391—21no—
157#391—21no—
158#391—21no—
159#391—24no—
160#391C2312yes—
161#391C2314yes—
162#390—22no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390C2312yes—
168#390C2312yes—
169#390C2312yes—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2312yes—
179#388—24no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—24no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2412yes—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D411yescharacteristic
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8—28no—
252C22×C8—44no—
253C22×C8—44no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—41no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265#127—24no—
266#127—44no—
267#124—44no—
268#124—48no—
269#124—44no—
270#124—44no—
271#123—48no—
272#123—48no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275#121—24no—
276#121—42no—
277#121—44no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4SD1618yescomplemented
288#117—84no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
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—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#106—28no—
350#106—28no—
351#106—44no—
352C4⋊C8#4118yes—
353#103—28no—
354#103—24no—
355#103—42no—
356#103—42no—
357#103—42no—
358#102—28no—
359#102—44no—
360#102—44no—
361C22⋊C8—28no—
362C4×C8—24no—
363C4×C8—44no—
364C4×C8—42no—
365#95—21no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24—22no—
383C24C22×D412yes—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—81no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—81no—
413C2×D4—81no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4C2×SD1614yes—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4C2×SD1618yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2511yescharacteristic
449SD16—88no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—88no—
459C2×C8C2×#4118yes—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—81no—
487C4⋊C4—84no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4C2×SD1614yes—
491C4⋊C4H(5,𝔽2)12yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4C2×SD1618yes—
508C42—21no—
509C42—24no—
510C42—22no—
511C42—21no—
512C42—42no—
513C42—42no—
514C42—41no—
515C42—41no—
516C42—44no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—84no—
521C42C2×SD1614yes—
522C42#13514yes—
523C42C2×#4114yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23C22×SD1614yes—
538C23C23×D411yescharacteristic
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—84no—
544Q8—84no—
545Q8—164no—
546D4—28no—
547D4—44no—
548D4—44no—
549D4—88no—
550D4—84no—
551D4—162no—
552D4—164no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—84no—
599C2×C4—164no—
600C2×C4#53212yes—
601C2×C4#53212yes—
602C2×C4C22×SD1614yes—
603C2×C4C22×SD1612yes—
604C2×C4C22×SD1612yes—
605C2×C4C22×SD1612yes—
606C2×C4C2×#13512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4C22×#4112yes—
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4#58512yes—
612C2×C4#58511yescharacteristic
613C8—28no—
614C8—42no—
615C8—42no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22C2×#53211yescharacteristic
631C22#348511yescharacteristic
632C22#348811yescharacteristic
633C22C23×SD1611yescharacteristic
634C22C22×#13511yescharacteristic
635C22#358511yescharacteristic
636C22#358611yescharacteristic
637C4—22no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—84no—
649C4—82no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4#316714yes—
654C4#351412yes—
655C4#352412yes—
656C2—44no—
657C2—162no—
658C2#3605111yescharacteristic
659C2C2×#316711yescharacteristic
660C2#6072711yescharacteristic
661C2#6169711yescharacteristic
662C2#6170111yescharacteristic
663C2#6194811yescharacteristic
664C2#6195311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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
2. #36194C2semidirect
3. #30317C2semidirect
4. #30203C2semidirect
5. #28979C2semidirect
6. #28687C2semidirect
7. #28571C2semidirect
8. #28374C2semidirect
9. #27857C2semidirect
10. #27787C2semidirect
12. #25341C2semidirect
13. #23816C2semidirect
14. #23755C2semidirect
15. #23095C2semidirect
16. #23062C2semidirect
17. #20555C2semidirect
287. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#364692
#361942
#303172
#302032
#289792
#286872
#285712
#283742
#278572
#277872
#253872
#253412
#238162
#237552
#230952
#230622
#205552