Finite Group Databasebeta

← #6570141

Subgroups — #6570141

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36477C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#30319C212yesmaximal, complemented
4#30305C211yesmaximal, characteristic, complemented
5#29121C211yesmaximal, characteristic, complemented
6#28909C214yesmaximal, complemented
7#28717C211yesmaximal, characteristic, complemented
8#28574C212yesmaximal, complemented
9#28223C212yesmaximal, complemented
10#28141C212yesmaximal, complemented
11#28062C214yesmaximal, complemented
12#27855C212yesmaximal, complemented
13#23799C211yesmaximal, characteristic, complemented
14#23756C212yesmaximal, complemented
15#22805C212yesmaximal, complemented
16#22767C212yesmaximal, complemented
17#20555C211yesmaximal, characteristic, complemented
18C2×#555—21no—
19C2×#487C2212yes—
20C2×#465C2214yes—
21C2×C4×SD16C2212yes—
22C2×C4×SD16C2212yes—
23C2×C8×D4C2212yes—
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2856C2212yes—
27#2855C2212yes—
28#2608—22no—
29#2587—22no—
30#2570—21no—
31#2566—21no—
32#2563—21no—
33#2538C2211yescharacteristic
34#2538C2211yescharacteristic
35#2537—21no—
36#2537C2212yes—
37#2500—22no—
38#2497C2212yes—
39#2491—22no—
40#2491C2214yes—
41#2429C2212yes—
42#2370—21no—
43#2357—21no—
44#2355C2211yescharacteristic
45#2354—21no—
46#2354C2212yes—
47Q8×C4⋊C4C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53#2099C2212yes—
54#2096C2211yescharacteristic
55#2096C2211yescharacteristic
56#2094C2212yes—
57#2084C2212yes—
58#2084C2212yes—
59#2077C2214yes—
60#2072C2212yes—
61#2072C2212yes—
62#2058C2214yes—
63#2041C2212yes—
64#2035C2214yes—
65#2034C2214yes—
66#2014C2211yescharacteristic
67#2014C2211yescharacteristic
68#2013C2212yes—
69#2000C2214yes—
70#1990C2212yes—
71#1968C2214yes—
72#1934C2214yes—
73#1934C2214yes—
74#1927C2214yes—
75#1926C2214yes—
76#1920C2214yes—
77#1893C2214yes—
78#1892C2214yes—
79#1880C2214yes—
80#1844C2212yes—
81#1834C2212yes—
82#1807C2214yes—
83#1803C2212yes—
84#1803C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87C4×#106C2214yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2214yes—
91C8×C22⋊C4C2214yes—
92C4×C22⋊C8C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C22×SD16C2312yes—
96C22×SD16C2312yes—
97C23×C8C2314yes—
98#555—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#125—21no—
105C2×#124—24no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2314yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C42C2312yes—
124#487—44no—
125#465—48no—
126C4×SD16—28no—
127C4×SD16—28no—
128C8×D4—28no—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2314yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#398—22no—
148#396—22no—
149#396—22no—
150#396—22no—
151#396—22no—
152#394—24no—
153#394—24no—
154#393—24no—
155#391C2311yescharacteristic
156#391C2311yescharacteristic
157#391C2312yes—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—22no—
163#390—22no—
164#390C2312yes—
165#390C2314yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—24no—
180#388—24no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—21no—
198#382—22no—
199#382—21no—
200#381—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—22no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2314yes—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×D4—24no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2412yes—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×SD16—24no—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16—42no—
250C2×SD16—42no—
251C2×SD16—42no—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8C2414yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—41no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264#127—42no—
265#125—44no—
266#124—416no—
267C22⋊Q8—24no—
268C22⋊Q8—42no—
269C22⋊Q8—44no—
270#121—48no—
271#121—48no—
272#121—44no—
273#121—44no—
274#121—84no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4SD1618yescomplemented
285#117—48no—
286#117—48no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
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×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
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—24no—
350#106—24no—
351#106—44no—
352#106—44no—
353C4⋊C8—24no—
354#103—28no—
355#103—24no—
356#103—24no—
357#103—44no—
358#102—24no—
359#102—24no—
360#102—44no—
361#102—44no—
362C22⋊C8—28no—
363C4×C8—44no—
364C4×C8—42no—
365C4×C8#4118yes—
366#95—21no—
367#95—22no—
368#95—21no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24C22×D412yes—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q8H(5,𝔽2)12yes—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—48no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4C2×SD1614yes—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4C2×SD1618yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445SD16—48no—
446SD16—48no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—88no—
455C2×C8C2×#4114yes—
456C2×C8C2×#4114yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4C2×SD1614yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4C2×SD1618yes—
506C42—21no—
507C42—24no—
508C42—22no—
509C42—21no—
510C42—42no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42—42no—
515C42—42no—
516C42—48no—
517C42—82no—
518C42—82no—
519C42C2×SD1614yes—
520C42#13514yes—
521C42C2×#4114yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C22×SD1614yes—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—164no—
542D4—28no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—84no—
548D4—84no—
549D4—168no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—88no—
597C2×C4#53212yes—
598C2×C4#53212yes—
599C2×C4C22×SD1614yes—
600C2×C4C22×SD1612yes—
601C2×C4C22×SD1612yes—
602C2×C4C22×SD1612yes—
603C2×C4C2×#13512yes—
604C2×C4C2×#13512yes—
605C2×C4C2×#13512yes—
606C2×C4C22×#4112yes—
607C2×C4C2×H(5,𝔽2)11yescharacteristic
608C2×C4#58511yescharacteristic
609C2×C4#58512yes—
610C8—24no—
611C8—24no—
612C8—44no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22C2×#53211yescharacteristic
628C22#348511yescharacteristic
629C22#348811yescharacteristic
630C22C23×SD1611yescharacteristic
631C22C22×#13511yescharacteristic
632C22#358511yescharacteristic
633C22#358611yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—82no—
644C4—82no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—82no—
649C4—84no—
650C4#316714yes—
651C4#349312yes—
652C4#349412yes—
653C2—44no—
654C2—162no—
655C2#3606211yescharacteristic
656C2C2×#316711yescharacteristic
657C2#6072111yescharacteristic
658C2#6169511yescharacteristic
659C2#6170011yescharacteristic
660C2#6195111yescharacteristic
661C2#6195411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
1. #36477C2semidirect
3. #30319C2semidirect
4. #30305C2semidirect
5. #29121C2semidirect
6. #28909C2semidirect
7. #28717C2semidirect
8. #28574C2semidirect
9. #28223C2semidirect
10. #28141C2semidirect
11. #28062C2semidirect
12. #27855C2semidirect
13. #23799C2semidirect
14. #23756C2semidirect
15. #22805C2semidirect
16. #22767C2semidirect
17. #20555C2semidirect
284. C4×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#364772
#361872
#303192
#303052
#291212
#289092
#287172
#285742
#282232
#281412
#280622
#278552
#237992
#237562
#228052
#227672
#205552