Finite Group Databasebeta

← #6573020

Subgroups — #6573020

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36288C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30060C212yesmaximal, complemented
4#29663C211yesmaximal, characteristic
5#29366C212yesmaximal, complemented
6#28370C212yesmaximal, complemented
7#27905C211yesmaximal, characteristic, complemented
8#27680C211yesmaximal, characteristic, complemented
9#26928C214yesmaximal, complemented
10#26786C214yesmaximal, complemented
11#26044C212yesmaximal, complemented
12#24730C212yesmaximal, complemented
13#23776C212yesmaximal, complemented
14#23570C211yesmaximal, characteristic
15#23094C212yesmaximal, complemented
16#22805C212yesmaximal, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520—21no—
20C2×#483C2214yes—
21C2×#476C2214yes—
22C2×#461C2212yes—
23C2×#461C2214yes—
24C2×C4×SD16C2212yes—
25#2862C2212yes—
26#2857C2211yescharacteristic
27#2612—22no—
28#2574—22no—
29#2551C2211yescharacteristic
30#2546C2211yescharacteristic
31#2508—21no—
32#2492C2212yes—
33#2439—21no—
34#2429C2212yes—
35#2369—21no—
36#2365C2211yescharacteristic
37#2361—22no—
38#2361C2214yes—
39#2355C2211yescharacteristic
40#2354—21no—
41#2354C2212yes—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44#2346—21no—
45#2326—22no—
46#2311—21no—
47C4×C4⋊Q8C2212yes—
48C4×#127C2212yes—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C2×#384C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2099C2212yes—
54#2096C2212yes—
55#2096C2212yes—
56#2096C2212yes—
57#2088C2214yes—
58#2066C2214yes—
59#2058C2211yescharacteristic
60#2058C2214yes—
61#2043C2214yes—
62#2041C2211yescharacteristic
63#2034C2212yes—
64#2024C2211yescharacteristic
65#2024C2212yes—
66#2014C2211yescharacteristic
67#2014C2212yes—
68#2000C2212yes—
69#2000C2214yes—
70#1968C2214yes—
71#1968C2214yes—
72#1956C2212yes—
73#1948C2212yes—
74#1939C2212yes—
75#1927C2214yes—
76#1927C2214yes—
77#1926C2214yes—
78#1923C2214yes—
79#1880C2214yes—
80#1877C2214yes—
81#1872C2212yes—
82#1864C2214yes—
83#1853C2212yes—
84#1844C2212yes—
85#1839C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2211yescharacteristic
89C4×#106C2214yes—
90C8×C4⋊C4C2212yes—
91C4×C4⋊C8C2212yes—
92C4×#103C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C22×#41—21no—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#532—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#124—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121C2312yes—
110#520—48no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×D4—24no—
115C2×C4×D4—21no—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2314yes—
120C2×#117—22no—
121C2×#117D411yescharacteristic
122C2×#117D411yescharacteristic
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128#483—48no—
129#476—48no—
130#461—28no—
131#461—48no—
132C4×SD16—28no—
133C2×#106C2312yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#400—22no—
153#398—22no—
154#398—22no—
155#396—22no—
156#395—22no—
157#393—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2311yescharacteristic
165#391C2312yes—
166#391C2312yes—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389C2314yes—
179#389C2311yescharacteristic
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—24no—
184#388—24no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#386—24no—
191#385—21no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#382—21no—
200#382—22no—
201#382—22no—
202#380—21no—
203#380—21no—
204#380—22no—
205#380—22no—
206#380C2311yescharacteristic
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376—22no—
223#376D412yes—
224C43C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—28no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8C2411yescharacteristic
241C22×Q8C2412yes—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—24no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—24no—
253C2×SD16—28no—
254C2×SD16—24no—
255C2×SD16—42no—
256C2×SD16—44no—
257C2×SD16—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266#127—24no—
267#124—28no—
268C22⋊Q8—22no—
269C22⋊Q8—41no—
270C22⋊Q8—44no—
271#121—28no—
272#121—24no—
273#121—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—48no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—416no—
283C4×D4—48no—
284C4×D4—84no—
285C4×D4#4118yes—
286#117—42no—
287#117—48no—
288#117SD1618yescomplemented
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2412yes—
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#106—28no—
350#106—24no—
351#106—44no—
352#106—44no—
353C4⋊C8—28no—
354C4⋊C8—24no—
355#103—28no—
356#103—24no—
357#103—28no—
358#103—44no—
359#103—44no—
360#102—24no—
361#102—24no—
362#102—24no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—42no—
366C4×C8—24no—
367C4×C8—44no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—22no—
374#95—24no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—22no—
390C24C22×D411yescharacteristic
391#41—432no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4C2×#4114yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—41no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C2×SD1614yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2511yescharacteristic
458SD16—48no—
459SD16—88no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8H(5,𝔽2)14yes—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4C2×SD1614yes—
499C4⋊C4C2×SD1614yes—
500C4⋊C4C2×#4114yes—
501C4⋊C42−1+412yes—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C22⋊C4C2×SD1618yes—
512C42—22no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—42no—
523C42—48no—
524C42—42no—
525C42—82no—
526C42C2×SD1618yes—
527C42C2×#4114yes—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—24no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23C22×SD1612yes—
542C23C23×D411yescharacteristic
543Q8—48no—
544Q8—42no—
545Q8—44no—
546Q8—42no—
547Q8—44no—
548Q8—164no—
549D4—24no—
550D4—24no—
551D4—48no—
552D4—48no—
553D4—416no—
554D4—42no—
555D4—42no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—168no—
605C2×C4—164no—
606C2×C4#52512yes—
607C2×C4#52512yes—
608C2×C4C22×SD1614yes—
609C2×C4C22×SD1612yes—
610C2×C4C22×SD1614yes—
611C2×C4C22×SD1612yes—
612C2×C4#57512yes—
613C2×C4#57712yes—
614C2×C4C22×#4112yes—
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4C2×2−1+411yescharacteristic
617C2×C4#58511yescharacteristic
618C8—24no—
619C8—44no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C2×#52511yescharacteristic
636C22#348911yescharacteristic
637C22#349811yescharacteristic
638C22C23×SD1611yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57711yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—24no—
644C4—22no—
645C4—22no—
646C4—24no—
647C4—44no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—81no—
653C4—84no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4#310114yes—
659C4#349412yes—
660C4#351112yes—
661C2—42no—
662C2—84no—
663C2#3608711yescharacteristic
664C2C2×#310111yescharacteristic
665C2#6058511yescharacteristic
666C2#6209911yescharacteristic
667C2#6213411yescharacteristic
668C2#6241311yescharacteristic
669C2#6245611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36288C2semidirect
3. #30060C2semidirect
5. #29366C2semidirect
6. #28370C2semidirect
7. #27905C2semidirect
8. #27680C2semidirect
9. #26928C2semidirect
10. #26786C2semidirect
11. #26044C2semidirect
12. #24730C2semidirect
13. #23776C2semidirect
15. #23094C2semidirect
16. #22805C2semidirect
17. #22639C2semidirect
288. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#362882
#342072
#300602
#296632
#293662
#283702
#279052
#276802
#269282
#267862
#260442
#247302
#237762
#235702
#230942
#228052
#226392