Finite Group Databasebeta

← #6986479

Subgroups — #6986479

Order 512.

1517 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36460C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic, complemented
3#32840C212yesmaximal, complemented
4#31941C214yesmaximal, complemented
5#31845C214yesmaximal, complemented
6#30339C212yesmaximal, complemented
7#30046C212yesmaximal, complemented
8#29661C211yesmaximal, characteristic
9#29142C212yesmaximal, complemented
10#28402C212yesmaximal, complemented
11#27730C212yesmaximal, complemented
12#25903C211yesmaximal, characteristic
13#25325C212yesmaximal, complemented
14#24818C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic, complemented
16#23571C211yesmaximal, characteristic, complemented
17#22639C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#486C2212yes—
21C2×#475C2212yes—
22C2×#460C2212yes—
23C2×C4×Q16C2212yes—
24#2862C2212yes—
25#2862C2212yes—
26#2860C2212yes—
27#2856C2211yescharacteristic
28#2642—22no—
29#2571C2211yescharacteristic
30#2568—21no—
31#2568—21no—
32#2568C2212yes—
33#2568C2212yes—
34#2559—22no—
35#2548C2211yescharacteristic
36#2526—22no—
37#2512—21no—
38#2503—22no—
39#2503C2214yes—
40#2500—21no—
41#2426C2212yes—
42#2420C2212yes—
43#2366C2211yescharacteristic
44#2363—21no—
45#2357—21no—
46Q8×C4⋊C4C2211yescharacteristic
47#2310—21no—
48#2310—21no—
49C4×#124C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2092C2214yes—
54#2085C2214yes—
55#2074C2214yes—
56#2073C2214yes—
57#2066C2214yes—
58#2058C2212yes—
59#2057C2211yescharacteristic
60#2055C2214yes—
61#2043C2211yescharacteristic
62#2036C2212yes—
63#2024C2211yescharacteristic
64#2024C2212yes—
65#2023C2212yes—
66#2023C2212yes—
67#2023C2212yes—
68#2016C2211yescharacteristic
69#2015C2214yes—
70#2013C2212yes—
71#2001C2212yes—
72#2000C2214yes—
73#1990C2212yes—
74#1988C2212yes—
75#1968C2214yes—
76#1958C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1923C2214yes—
80#1878C2214yes—
81#1877C2214yes—
82#1874C2214yes—
83#1872C2212yes—
84#1853C2212yes—
85#1839C2212yes—
86#1835C2214yes—
87#1805C2214yes—
88#1804C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2312yes—
97C22×SD16C2314yes—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#127C2311yescharacteristic
107C2×#126—22no—
108C2×#125—21no—
109C2×#124C2312yes—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×#121—21no—
114C2×#121—21no—
115C2×#121C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C2×#117D411yescharacteristic
124C2×#117D411yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128#486—44no—
129#475—44no—
130#460—44no—
131C4×Q16—44no—
132C2×#107C2312yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#400—24no—
150#400—24no—
151#400—24no—
152#400—24no—
153#400—22no—
154#397—22no—
155#396—22no—
156#394—22no—
157#394—22no—
158#394—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391C2314yes—
170#391C2314yes—
171#391C2311yescharacteristic
172#390—21no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—22no—
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#387—21no—
193#386—22no—
194#386—22no—
195#385—21no—
196#384—21no—
197#384C2311yescharacteristic
198#382—21no—
199#382—22no—
200#382—22no—
201#382C2314yes—
202#382C2314yes—
203#382C2312yes—
204#381—24no—
205#381—24no—
206#381—22no—
207#380C2311yescharacteristic
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232#336C2314yes—
233#336C2311yescharacteristic
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8C2412yes—
241C22×Q8C2411yescharacteristic
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×Q16—24no—
252C2×Q16—42no—
253C2×Q16—42no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—44no—
266C4⋊Q8—44no—
267#127—24no—
268#126—84no—
269#125—44no—
270#124—24no—
271#124—42no—
272C22⋊Q8—22no—
273C22⋊Q8—41no—
274C22⋊Q8—42no—
275C22⋊Q8—42no—
276C22⋊Q8—44no—
277#121—44no—
278#121—44no—
279#121—82no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×D4—44no—
286C4×D4—44no—
287#117—24no—
288#117—42no—
289#117—44no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
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—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2412yes—
345C2×C42C2412yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—42no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—42no—
353#103—42no—
354#103—44no—
355#103—42no—
356#103—42no—
357#103—44no—
358#102—44no—
359#102—44no—
360#102—44no—
361C22⋊C8—42no—
362C22⋊C8—42no—
363C4×C8—24no—
364C4×C8—44no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—21no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—22no—
387C24C22×D411yescharacteristic
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—816no—
407C2×Q82−1+412yes—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4—82no—
420C2×D4—82no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4#13514yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455Q16—48no—
456SD16—88no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—81no—
510C42—22no—
511C42—22no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—42no—
519C42—42no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—82no—
524C42H(5,𝔽2)14yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C2×#13512yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—84no—
547Q8—88no—
548Q8—88no—
549D4—44no—
550D4—44no—
551D4—84no—
552D4—162no—
553D4—164no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—88no—
604C2×C4—164no—
605C2×C4#53512yes—
606C2×C4#53512yes—
607C2×C4C2×#13512yes—
608C2×C4C2×#13512yes—
609C2×C4#57512yes—
610C2×C4#57712yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×2−1+411yescharacteristic
613C2×C4#58512yes—
614C2×C4#58511yescharacteristic
615C8—42no—
616C8—42no—
617C8—44no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22C2×#53511yescharacteristic
633C22#348911yescharacteristic
634C22#350011yescharacteristic
635C22C22×#13511yescharacteristic
636C22C2×#57511yescharacteristic
637C22C2×#57711yescharacteristic
638C22#358611yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—82no—
649C4—84no—
650C4—81no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4#321114yes—
656C4#354112yes—
657C4#355112yes—
658C2—42no—
659C2—162no—
660C2#4291011yescharacteristic
661C2C2×#321111yescharacteristic
662C2#6081311yescharacteristic
663C2#6210111yescharacteristic
664C2#6213511yescharacteristic
665C2#6253111yescharacteristic
666C2#6259311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. #35320C2semidirect
3. #32840C2semidirect
4. #31941C2semidirect
5. #31845C2semidirect
6. #30339C2semidirect
7. #30046C2semidirect
9. #29142C2semidirect
10. #28402C2semidirect
11. #27730C2semidirect
13. #25325C2semidirect
14. #24818C2semidirect
15. #24301C2semidirect
16. #23571C2semidirect
17. #22639C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#353202
#328402
#319412
#318452
#303392
#300462
#296612
#291422
#284022
#277302
#259032
#253252
#248182
#243012
#235712
#226392