Finite Group Databasebeta

← #6535441

Subgroups — #6535441

Order 512.

1613 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#36460C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#30129C212yesmaximal, complemented
4#29661C211yesmaximal, characteristic
5#29521C212yesmaximal, complemented
6#29260C212yesmaximal, complemented
7#27906C211yesmaximal, characteristic
8#27681C211yesmaximal, characteristic, complemented
9#26637C214yesmaximal, complemented
10#26567C214yesmaximal, complemented
11#26089C212yesmaximal, complemented
12#24633C212yesmaximal, complemented
13#23807C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic, complemented
15#23003C212yesmaximal, complemented
16#22674C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#481C2214yes—
20C2×#460C2212yes—
21C2×#441C2214yes—
22C2×#440C2214yes—
23C2×C4×Q16C2212yes—
24#2860C2212yes—
25#2856C2211yescharacteristic
26#2571C2211yescharacteristic
27#2559—22no—
28#2548C2211yescharacteristic
29#2526—22no—
30#2512—21no—
31#2500—21no—
32#2426C2212yes—
33#2420C2212yes—
34#2366C2211yescharacteristic
35#2363—21no—
36#2360—22no—
37#2360C2214yes—
38#2357—21no—
39#2355—21no—
40#2355C2212yes—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2211yescharacteristic
43Q8×C4⋊C4C2212yes—
44#2330—22no—
45#2317—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2274—21no—
52C4×#117C2211yescharacteristic
53#2100C2212yes—
54#2097C2214yes—
55#2095C2212yes—
56#2094C2212yes—
57#2094C2212yes—
58#2066C2214yes—
59#2057C2211yescharacteristic
60#2055C2214yes—
61#2043C2211yescharacteristic
62#2036C2212yes—
63#2024C2211yescharacteristic
64#2023C2212yes—
65#2016C2211yescharacteristic
66#2015C2214yes—
67#2013C2212yes—
68#2001C2212yes—
69#2000C2214yes—
70#1983C2214yes—
71#1957C2212yes—
72#1949C2214yes—
73#1947C2212yes—
74#1943C2214yes—
75#1939C2212yes—
76#1936C2214yes—
77#1925C2214yes—
78#1924C2214yes—
79#1920C2214yes—
80#1880C2214yes—
81#1872C2212yes—
82#1865C2214yes—
83#1865C2214yes—
84#1862C2214yes—
85#1852C2212yes—
86#1844C2212yes—
87#1831C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C22×Q16C2312yes—
96C22×SD16C2314yes—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#126—22no—
102C2×#125—21no—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117D411yescharacteristic
120C2×#117D411yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#481—48no—
126#460—28no—
127#441—48no—
128#440—48no—
129C4×Q16—28no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#397—22no—
151#396—22no—
152#394—22no—
153#394—22no—
154#394—22no—
155#391—22no—
156#391C2314yes—
157#391C2311yescharacteristic
158#390—21no—
159#390—21no—
160#390—21no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389—22no—
174#389C2314yes—
175#389C2311yescharacteristic
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—21no—
185#387—24no—
186#387—24no—
187#386—22no—
188#386—22no—
189#385—21no—
190#385—24no—
191#384—22no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2312yes—
198#382—21no—
199#382—22no—
200#382C2312yes—
201#381—22no—
202#380C2311yescharacteristic
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4C2312yes—
222#376D412yes—
223C43C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8C2412yes—
236C22×Q8C2411yescharacteristic
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2412yes—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×Q16—24no—
247C2×Q16—24no—
248C2×Q16—24no—
249C2×SD16—28no—
250C2×SD16—44no—
251C2×SD16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257#127—24no—
258#126—84no—
259#125—28no—
260#124—24no—
261#124—42no—
262C22⋊Q8—28no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—22no—
266C22⋊Q8—41no—
267#121—44no—
268#121—44no—
269#121—82no—
270C4×Q8—28no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—84no—
277C4×Q8#4118yes—
278C4×D4—44no—
279C4×D4—44no—
280#117—28no—
281#117—24no—
282#117—42no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—24no—
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—24no—
349#107—44no—
350#106—28no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—24no—
355#103—24no—
356#103—24no—
357#103—42no—
358#103—44no—
359#103—44no—
360#102—28no—
361#102—28no—
362#102—44no—
363C22⋊C8—24no—
364C22⋊C8—42no—
365#97—28no—
366C4×C8—24no—
367#95—21no—
368#95—21no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—21no—
379#95—22no—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384C24—22no—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8C2×#4114yes—
398C2×Q82−1+412yes—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—82no—
411C2×D4—82no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446Q16—216no—
447SD16—416no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8H(5,𝔽2)14yes—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—81no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×#4114yes—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—81no—
505C42—22no—
506C42—22no—
507C42—28no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—84no—
519C42—82no—
520C42—84no—
521C42C2×#4114yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C2×#13712yes—
536C23C23×D411yescharacteristic
537Q8—24no—
538Q8—24no—
539Q8—24no—
540Q8—48no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544D4—44no—
545D4—44no—
546D4—84no—
547D4—162no—
548D4—164no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—164no—
599C2×C4—164no—
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4#57512yes—
605C2×C4#57712yes—
606C2×C4C22×#4112yes—
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4C2×2−1+411yescharacteristic
609C2×C4#58511yescharacteristic
610C8—24no—
611C8—42no—
612C8—44no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22C2×#52511yescharacteristic
628C22#348911yescharacteristic
629C22#349811yescharacteristic
630C22C22×#13711yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57711yescharacteristic
633C22#358511yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—82no—
648C4—84no—
649C4—81no—
650C4#311714yes—
651C4#349212yes—
652C4#350612yes—
653C2—42no—
654C2—162no—
655C2#3568911yescharacteristic
656C2C2×#311711yescharacteristic
657C2#6058811yescharacteristic
658C2#6211211yescharacteristic
659C2#6213511yescharacteristic
660C2#6242211yescharacteristic
661C2#6245511yescharacteristic
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. #34214C2semidirect
3. #30129C2semidirect
5. #29521C2semidirect
6. #29260C2semidirect
8. #27681C2semidirect
9. #26637C2semidirect
10. #26567C2semidirect
11. #26089C2semidirect
12. #24633C2semidirect
13. #23807C2semidirect
14. #23571C2semidirect
15. #23003C2semidirect
16. #22674C2semidirect
17. #22636C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#342142
#301292
#296612
#295212
#292602
#279062
#276812
#266372
#265672
#260892
#246332
#238072
#235712
#230032
#226742
#226362