Finite Group Databasebeta

← #6563994

Subgroups — #6563994

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36315C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30043C212yesmaximal, complemented
4#29923C211yesmaximal, characteristic, complemented
5#29370C212yesmaximal, complemented
6#28380C212yesmaximal, complemented
7#27242C214yesmaximal, complemented
8#26465C212yesmaximal, complemented
9#25722C211yesmaximal, characteristic, complemented
10#25365C212yesmaximal, complemented
11#25169C214yesmaximal
12#24209C211yesmaximal, characteristic
13#23755C211yesmaximal, characteristic
14#23685C212yesmaximal, complemented
15#23187C212yesmaximal, complemented
16#23100C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#494C2212yes—
20C2×#479C2214yes—
21C2×#464C2214yes—
22C2×#461C2212yes—
23C2×#451C2214yes—
24C2×#448C2212yes—
25C2×#441C2214yes—
26#2861C2212yes—
27#2857C2211yescharacteristic
28#2552C2211yescharacteristic
29#2547C2211yescharacteristic
30#2531—22no—
31#2529—22no—
32#2497—21no—
33#2489C2212yes—
34#2485—21no—
35#2426C2212yes—
36#2369—21no—
37#2369—21no—
38#2365C2212yes—
39#2365C2212yes—
40#2365C2211yescharacteristic
41#2354—21no—
42#2354C2211yescharacteristic
43#2346—21no—
44#2345—22no—
45Q8×C22⋊C4C2214yes—
46#2331—22no—
47#2316—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2212yes—
50C4×#121C2211yescharacteristic
51#2295C2212yes—
52#2285C2211yescharacteristic
53#2274—21no—
54C4×#117C2211yescharacteristic
55#2100C2211yescharacteristic
56#2095C2212yes—
57#2094C2211yescharacteristic
58#2084C2211yescharacteristic
59#2077C2212yes—
60#2072C2212yes—
61#2071C2211yescharacteristic
62#2066C2214yes—
63#2056C2214yes—
64#2043C2212yes—
65#2036C2214yes—
66#2034C2212yes—
67#2026C2214yes—
68#2016C2212yes—
69#2016C2212yes—
70#2015C2214yes—
71#2013C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1971C2214yes—
76#1956C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1928C2214yes—
80#1924C2214yes—
81#1880C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1863C2214yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1787C2212yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C22×Q16C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125C2312yes—
106C2×#124C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2314yes—
112C2×C4×D4—21no—
113C2×C4×D4C2311yescharacteristic
114C2×#117—21no—
115C2×#117—21no—
116C2×#117D411yescharacteristic
117C2×#117D411yescharacteristic
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C42C2311yescharacteristic
121#494—28no—
122#479—48no—
123#464—48no—
124#461—28no—
125#451—48no—
126#448—44no—
127#441—48no—
128C2×#107C2314yes—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#400—22no—
147#398—22no—
148#397—22no—
149#397—22no—
150#396—22no—
151#394—22no—
152#391—24no—
153#391—24no—
154#391C2314yes—
155#391C2314yes—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#390—22no—
160#390C2311yescharacteristic
161#390C2312yes—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389C2312yes—
173#389C2311yescharacteristic
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#387—22no—
187#386—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#382—21no—
193#382—22no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380—24no—
201#380C2311yescharacteristic
202#380C2312yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—24no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376—22no—
224#376D412yes—
225C43C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235#336C2314yes—
236C22×Q8—28no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8—44no—
241C22×Q8C2412yes—
242C22×Q8C2414yes—
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×Q16—28no—
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—44no—
255C2×Q16—44no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—44no—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—28no—
263C4⋊Q8—28no—
264C4⋊Q8—28no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8#4118yes—
268#125—44no—
269#124—44no—
270C22⋊Q8—44no—
271#121—22no—
272#121—41no—
273C4×Q8—24no—
274C4×Q8—42no—
275C4×Q8—44no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×D4—24no—
280C4×D4—44no—
281#117—24no—
282#117—42no—
283#117—44no—
284#117—44no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
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—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—44no—
348#106—24no—
349#106—24no—
350C4⋊C8—28no—
351C4⋊C8—24no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—44no—
356#103—44no—
357#103—44no—
358#103—44no—
359#102—24no—
360#102—42no—
361C22⋊C8—24no—
362C22⋊C8—42no—
363#97—44no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24C22×D411yescharacteristic
388C2×Q8—28no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—82no—
399C2×Q8—82no—
400C2×Q8—88no—
401C2×Q8—88no—
402C2×Q8C2×#4118yes—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4#13714yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444Q16—416no—
445Q16—88no—
446SD16—88no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8H(5,𝔽2)14yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4H(5,𝔽2)12yes—
489C22⋊C4—24no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C42—22no—
503C42—22no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—22no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—44no—
514C42—42no—
515C42—44no—
516C42—84no—
517C42—84no—
518C42C2×#4114yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×#13712yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—28no—
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—88no—
541D4—42no—
542D4—42no—
543D4—84no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4#52512yes—
596C2×C4#52512yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13712yes—
599C2×C4#57512yes—
600C2×C4#57812yes—
601C2×C4C22×#4112yes—
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4#58511yescharacteristic
605C8—24no—
606C8—42no—
607C8—44no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22C2×#52511yescharacteristic
623C22#348811yescharacteristic
624C22#349811yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57511yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—41no—
636C4—48no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—81no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4#311714yes—
646C4#351412yes—
647C4#351912yes—
648C2—42no—
649C2—161no—
650C2#3600411yescharacteristic
651C2C2×#311711yescharacteristic
652C2#6058311yescharacteristic
653C2#6196011yescharacteristic
654C2#6199811yescharacteristic
655C2#6242411yescharacteristic
656C2#6245211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. #34216C2semidirect
3. #30043C2semidirect
4. #29923C2semidirect
5. #29370C2semidirect
6. #28380C2semidirect
7. #27242C2semidirect
8. #26465C2semidirect
9. #25722C2semidirect
10. #25365C2semidirect
14. #23685C2semidirect
15. #23187C2semidirect
16. #23100C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363152
#342162
#300432
#299232
#293702
#283802
#272422
#264652
#257222
#253652
#251692
#242092
#237552
#236852
#231872
#231002
#226432