Finite Group Databasebeta

← #6495340

Subgroups — #6495340

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33305C211yesmaximal, characteristic, complemented
4#33294C211yesmaximal, characteristic, complemented
5#33238C211yesmaximal, characteristic, complemented
6#33232C211yesmaximal, characteristic, complemented
7#29815C214yesmaximal, complemented
8#28619C214yesmaximal, complemented
9#28616C212yesmaximal, complemented
10#28613C212yesmaximal, complemented
11#25604C214yesmaximal, complemented
12#25600C212yesmaximal, complemented
13#25598C212yesmaximal
14#22273C214yesmaximal, complemented
15C4×#432C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#529C2212yes—
18C2×#497C2214yes—
19C2×#497C2214yes—
20C2×#494C2212yes—
21C2×#492C2212yes—
22C2×#491C2214yes—
23C2×#487C2212yes—
24C2×#486C2212yes—
25C2×#432C2212yes—
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2212yes—
29#2861C2212yes—
30#2860—21no—
31#2860—21no—
32#2859—21no—
33#2859—21no—
34#2859C2211yescharacteristic
35#2858C2211yescharacteristic
36#2857—21no—
37#2857—21no—
38#2857C2211yescharacteristic
39#2857C2212yes—
40#2857C2212yes—
41#2856C2211yescharacteristic
42#2854—21no—
43#2854—21no—
44#2850C2212yes—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8—21no—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2212yes—
50C4×#127C2211yescharacteristic
51C4×#124C2212yes—
52C4×#121—21no—
53C4×#121—21no—
54#2293—21no—
55#2293—21no—
56#2285—21no—
57#2285—21no—
58C2×C43C2211yescharacteristic
59#2056C2218yes—
60#2055C2218yes—
61#2001C2218yes—
62#2000C2218yes—
63#1992C2214yes—
64#1990C2212yes—
65#1990C2212yes—
66#1960C2214yes—
67#1959C2214yes—
68#1959C2214yes—
69#1956C2212yes—
70#1956C2212yes—
71#1880C2218yes—
72#1878C2218yes—
73#1878C2218yes—
74#1877C2218yes—
75#1836C2214yes—
76#1767C2212yes—
77C4×C4⋊C8C2212yes—
78#1749C2212yes—
79C4×C22⋊C8C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C22×Q16C2318yes—
84C22×SD16C2318yes—
85#533—44no—
86#533—44no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#127C2311yescharacteristic
95#529—48no—
96C2×#126—22no—
97C2×#125—22no—
98C2×#125C2312yes—
99C2×#124C2312yes—
100C2×#124C2312yes—
101C2×#123—22no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—22no—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2312yes—
115C2×#117—22no—
116C2×#117—22no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#497—216no—
125#497—48no—
126#494—44no—
127#492—44no—
128#491—48no—
129#487—44no—
130#486—44no—
131#432—28no—
132C2×C4⋊C8C2314yes—
133C2×#103C2318yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#102C2318yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#391—22no—
145#391—22no—
146#391C2314yes—
147#391C2312yes—
148#391C2314yes—
149#391C2312yes—
150#390—21no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—21no—
155#390—21no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—22no—
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#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#386—22no—
185#386—22no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382—22no—
198#382C2312yes—
199#380—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C43D411yescharacteristic
214C43D411yescharacteristic
215C43D411yescharacteristic
216C43D411yescharacteristic
217C43C2311yescharacteristic
218C43C2311yescharacteristic
219C43C2311yescharacteristic
220C43C2311yescharacteristic
221#340C2318yes—
222#340C2318yes—
223#336C2314yes—
224#336C2314yes—
225C22×Q8—22no—
226C22×Q8C2412yes—
227C22×Q8C2414yes—
228C22×D4—22no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2412yes—
232C23×C4—22no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—216no—
241C2×Q16—48no—
242C2×Q16—48no—
243C2×SD16—216no—
244C2×SD16—48no—
245C2×SD16—48no—
246C22×C8C2414yes—
247C22×C8C2414yes—
248C4⋊Q8—28no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—28no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257#127—24no—
258#126—48no—
259#125—28no—
260#125—48no—
261#124—28no—
262#124—28no—
263#123—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268#121—44no—
269#121—44no—
270#121—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—48no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—48no—
278C4×D4—44no—
279#117—48no—
280#117—48no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C4⋊C8—28no—
357#103—48no—
358#103—48no—
359#103—48no—
360#102—48no—
361C22⋊C8—28no—
362#97—24no—
363#97—24no—
364#97—24no—
365#97#4118yes—
366C4×C8—24no—
367C4×C8—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378C24—22no—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—28no—
382C2×Q8—28no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4#13714yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C2511yescharacteristic
445Q16—816no—
446SD16—816no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8C2×#4118yes—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—44no—
517C42—48no—
518C42#13714yes—
519C42C2×#4114yes—
520C42H(5,𝔽2)14yes—
521C42H(5,𝔽2)14yes—
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—48no—
538Q8—48no—
539Q8—48no—
540Q8—88no—
541D4—48no—
542D4—88no—
543D4—84no—
544D4—84no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—28no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4#53212yes—
595C2×C4#53212yes—
596C2×C4#53512yes—
597C2×C4#53512yes—
598C2×C4#53512yes—
599C2×C4#53512yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4#57512yes—
606C2×C4#57512yes—
607C2×C4C22×#4112yes—
608C2×C4C2×H(5,𝔽2)12yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C8—28no—
611C8—44no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22C2×#53211yescharacteristic
627C22C2×#53511yescharacteristic
628C22C2×#53511yescharacteristic
629C22C22×#13711yescharacteristic
630C22C22×#13711yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57511yescharacteristic
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—84no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—84no—
649C4#321814yes—
650C4#321914yes—
651C4#350814yes—
652C2—42no—
653C2—162no—
654C2#3529311yescharacteristic
655C2#6070111yescharacteristic
656C2#6070111yescharacteristic
657C2C2×#321811yescharacteristic
658C2C2×#321911yescharacteristic
659C2#6080411yescharacteristic
660C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33305C2semidirect
4. #33294C2semidirect
5. #33238C2semidirect
6. #33232C2semidirect
7. #29815C2semidirect
8. #28619C2semidirect
9. #28616C2semidirect
10. #28613C2semidirect
11. #25604C2semidirect
12. #25600C2semidirect
14. #22273C2semidirect
15. C4×#432C2semidirect

Maximal subgroups

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

subgroupindex
#366452
#366312
#333052
#332942
#332382
#332322
#298152
#286192
#286162
#286132
#256042
#256002
#255982
#222732
C4×#4322