Finite Group Databasebeta

← #6644659

Subgroups — #6644659

Order 512.

1614 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2566C211yesmaximal, characteristic
3#33009C212yesmaximal
4#32433C214yesmaximal, complemented
5#32347C214yesmaximal, complemented
6#31639C214yesmaximal, complemented
7#31525C214yesmaximal, complemented
8#29643C212yesmaximal, complemented
9#29103C211yesmaximal, characteristic
10#28302C211yesmaximal, characteristic
11#28123C212yesmaximal
12#27961C211yesmaximal, characteristic, complemented
13#27910C211yesmaximal, characteristic, complemented
14#27653C212yesmaximal, complemented
15#21912C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#483C2212yes—
21C2×#450C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2646—21no—
26#2627—21no—
27#2590—22no—
28#2588—22no—
29#2566—22no—
30#2566—22no—
31#2566C2214yes—
32#2566C2214yes—
33#2511C2212yescomplemented
34#2507C2212yescomplemented
35#2451C2212yescomplemented
36#2446C2212yescomplemented
37C2×#396—21no—
38C2×#396—21no—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#385—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2214yescomplemented
50#2095C2214yescomplemented
51#2094C2212yes—
52#2085C2214yescomplemented
53#2074C2214yescomplemented
54#2071C2212yes—
55#2067C2214yes—
56#2059C2214yescomplemented
57#2059C2214yescomplemented
58#2055C2214yes—
59#2026C2214yescomplemented
60#2026C2214yescomplemented
61#2023C2212yes—
62#2013C2212yes—
63#1990C2212yescomplemented
64#1968C2214yes—
65#1956C2212yescomplemented
66#1949C2214yescomplemented
67#1934C2214yes—
68#1925C2214yes—
69#1916C2214yescomplemented
70#1916C2214yescomplemented
71#1880C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1845C2212yescomplemented
77#1834C2212yescomplemented
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1805C2214yescomplemented
83#1797C2212yes—
84#1786C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×SD16C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—22no—
93C2×#125—44no—
94C2×#125C2314yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4—22no—
99C2×C4×D4—44no—
100C2×C4×D4C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#484—44no—
119#484—44no—
120#483—44no—
121#450—44no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2312yes—
136#400—22no—
137#400—22no—
138#400—22no—
139#398—22no—
140#398—22no—
141#398—22no—
142#398—22no—
143#396—24no—
144#396—24no—
145#396—24no—
146#396—24no—
147#395—22no—
148#391—21no—
149#391—22no—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#390—22no—
155#390—21no—
156#390—21no—
157#390C2314yes—
158#390C2312yes—
159#390C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#385—22no—
178#385—22no—
179#385—24no—
180#385—24no—
181#385—24no—
182#385—24no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#382—22no—
190#382C2314yes—
191#381—22no—
192#380—21no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C25—21no—
219C22×Q8—21no—
220C22×Q8C2412yes—
221C22×D4—24no—
222C22×D4—21no—
223C22×D4—24no—
224C22×D4—42no—
225C22×D4C2412yes—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—44no—
231C23×C4—82no—
232C23×C4—82no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×SD16—28no—
241C2×SD16—44no—
242C2×SD16—44no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247#125—48no—
248#125—48no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8—44no—
252C4×Q8—44no—
253C4×D4—48no—
254C4×D4—48no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—82no—
285C2×C4⋊C4—82no—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2412yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323#107—44no—
324#107—44no—
325#106—44no—
326C4⋊C8—44no—
327#103—44no—
328#103—44no—
329#103—44no—
330#102—28no—
331#102—44no—
332#102—44no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335C4×C8—42no—
336C4×C8—42no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356#95C2×D412yes—
357C24—24no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24C22×D411yescharacteristic
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—416no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D42−1+414yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—816no—
419C22×C4#13514yes—
420C22×C4#13714yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4H(5,𝔽2)14yes—
431C22×C4C2511yescharacteristic
432SD16—88no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—24no—
482C42—24no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—41no—
487C42—41no—
488C42—42no—
489C42—82no—
490C42—82no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—24no—
496C23—48no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23#53512yes—
510C23C2×#13512yes—
511C23C2×#13712yes—
512C23C23×D411yescharacteristic
513Q8—44no—
514Q8—44no—
515Q8—162no—
516D4—28no—
517D4—48no—
518D4—44no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—816no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4#53512yes—
574C2×C4C2×#13512yes—
575C2×C4C2×#13512yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4C2×H(5,𝔽2)12yes—
579C2×C4C2×2−1+412yes—
580C2×C4#58512yes—
581C8—44no—
582C8—44no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—816no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#321014yes—
598C22C2×#53511yescharacteristic
599C22#348511yescharacteristic
600C22#348611yescharacteristic
601C22C22×#13511yescharacteristic
602C22C22×#13711yescharacteristic
603C22#358511yescharacteristic
604C22#358511yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4#353514yes—
622C2—24no—
623C2—84no—
624C2#3711811yescharacteristic
625C2C2×#321011yescharacteristic
626C2#6080611yescharacteristic
627C2#6169911yescharacteristic
628C2#6170811yescharacteristic
629C2#6181911yescharacteristic
630C2#6182611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35015C2semidirect
4. #32433C2semidirect
5. #32347C2semidirect
6. #31639C2semidirect
7. #31525C2semidirect
8. #29643C2semidirect
12. #27961C2semidirect
13. #27910C2semidirect
14. #27653C2semidirect
15. #21912C2semidirect
33. #2511C22semidirect
34. #2507C22semidirect
35. #2451C22semidirect
36. #2446C22semidirect
49. #2100C22semidirect
50. #2095C22semidirect
52. #2085C22semidirect
53. #2074C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
59. #2026C22semidirect
60. #2026C22semidirect
63. #1990C22semidirect
65. #1956C22semidirect
66. #1949C22semidirect
69. #1916C22semidirect
70. #1916C22semidirect
76. #1845C22semidirect
77. #1834C22semidirect
82. #1805C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#25662
#330092
#324332
#323472
#316392
#315252
#296432
#291032
#283022
#281232
#279612
#279102
#276532
#219122