Finite Group Databasebeta

← #6508500

Subgroups — #6508500

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2454C211yesmaximal, characteristic
3#29109C211yesmaximal, characteristic
4#29084C214yesmaximal, complemented
5#28957C214yesmaximal, complemented
6#28311C211yesmaximal, characteristic
7#26845C214yesmaximal, complemented
8#26498C214yesmaximal, complemented
9#25912C211yesmaximal, characteristic, complemented
10#25819C211yesmaximal, characteristic, complemented
11#24602C212yesmaximal, complemented
12#24501C212yesmaximal
13#22085C212yesmaximal, complemented
14#21453C212yesmaximal
15#20536C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#501C2212yescomplemented
19C2×#491C2212yescomplemented
20C2×#484C2212yes—
21C2×#450C2212yes—
22C2×#440C2214yes—
23C2×#431C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yescomplemented
32#2507C2212yescomplemented
33#2454—22no—
34#2454—22no—
35#2454C2214yes—
36#2454C2214yes—
37#2451C2212yescomplemented
38#2446C2212yescomplemented
39C2×#399—21no—
40C2×#396—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#385—21no—
49C2×#380—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2093C2212yes—
52#2087C2214yescomplemented
53#2071C2212yes—
54#2068C2214yescomplemented
55#2065C2214yes—
56#2059C2214yescomplemented
57#2055C2214yes—
58#2026C2214yescomplemented
59#2023C2212yes—
60#2013C2212yes—
61#1992C2212yescomplemented
62#1992C2212yescomplemented
63#1973C2214yes—
64#1968C2214yes—
65#1943C2214yescomplemented
66#1942C2214yescomplemented
67#1936C2214yes—
68#1935C2214yescomplemented
69#1934C2214yes—
70#1933C2214yescomplemented
71#1918C2214yescomplemented
72#1877C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1865C2214yescomplemented
76#1862C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1786C2212yes—
81#1770C2214yescomplemented
82#1770C2214yescomplemented
83#1762C2214yes—
84#1749C2211yescharacteristic
85#1749C2212yes—
86#1749C2211yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C22×SD16C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—21no—
93C2×#125—21no—
94C2×#125—44no—
95C2×#125C2312yes—
96C2×#125C2312yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×D4—22no—
101C2×C4×D4—44no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#501—44no—
121#491—44no—
122#484—28no—
123#450—28no—
124#440—48no—
125#431—28no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2314yes—
142#400—22no—
143#400—22no—
144#400—22no—
145#399—24no—
146#399—24no—
147#398—22no—
148#398—22no—
149#398—22no—
150#398—22no—
151#396—24no—
152#396—24no—
153#395—22no—
154#391—21no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#390—22no—
161#390—21no—
162#390—21no—
163#390C2314yes—
164#390C2312yes—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2314yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#385—22no—
188#385—22no—
189#385—24no—
190#385—24no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#381—22no—
194#380—21no—
195#380—21no—
196#380—21no—
197#380—24no—
198#380—24no—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95D411yescharacteristic
210C2×#95D411yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C22×Q8—21no—
222C22×Q8C2412yes—
223C22×D4—24no—
224C22×D4—21no—
225C22×D4—24no—
226C22×D4—42no—
227C22×D4C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—82no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×SD16—28no—
243C2×SD16—28no—
244C2×SD16—44no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—42no—
248C4⋊Q8—42no—
249#125—24no—
250#125—24no—
251#125—42no—
252#125—42no—
253#125—44no—
254#125—44no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—44no—
258C4×Q8—44no—
259C4×D4—28no—
260C4×D4—44no—
261C4×D4—48no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4—82no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2412yes—
330#107—28no—
331#107—28no—
332#106—44no—
333C4⋊C8—28no—
334#103—28no—
335#103—28no—
336#103—44no—
337#102—28no—
338#102—28no—
339#102—44no—
340C22⋊C8—24no—
341C22⋊C8#4118yes—
342#97—44no—
343#97—42no—
344#97—42no—
345C4×C8—28no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—24no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24C22×D411yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—42no—
380C2×Q8—81no—
381C2×Q8—81no—
382C2×D4—24no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—22no—
386C2×D4—416no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D42−1+414yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—84no—
428C22×C4—816no—
429C22×C4#13714yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2511yescharacteristic
441SD16—416no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8C2×#4118yes—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4H(5,𝔽2)14yes—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—41no—
500C42—41no—
501C42—82no—
502C42—82no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—24no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—84no—
518C23—84no—
519C23—84no—
520C23—84no—
521C23#53212yes—
522C23C2×#13712yes—
523C23#57512yes—
524C23C23×D411yescharacteristic
525Q8—44no—
526Q8—44no—
527Q8—162no—
528D4—28no—
529D4—48no—
530D4—44no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—816no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4#53212yes—
589C2×C4C2×#13714yes—
590C2×C4C22×#4112yes—
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×2−1+412yes—
593C8—28no—
594C8—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—816no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22#318114yes—
610C22C2×#53211yescharacteristic
611C22#348411yescharacteristic
612C22#348611yescharacteristic
613C22C22×#13711yescharacteristic
614C22C2×#57511yescharacteristic
615C22#358511yescharacteristic
616C22#358511yescharacteristic
617C4—24no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—44no—
624C4—42no—
625C4—41no—
626C4—41no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#352014yes—
634C2—24no—
635C2—84no—
636C2#3539711yescharacteristic
637C2C2×#318111yescharacteristic
638C2#6070411yescharacteristic
639C2#6158111yescharacteristic
640C2#6158811yescharacteristic
641C2#6182611yescharacteristic
642C2#6183111yescharacteristic
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. #29084C2semidirect
5. #28957C2semidirect
7. #26845C2semidirect
8. #26498C2semidirect
9. #25912C2semidirect
10. #25819C2semidirect
11. #24602C2semidirect
13. #22085C2semidirect
15. #20536C2semidirect
18. C2×#501C22semidirect
19. C2×#491C22semidirect
31. #2511C22semidirect
32. #2507C22semidirect
37. #2451C22semidirect
38. #2446C22semidirect
52. #2087C22semidirect
54. #2068C22semidirect
56. #2059C22semidirect
58. #2026C22semidirect
61. #1992C22semidirect
62. #1992C22semidirect
65. #1943C22semidirect
66. #1942C22semidirect
68. #1935C22semidirect
70. #1933C22semidirect
71. #1918C22semidirect
75. #1865C22semidirect
81. #1770C22semidirect
82. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#24542
#291092
#290842
#289572
#283112
#268452
#264982
#259122
#258192
#246022
#245012
#220852
#214532
#205362