Finite Group Databasebeta

← #6660843

Subgroups — #6660843

Order 512.

1953 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic, complemented
2#35272C211yesmaximal, characteristic
3#32731C214yesmaximal, complemented
4#30327C212yesmaximal, complemented
5#30319C212yesmaximal, complemented
6#29911C212yesmaximal, complemented
7#29758C212yesmaximal, complemented
8#29713C212yesmaximal, complemented
9#29625C212yesmaximal, complemented
10#29340C212yesmaximal, complemented
11#28323C212yesmaximal, complemented
12#24516C214yesmaximal, complemented
13#24015C212yesmaximal, complemented
14#23926C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic, complemented
16C2×#536C2211yescharacteristic
17C2×#535C2212yes—
18C22×#127C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#482C2214yescomplemented
21C2×#448C2214yescomplemented
22C2×#450C2214yes—
23C2×#433C2212yes—
24#2856C2212yes—
25#2856C2212yes—
26#2855C2212yes—
27#2855C2212yescomplemented
28#2625C2212yes—
29#2625C2212yescomplemented
30#2619C2212yes—
31#2619C2212yes—
32#2548—22no—
33#2541—22no—
34#2447—22no—
35#2446—22no—
36#2419—22no—
37#2419—22no—
38#2408—22no—
39#2408—22no—
40#2316C2211yescharacteristic, complemented
41#2311C2211yescharacteristic
42#2310C2211yescharacteristic, complemented
43#2309C2212yes—
44#2296C2211yescharacteristic, complemented
45C2×#391C2211yescharacteristic, complemented
46#2285C2211yescharacteristic
47#2101C2214yescomplemented
48#2097C2214yescomplemented
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2072C2212yescomplemented
52#2072C2212yescomplemented
53#2067C2214yescomplemented
54#2065C2214yescomplemented
55#2055C2214yescomplemented
56#2055C2214yescomplemented
57#2041C2212yes—
58#2041C2212yes—
59#2041C2214yes—
60#2034C2214yescomplemented
61#2033C2214yescomplemented
62#2014C2212yes—
63#2014C2212yes—
64#2014C2214yes—
65#2000C2214yescomplemented
66#2000C2214yescomplemented
67#1973C2214yescomplemented
68#1968C2214yescomplemented
69#1936C2214yescomplemented
70#1934C2214yescomplemented
71#1875C2214yescomplemented
72#1874C2214yescomplemented
73#1873C2214yes—
74#1854C2214yes—
75#1817C2212yes—
76#1798C2212yescomplemented
77#1797C2212yescomplemented
78#1786C2214yes—
79#1780C2212yes—
80#1750C2212yescomplemented
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×D4—22no—
86C23×D4C2312yes—
87C22×SD16C2314yes—
88C22×SD16C2314yes—
89#536—44no—
90#535—48no—
91C2×C4⋊Q8—21no—
92C2×#127—22no—
93C2×#127—22no—
94C2×#127—44no—
95C2×#127C2312yes—
96C2×#127C2312yes—
97C2×#124—22no—
98C2×#124—21no—
99C2×#123—21no—
100C2×#123—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2312yes—
103C2×#121—22no—
104C2×#121—22no—
105C2×#121—22no—
106C2×#121—22no—
107C2×#121C2314yes—
108C2×#121C2314yes—
109C2×#120—22no—
110C2×#120—22no—
111C4×#41—44no—
112C2×C4×Q8D412yes—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4D412yes—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#482—48no—
127#450—48no—
128#448—48no—
129#433—44no—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×#97C2312yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#400—22no—
149#398—22no—
150#398—22no—
151#397—22no—
152#397—22no—
153#397—22no—
154#397—22no—
155#394—24no—
156#394—24no—
157#392—24no—
158#392—24no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—24no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—21no—
184#386—21no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384C2312yes—
189#382C2312yes—
190#381—22no—
191#381—22no—
192#381—22no—
193#380—22no—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#379—24no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C2×#95—22no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#336C2312yes—
206#336C2312yes—
207#336C2314yes—
208C25—22no—
209C2×#41—24no—
210C2×#41—24no—
211C2×#41—42no—
212C22×Q8—21no—
213C22×Q8—24no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8C2×D411yescharacteristic
217C22×Q8C2412yes—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—24no—
224C22×D4—28no—
225C22×D4—44no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—48no—
229C22×D4—84no—
230C22×D4—84no—
231C22×D4C2×D411yescharacteristic
232C22×D4C2×D411yescharacteristic
233C22×D4C2×D411yescharacteristic
234C22×D4C2414yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×SD16—44no—
244C2×SD16—44no—
245C2×SD16—44no—
246C2×SD16—44no—
247C2×SD16—44no—
248C2×SD16—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—44no—
252#127—44no—
253#127—42no—
254#127—42no—
255#127—48no—
256#127—48no—
257#124—44no—
258#124—48no—
259#123—44no—
260#123—44no—
261C22⋊Q8—28no—
262C22⋊Q8—44no—
263#121—28no—
264#121—28no—
265#121—44no—
266#121—44no—
267#121—84no—
268#121—84no—
269#121—84no—
270#121—84no—
271#120—48no—
272#120—48no—
273C4×Q8—44no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—44no—
277#117—24no—
278#117—24no—
279#117—42no—
280#117—42no—
281#117—44no—
282#117—42no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—44no—
330#106—44no—
331C4⋊C8—44no—
332#103—44no—
333#103—44no—
334#102—44no—
335#102—44no—
336#102—44no—
337#102—44no—
338#102—44no—
339#102—44no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344#97—42no—
345C4×C8—42no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—24no—
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—
361C24—24no—
362C24—44no—
363C24—44no—
364C24—44no—
365C24—84no—
366C24—84no—
367C24C22×D411yescharacteristic
368C24C22×D411yescharacteristic
369C24C22×D411yescharacteristic
370#41—416no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—41no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—88no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—84no—
412C2×D4—84no—
413C2×D4—832no—
414C2×D4—84no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—84no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C4C2511yescharacteristic
454SD16—88no—
455SD16—88no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—42no—
503C42—44no—
504C42—42no—
505C42—42no—
506C23—22no—
507C23—22no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—22no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—88no—
527C23—816no—
528C23—164no—
529C23—164no—
530C23—168no—
531C23#53412yes—
532C23#53512yes—
533C23C23×D411yescharacteristic
534Q8—44no—
535Q8—44no—
536Q8—44no—
537Q8—84no—
538D4—44no—
539D4—44no—
540D4—44no—
541D4—44no—
542D4—44no—
543D4—44no—
544D4—44no—
545D4—88no—
546D4—88no—
547D4—88no—
548D4—816no—
549D4—168no—
550D4—168no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—162no—
592C2×C4#53412yes—
593C2×C4#53512yes—
594C2×C4#57714yes—
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4C2×H(5,𝔽2)12yes—
597C2×C4#58512yes—
598C8—44no—
599C8—44no—
600C22—22no—
601C22—22no—
602C22—24no—
603C22—24no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—41no—
614C22—42no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—84no—
622C22—1616no—
623C22C2×#53411yescharacteristic
624C22C2×#53511yescharacteristic
625C22#344911yescharacteristic
626C22C2×#57712yes—
627C22#358512yes—
628C4—22no—
629C4—22no—
630C4—24no—
631C4—41no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4#353414yes—
641C2—42no—
642C2—42no—
643C2—81no—
644C2—164no—
645C2#3744011yescharacteristic
646C2#6075712yes—
647C2#6083312yes—
648C2#6101211yescharacteristic
649C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35272C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35307C2semidirect
3. #32731C2semidirect
4. #30327C2semidirect
5. #30319C2semidirect
6. #29911C2semidirect
7. #29758C2semidirect
8. #29713C2semidirect
9. #29625C2semidirect
10. #29340C2semidirect
11. #28323C2semidirect
12. #24516C2semidirect
13. #24015C2semidirect
14. #23926C2semidirect
15. #20720C2semidirect
20. C2×#482C22semidirect
21. C2×#448C22semidirect
27. #2855C22semidirect
29. #2625C22semidirect
40. #2316C22semidirect
42. #2310C22semidirect
44. #2296C22semidirect
45. C2×#391C22semidirect
47. #2101C22semidirect
48. #2097C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
51. #2072C22semidirect
52. #2072C22semidirect
53. #2067C22semidirect
54. #2065C22semidirect
55. #2055C22semidirect
56. #2055C22semidirect
60. #2034C22semidirect
61. #2033C22semidirect
65. #2000C22semidirect
66. #2000C22semidirect
67. #1973C22semidirect
68. #1968C22semidirect
69. #1936C22semidirect
70. #1934C22semidirect
71. #1875C22semidirect
72. #1874C22semidirect
76. #1798C22semidirect
77. #1797C22semidirect
80. #1750C22semidirect
81. #1749C22semidirect
82. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#353072
#352722
#327312
#303272
#303192
#299112
#297582
#297132
#296252
#293402
#283232
#245162
#240152
#239262
#207202