Finite Group Databasebeta

← #6508735

Subgroups — #6508735

Order 512.

2087 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3194C211yesmaximal, characteristic
2C22×#480C211yesmaximal, characteristic, complemented
3#35227C211yesmaximal, characteristic, complemented
4#34363C211yesmaximal, characteristic, complemented
5#28129C214yesmaximal, complemented
6#27655C214yesmaximal, complemented
7#25227C214yesmaximal, complemented
8#25223C214yesmaximal, complemented
9#24577C212yesmaximal, complemented
10#24576C212yesmaximal, complemented
11#24527C212yesmaximal, complemented
12#24526C212yesmaximal, complemented
13#21933C211yesmaximal, characteristic, complemented
14#21760C211yesmaximal, characteristic, complemented
15#20700C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3194—28no—
20C2×#482C2212yes—
21C2×#483C2211yescharacteristic
22C2×#480—28no—
23C2×#480C2211yescharacteristic
24C2×#480C2214yes—
25C2×#480C2214yes—
26C2×#469C2212yes—
27C2×#468C2212yes—
28C2×#449C2212yes—
29C2×#450C2211yescharacteristic
30C2×#447C2211yescharacteristic
31C22×#107C2212yes—
32C22×#102C2212yes—
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2849—21no—
36#2849—21no—
37#2623—22no—
38#2504C2212yescomplemented
39#2503C2212yes—
40#2502C2212yescomplemented
41#2501C2212yes—
42#2487—22no—
43#2486—22no—
44#2377—22no—
45#2363C2214yes—
46#2344C2212yes—
47#2342—22no—
48D4×C22⋊C4C2212yes—
49#2333—22no—
50#2324C2214yes—
51#2317C2212yescomplemented
52#2311C2212yescomplemented
53#2309—21no—
54#2308—21no—
55#2280C2211yescharacteristic
56#2280C2211yescharacteristic
57#2085C2214yescomplemented
58#2083C2214yescomplemented
59#2073C2214yescomplemented
60#2071C2214yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2055C2214yescomplemented
64#2054C2214yescomplemented
65#1949C2214yescomplemented
66#1947C2214yescomplemented
67#1928C2214yescomplemented
68#1925C2214yescomplemented
69#1893C2212yes—
70#1892C2212yes—
71#1879C2214yescomplemented
72#1877C2214yescomplemented
73#1875C2212yescomplemented
74#1873C2212yescomplemented
75#1851C2214yescomplemented
76#1850C2212yes—
77#1850C2212yes—
78#1825C2212yescomplemented
79#1824C2214yescomplemented
80#1823C2212yes—
81#1823C2212yes—
82#1804C2214yescomplemented
83#1802C2214yescomplemented
84#1798C2214yescomplemented
85#1788C2212yescomplemented
86#1736C2214yescomplemented
87C22×#41C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×SD16C2312yes—
93C22×D8C2312yes—
94C22×M16C2312yes—
95C23×C8C2312yes—
96#525—24no—
97#525—24no—
98C2×#123—21no—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8C2311yescharacteristic
102C2×C22⋊Q8C2311yescharacteristic
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121D418yes—
107C2×#121C2311yescharacteristic
108C2×#121C2311yescharacteristic
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4C2314yes—
115C2×#117—22no—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C2×#117C2312yes—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—22no—
130#483—24no—
131#482—28no—
132#480—216no—
133#480—216no—
134#480—24no—
135#469—28no—
136#468—28no—
137#450—24no—
138#449—28no—
139#447—24no—
140C2×#107—28no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2312yes—
148C2×#103C2314yes—
149C2×#102—28no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156#407—24no—
157C2×C22⋊C8—24no—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163#400—24no—
164#399—24no—
165#398—24no—
166#397—24no—
167#391C2314yes—
168#390—24no—
169#390—24no—
170#390C2314yes—
171#390C2314yes—
172#389—24no—
173#389C2314yes—
174#388—24no—
175#388—24no—
176#387—24no—
177#386—24no—
178#386—24no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—24no—
184#385—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—22no—
189#384C2314yes—
190#382C2314yes—
191#381—21no—
192#381—22no—
193#381—22no—
194#381—22no—
195#381D412yes—
196#380—24no—
197#380D412yes—
198#380D412yes—
199#380C2312yes—
200#380C2312yes—
201#379—21no—
202#379—22no—
203#379D411yescharacteristic
204#379D411yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—24no—
209C4×C22⋊C4C2312yes—
210C4×C22⋊C4C2312yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—44no—
224C22×Q8C2411yescharacteristic
225C22×D4—28no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—44no—
234C22×D4—84no—
235C22×D4C2×D414yes—
236C22×D4C2411yescharacteristic
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—84no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D414yes—
251C23×C4#4114yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×SD16—24no—
256C2×SD16—44no—
257C2×D8—24no—
258C2×D8—44no—
259C2×M16—24no—
260C2×M16—44no—
261C22×C8—24no—
262C22×C8—44no—
263C22×C8C2414yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266#123—44no—
267#123—84no—
268#123—84no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273#121—28no—
274#121—28no—
275#121—24no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—22no—
281#121C2×D418yes—
282#121C2×D418yes—
283#120—44no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288#117—22no—
289#117—22no—
290#117—44no—
291#117—44no—
292#117—84no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—84no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42C2414yes—
346#107—28no—
347#107—28no—
348#107—24no—
349#106—24no—
350C4⋊C8—44no—
351C4⋊C8—44no—
352#103—24no—
353#103—44no—
354#102—28no—
355#102—28no—
356#102—24no—
357#102—24no—
358#102—24no—
359#102—44no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—22no—
363C22⋊C8—24no—
364C22⋊C8—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D414yes—
374#95C2×D414yes—
375C24—24no—
376C24—22no—
377C24—24no—
378C24—21no—
379C24—42no—
380C24—42no—
381C24—44no—
382C24—82no—
383C24—82no—
384C24#12114yes—
385C24C22×D411yescharacteristic
386C24C22×D411yescharacteristic
387C24C22×D411yescharacteristic
388#41—84no—
389#41—84no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—24no—
393C2×D4—28no—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—48no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—84no—
410C2×D4—82no—
411C2×D4—816no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4C22×D414yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—816no—
446C22×C4#12114yes—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2511yescharacteristic
461SD16—84no—
462D8—84no—
463M16—84no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—48no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4C22×D414yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—24no—
514C42—24no—
515C42—42no—
516C42—82no—
517C42—84no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—44no—
529C23—44no—
530C23—416no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—48no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—84no—
541C23—84no—
542C23—168no—
543C23C2×#12112yes—
544C23C2×#12112yes—
545C23C2×#12112yes—
546C23#57614yes—
547C23C23×D411yescharacteristic
548Q8—42no—
549Q8—42no—
550D4—24no—
551D4—24no—
552D4—42no—
553D4—44no—
554D4—42no—
555D4—84no—
556D4—88no—
557D4—88no—
558D4—84no—
559D4—84no—
560D4—84no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—84no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4—168no—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C23×D411yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4#58514yes—
606C8—44no—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—28no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—416no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—84no—
625C22—84no—
626C22—81no—
627C22—81no—
628C22—84no—
629C22—81no—
630C22—168no—
631C22#304114yes—
632C22C22×#12111yescharacteristic
633C22#348512yes—
634C22C2×#57612yes—
635C22#358612yes—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—84no—
643C4—81no—
644C4—81no—
645C4—84no—
646C4—81no—
647C4#348512yes—
648C4#349112yes—
649C2—24no—
650C2—42no—
651C2—42no—
652C2—84no—
653C2—161no—
654C2#3544711yescharacteristic
655C2C2×#304111yescharacteristic
656C2#6042711yescharacteristic
657C2#6174414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3194C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#480C2semidirect
3. #35227C2semidirect
4. #34363C2semidirect
5. #28129C2semidirect
6. #27655C2semidirect
7. #25227C2semidirect
8. #25223C2semidirect
9. #24577C2semidirect
10. #24576C2semidirect
11. #24527C2semidirect
12. #24526C2semidirect
13. #21933C2semidirect
14. #21760C2semidirect
15. #20700C2semidirect
38. #2504C22semidirect
40. #2502C22semidirect
51. #2317C22semidirect
52. #2311C22semidirect
57. #2085C22semidirect
58. #2083C22semidirect
59. #2073C22semidirect
60. #2071C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2055C22semidirect
64. #2054C22semidirect
65. #1949C22semidirect
66. #1947C22semidirect
67. #1928C22semidirect
68. #1925C22semidirect
71. #1879C22semidirect
72. #1877C22semidirect
73. #1875C22semidirect
74. #1873C22semidirect
75. #1851C22semidirect
78. #1825C22semidirect
79. #1824C22semidirect
82. #1804C22semidirect
83. #1802C22semidirect
84. #1798C22semidirect
85. #1788C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31942
C22×#4802
#352272
#343632
#281292
#276552
#252272
#252232
#245772
#245762
#245272
#245262
#219332
#217602
#207002