Finite Group Databasebeta

← #6508784

Subgroups — #6508784

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3186C211yesmaximal, characteristic
2C2×#3094C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic, complemented
4#34379C211yesmaximal, characteristic, complemented
5#28126C214yesmaximal, complemented
6#27655C214yesmaximal, complemented
7#25214C214yesmaximal, complemented
8#25200C214yesmaximal, complemented
9#24591C214yesmaximal, complemented
10#24539C214yesmaximal, complemented
11#21933C211yesmaximal, characteristic, complemented
12#21764C211yesmaximal, characteristic, complemented
13C2×#1825C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3186—28no—
18#3094—28no—
19C2×#481C2212yescomplemented
20C2×#482C2214yes—
21C2×#480C2212yes—
22C2×#480C2212yes—
23C2×#480C2212yescomplemented
24C2×#466C2214yes—
25C2×#447C2212yes—
26C2×#447C2212yes—
27C2×#428C2212yescomplemented
28C2×#418C2212yescomplemented
29C2×#407C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2854—22no—
32#2624—21no—
33#2624—21no—
34#2504C2212yescomplemented
35#2501C2212yescomplemented
36#2500C2214yes—
37#2482—22no—
38#2405—22no—
39#2377—22no—
40#2361C2214yescomplemented
41#2345C2212yescomplemented
42D4×C22⋊C4—22no—
43D4×C22⋊C4C2212yescomplemented
44#2325—22no—
45#2324C2214yes—
46#2317C2212yescomplemented
47#2315—21no—
48#2312—21no—
49#2309C2212yescomplemented
50#2280C2211yescharacteristic
51C2×#380C2211yescharacteristic
52#2085C2214yescomplemented
53#2083C2214yescomplemented
54#2073C2214yescomplemented
55#2071C2214yescomplemented
56#2067C2214yescomplemented
57#2065C2214yescomplemented
58#2054C2214yescomplemented
59#2054C2214yescomplemented
60#1949C2214yescomplemented
61#1947C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1895C2214yes—
65#1879C2214yescomplemented
66#1879C2214yescomplemented
67#1874C2214yescomplemented
68#1851C2214yescomplemented
69#1850C2214yescomplemented
70#1825C2214yescomplemented
71#1825C2214yescomplemented
72#1824C2212yescomplemented
73#1804C2214yescomplemented
74#1802C2214yescomplemented
75#1798C2214yescomplemented
76#1786C2212yescomplemented
77C2×#340C2212yescomplemented
78C2×#340C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×D8C2314yes—
85C22×M16C2312yes—
86C23×C8C2312yes—
87C2×#123—24no—
88C2×#121—21no—
89C2×#121—21no—
90C2×#121—24no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121D418yes—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×C4×D4—24no—
101C2×C4×D4C2314yes—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—24no—
105C2×#117—24no—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—22no—
119C22×C42—21no—
120#482—216no—
121#481—28no—
122#480—28no—
123#480—28no—
124#480—28no—
125#466—216no—
126#447—28no—
127#447—28no—
128#428—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134#418—28no—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#407—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#399—22no—
148#399—22no—
149#397—22no—
150#397—22no—
151#394—24no—
152#394—24no—
153#390—22no—
154#390—22no—
155#390—24no—
156#390C2314yes—
157#390C2314yes—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#387—24no—
167#386—24no—
168#386—24no—
169#386—24no—
170#386—24no—
171#385—24no—
172#385—24no—
173#385—24no—
174#384C2314yes—
175#382C2314yes—
176#381D412yes—
177#380—21no—
178#380—21no—
179#380—22no—
180#380D411yescharacteristic
181#380D411yescharacteristic
182#380D412yes—
183#380C2312yes—
184#380C2312yes—
185#379—24no—
186#379—22no—
187#379D412yes—
188C4×C4⋊C4C2314yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4C2312yes—
191C4×C22⋊C4C2312yes—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C25C2×D411yescharacteristic
204C22×D4—22no—
205C22×D4—28no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—42no—
218C22×D4—44no—
219C22×D4—84no—
220C22×D4—84no—
221C22×D4C2×D414yes—
222C22×D4C2412yes—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D414yes—
236C23×C4#4114yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×D8—28no—
241C2×D8—44no—
242C2×D8—44no—
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C22×C8—24no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251#123—88no—
252#121—28no—
253#121—28no—
254#121—24no—
255#121—24no—
256#121—24no—
257#121—24no—
258#121—24no—
259#121—24no—
260#121—44no—
261#121—44no—
262#121C2×D418yes—
263#121C2×D418yes—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—88no—
267#117—24no—
268#117—24no—
269#117—24no—
270#117—24no—
271#117—42no—
272#117—42no—
273#117—44no—
274#117—44no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
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⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—84no—
319C2×C22⋊C4—82no—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42C2414yes—
329#107—28no—
330#107—28no—
331#106—28no—
332C4⋊C8—44no—
333C4⋊C8—44no—
334#102—28no—
335#102—28no—
336#102—28no—
337#102—28no—
338#102—44no—
339#102—44no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95C2×D412yes—
351#95C2×D414yes—
352#95C2×D412yes—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—41no—
358C24—42no—
359C24—41no—
360C24—42no—
361C24—42no—
362C24—44no—
363C24—42no—
364C24—82no—
365C24—82no—
366C24—82no—
367C24—81no—
368C24—81no—
369C24#12114yes—
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C24C22×D411yescharacteristic
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—24no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—42no—
390C2×D4—84no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—816no—
399C2×D4—816no—
400C2×D4C22×D414yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4#12114yes—
434C22×C4#12114yes—
435C22×C4#12114yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2511yescharacteristic
448D8—88no—
449M16—48no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—84no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4C22×D414yes—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—82no—
496C42—24no—
497C42—24no—
498C42—44no—
499C42—44no—
500C42—84no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—24no—
514C23—44no—
515C23—44no—
516C23—48no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—164no—
534C23—164no—
535C23—164no—
536C23—164no—
537C23C2×#12112yes—
538C23C2×#12112yes—
539C23C2×#12112yes—
540C23#57614yes—
541C23C23×D411yescharacteristic
542D4—28no—
543D4—416no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—84no—
550D4—84no—
551D4—88no—
552D4—88no—
553D4—88no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—168no—
593C2×C4—164no—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4C2×#12112yes—
597C2×C4C23×D411yescharacteristic
598C2×C4C22×#4111yescharacteristic
599C2×C4#58512yes—
600C2×C4#58512yes—
601C8—44no—
602C8—44no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—88no—
621C22—84no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—82no—
628C22—168no—
629C22—168no—
630C22#304114yes—
631C22C22×#12111yescharacteristic
632C22#348512yes—
633C22C2×#57612yes—
634C22#358511yescharacteristic
635C22#358611yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—84no—
646C4#348512yes—
647C4#349111yescharacteristic
648C4#349111yescharacteristic
649C2—24no—
650C2—42no—
651C2—42no—
652C2—84no—
653C2—162no—
654C2#3544711yescharacteristic
655C2C2×#304111yescharacteristic
656C2#6042511yescharacteristic
657C2#6175212yes—
658C2#6175912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3186C2
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. C2×#3094C2semidirect
3. #35262C2semidirect
4. #34379C2semidirect
5. #28126C2semidirect
6. #27655C2semidirect
7. #25214C2semidirect
8. #25200C2semidirect
9. #24591C2semidirect
10. #24539C2semidirect
11. #21933C2semidirect
12. #21764C2semidirect
13. C2×#1825C2semidirect
19. C2×#481C22semidirect
23. C2×#480C22semidirect
27. C2×#428C22semidirect
28. C2×#418C22semidirect
34. #2504C22semidirect
35. #2501C22semidirect
40. #2361C22semidirect
41. #2345C22semidirect
43. D4×C22⋊C4C22semidirect
46. #2317C22semidirect
49. #2309C22semidirect
52. #2085C22semidirect
53. #2083C22semidirect
54. #2073C22semidirect
55. #2071C22semidirect
56. #2067C22semidirect
57. #2065C22semidirect
58. #2054C22semidirect
59. #2054C22semidirect
60. #1949C22semidirect
61. #1947C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
65. #1879C22semidirect
66. #1879C22semidirect
67. #1874C22semidirect
68. #1851C22semidirect
69. #1850C22semidirect
70. #1825C22semidirect
71. #1825C22semidirect
72. #1824C22semidirect
73. #1804C22semidirect
74. #1802C22semidirect
75. #1798C22semidirect
76. #1786C22semidirect
77. C2×#340C22semidirect
78. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
C2×#30942
#352622
#343792
#281262
#276552
#252142
#252002
#245912
#245392
#219332
#217642
C2×#18252