Finite Group Databasebeta

← #6508792

Subgroups — #6508792

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3192C211yesmaximal, characteristic
2C2×#3094C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic, complemented
4#34363C211yesmaximal, characteristic, complemented
5#28126C214yesmaximal, complemented
6#27654C214yesmaximal, complemented
7#25227C214yesmaximal, complemented
8#25221C214yesmaximal, complemented
9#24589C214yesmaximal, complemented
10#24539C214yesmaximal, complemented
11#21935C211yesmaximal, characteristic, complemented
12#21767C211yesmaximal, characteristic, complemented
13#20701C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3192—28no—
18#3094—28no—
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#481C2212yescomplemented
22C2×#482C2214yes—
23C2×#480C2212yescomplemented
24C2×#469C2214yes—
25C2×#447C2212yes—
26C2×#447C2212yes—
27C2×#428C2212yescomplemented
28C2×#418C2212yescomplemented
29C2×#407C2211yescharacteristic
30C2×#407C2211yescharacteristic
31#2854—22no—
32#2624—22no—
33#2504C2212yescomplemented
34#2501C2212yescomplemented
35#2500C2214yes—
36#2482—22no—
37#2405—21no—
38#2405—21no—
39#2377—22no—
40#2363C2214yescomplemented
41#2344C2212yescomplemented
42#2342—22no—
43D4×C22⋊C4C2212yescomplemented
44#2333—22no—
45#2324C2214yes—
46#2317C2212yescomplemented
47#2309—21no—
48#2309C2212yescomplemented
49#2308—21no—
50#2280C2211yescharacteristic
51#2280C2211yescharacteristic
52#2085C2214yescomplemented
53#2084C2214yescomplemented
54#2073C2214yescomplemented
55#2071C2214yescomplemented
56#2067C2214yescomplemented
57#2065C2214yescomplemented
58#2054C2214yescomplemented
59#2054C2214yescomplemented
60#1949C2214yescomplemented
61#1948C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1895C2214yes—
65#1879C2214yescomplemented
66#1879C2214yescomplemented
67#1875C2214yescomplemented
68#1851C2214yescomplemented
69#1851C2214yescomplemented
70#1825C2214yescomplemented
71#1825C2214yescomplemented
72#1823C2212yescomplemented
73#1804C2214yescomplemented
74#1802C2214yescomplemented
75#1798C2214yescomplemented
76#1786C2212yescomplemented
77#1736C2212yescomplemented
78#1736C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×D8C2314yes—
85C22×M16C2312yes—
86C22×M16C2312yes—
87C2×#123—22no—
88C2×#123—22no—
89C2×#121—21no—
90C2×#121—21no—
91C2×#121—24no—
92C2×#121—21no—
93C2×#121—21no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121D418yes—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×C4×D4C2314yes—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—22no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
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#481—28no—
123#481—28no—
124#480—28no—
125#469—216no—
126#447—28no—
127#447—28no—
128#428—28no—
129C2×#107C2314yes—
130C2×#106C2314yes—
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—
142#407—24no—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147#399—24no—
148#397—24no—
149#394—24no—
150#394—24no—
151#390—24no—
152#390—24no—
153#390—24no—
154#390C2314yes—
155#390C2314yes—
156#390C2314yes—
157#389—24no—
158#389C2314yes—
159#388—24no—
160#388—24no—
161#388—24no—
162#388—24no—
163#387—24no—
164#386—24no—
165#386—24no—
166#386—22no—
167#386—22no—
168#386—24no—
169#385—24no—
170#385—22no—
171#385—22no—
172#384C2314yes—
173#382C2314yes—
174#381—21no—
175#381—22no—
176#381D412yes—
177#380D412yes—
178#380D412yes—
179#380C2312yes—
180#380C2312yes—
181#379—21no—
182#379—22no—
183#379—22no—
184#379—22no—
185#379D411yescharacteristic
186#379D411yescharacteristic
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—24no—
191C4×C22⋊C4C2312yes—
192C4×C22⋊C4C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25—21no—
203C25C2×D411yescharacteristic
204C22×D4—28no—
205C22×D4—22no—
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—21no—
227C23×C4—21no—
228C23×C4—24no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D414yes—
235C23×C4#4114yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×D8—28no—
240C2×D8—44no—
241C2×D8—44no—
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—42no—
245C2×M16—42no—
246C2×M16—42no—
247C2×M16—42no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250#123—84no—
251#123—84no—
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—
266#117—24no—
267#117—24no—
268#117—24no—
269#117—24no—
270#117—42no—
271#117—42no—
272#117—44no—
273#117—44no—
274#117—42no—
275#117—42no—
276#117—84no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
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—24no—
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—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—84no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—44no—
329C2×C42C2414yes—
330#107—28no—
331#106—28no—
332#106—28no—
333C4⋊C8—44no—
334C4⋊C8—44no—
335#102—28no—
336#102—28no—
337#102—28no—
338#102—28no—
339#102—44no—
340#102—44no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—44no—
360C24—44no—
361C24—42no—
362C24—82no—
363C24—82no—
364C24—82no—
365C24—82no—
366C24#12114yes—
367C24C22×D411yescharacteristic
368C24C22×D411yescharacteristic
369C24C22×D411yescharacteristic
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—84no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4—816no—
400C2×D4—816no—
401C2×D4C22×D414yes—
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—22no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4#12114yes—
431C22×C4#12114yes—
432C22×C4#12114yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445D8—88no—
446M16—48no—
447M16—84no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4C22×D414yes—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C42—24no—
495C42—24no—
496C42—44no—
497C42—44no—
498C42—82no—
499C42—84no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—48no—
514C23—42no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23—84no—
527C23—84no—
528C23—88no—
529C23—168no—
530C23—168no—
531C23C2×#12112yes—
532C23C2×#12112yes—
533C23C2×#12112yes—
534C23C23×D411yescharacteristic
535D4—28no—
536D4—44no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—84no—
542D4—84no—
543D4—88no—
544D4—88no—
545D4—88no—
546D4—88no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—168no—
586C2×C4C2×#12112yes—
587C2×C4C2×#12112yes—
588C2×C4C2×#12112yes—
589C2×C4C23×D411yescharacteristic
590C2×C4C22×#4111yescharacteristic
591C2×C4#58512yes—
592C2×C4#58512yes—
593C8—44no—
594C8—44no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—88no—
611C22—84no—
612C22—84no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—84no—
617C22—82no—
618C22—168no—
619C22—168no—
620C22C22×#12111yescharacteristic
621C22#348512yes—
622C22#358512yes—
623C22#358512yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4#348512yes—
635C4#349111yescharacteristic
636C4#349111yescharacteristic
637C2—42no—
638C2—42no—
639C2—42no—
640C2—84no—
641C2—162no—
642C2#3544711yescharacteristic
643C2#6042311yescharacteristic
644C2#6042611yescharacteristic
645C2#6178314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3192C2
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. #34363C2semidirect
5. #28126C2semidirect
6. #27654C2semidirect
7. #25227C2semidirect
8. #25221C2semidirect
9. #24589C2semidirect
10. #24539C2semidirect
11. #21935C2semidirect
12. #21767C2semidirect
13. #20701C2semidirect
21. C2×#481C22semidirect
23. C2×#480C22semidirect
27. C2×#428C22semidirect
28. C2×#418C22semidirect
33. #2504C22semidirect
34. #2501C22semidirect
40. #2363C22semidirect
41. #2344C22semidirect
43. D4×C22⋊C4C22semidirect
46. #2317C22semidirect
48. #2309C22semidirect
52. #2085C22semidirect
53. #2084C22semidirect
54. #2073C22semidirect
55. #2071C22semidirect
56. #2067C22semidirect
57. #2065C22semidirect
58. #2054C22semidirect
59. #2054C22semidirect
60. #1949C22semidirect
61. #1948C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
65. #1879C22semidirect
66. #1879C22semidirect
67. #1875C22semidirect
68. #1851C22semidirect
69. #1851C22semidirect
70. #1825C22semidirect
71. #1825C22semidirect
72. #1823C22semidirect
73. #1804C22semidirect
74. #1802C22semidirect
75. #1798C22semidirect
76. #1786C22semidirect
77. #1736C22semidirect
78. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31922
C2×#30942
#352622
#343632
#281262
#276542
#252272
#252212
#245892
#245392
#219352
#217672
#207012