Finite Group Databasebeta

← #6477711

Subgroups — #6477711

Order 512.

1938 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C2×#2921C211yesmaximal, characteristic, complemented
3#35311C211yesmaximal, characteristic, complemented
4#35262C211yesmaximal, characteristic
5#33145C214yesmaximal, complemented
6#29784C214yesmaximal, complemented
7#24614C214yesmaximal, complemented
8#24612C214yesmaximal, complemented
9#24518C214yesmaximal, complemented
10#24516C214yesmaximal, complemented
11#21943C211yesmaximal, characteristic, complemented
12#21768C211yesmaximal, characteristic, complemented
13#21767C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3193—28no—
18C2×#484C2212yes—
19C2×#480C2212yes—
20C2×#468C2214yes—
21C2×#448C2212yes—
22C2×#450C2212yes—
23#2921—28no—
24C2×#433C2214yes—
25C2×#433C2214yescomplemented
26C2×#423C2212yescomplemented
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×C4×M16C2212yescomplemented
30#2855C2212yescomplemented
31#2855C2212yescomplemented
32#2854C2212yes—
33#2854C2212yes—
34#2625C2214yes—
35#2624C2214yes—
36#2504—21no—
37#2503—21no—
38#2502—21no—
39#2501—21no—
40#2500—22no—
41#2499—22no—
42#2484—22no—
43#2483—21no—
44#2483—21no—
45#2482—22no—
46#2405—21no—
47#2405—21no—
48#2317C2212yes—
49#2317C2212yescomplemented
50#2310C2212yescomplemented
51#2309C2212yes—
52#2280C2211yescharacteristic
53#2280C2211yescharacteristic
54#2087C2214yescomplemented
55#2086C2214yescomplemented
56#2068C2214yescomplemented
57#2067C2214yescomplemented
58#2066C2214yescomplemented
59#2065C2214yescomplemented
60#2055C2214yescomplemented
61#2055C2214yescomplemented
62#1973C2214yescomplemented
63#1968C2214yescomplemented
64#1944C2214yescomplemented
65#1942C2214yescomplemented
66#1936C2214yescomplemented
67#1935C2214yescomplemented
68#1934C2214yescomplemented
69#1933C2214yescomplemented
70#1895C2214yes—
71#1894C2214yescomplemented
72#1875C2214yes—
73#1873C2214yes—
74#1854C2214yes—
75#1854C2214yescomplemented
76#1824C2212yescomplemented
77#1823C2212yes—
78#1809C2214yescomplemented
79#1787C2212yescomplemented
80#1786C2212yes—
81C23×Q8C2311yescharacteristic
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×SD16C2314yes—
86C22×M16C2312yes—
87C22×M16C2312yes—
88C2×#123—22no—
89C2×#123—22no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×#121—21no—
97C2×#121—21no—
98C2×#121—24no—
99C2×#121—24no—
100C2×#121C2312yes—
101C2×#121C2312yes—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117D412yes—
105C2×#117D412yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4D411yescharacteristic
118C22×C22⋊C4D411yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#484—28no—
122#480—28no—
123#468—216no—
124#450—28no—
125#448—28no—
126#433—216no—
127#433—216no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131#423—28no—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140#407—24no—
141#407D418yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C4×M16—28no—
147C2×#97C2314yes—
148C2×C4×C8C2314yes—
149#400—24no—
150#399—24no—
151#398—24no—
152#397—24no—
153#397—24no—
154#397—24no—
155#394—24no—
156#394—24no—
157#391—22no—
158#391—22no—
159#391C2314yes—
160#391C2314yes—
161#390—22no—
162#390—22no—
163#390C2314yes—
164#390C2314yes—
165#389C2314yes—
166#389C2314yes—
167#389C2314yes—
168#389C2314yes—
169#388—24no—
170#388—24no—
171#387—24no—
172#387—24no—
173#386—22no—
174#386—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384—22no—
183#382—22no—
184#382—22no—
185#381—22no—
186#381—22no—
187#381—21no—
188#381—21no—
189#381—22no—
190#381—22no—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—21no—
196#379—21no—
197#379—22no—
198#379—22no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209C25—21no—
210C25C2×D411yescharacteristic
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8C2412yes—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—21no—
221C22×D4—22no—
222C22×D4—42no—
223C22×D4—42no—
224C22×D4—44no—
225C22×D4—84no—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D414yes—
240C23×C4#4114yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×SD16—28no—
245C2×SD16—44no—
246C2×SD16—44no—
247C2×M16—24no—
248C2×M16—42no—
249C2×M16—42no—
250C2×M16—42no—
251C2×M16—42no—
252C2×M16C2×D418yes—
253C22×C8C2414yes—
254C22×C8C2414yes—
255#123—84no—
256#123—84no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263#121—24no—
264#121—24no—
265#121—24no—
266#121—24no—
267#121—44no—
268#121—44no—
269#117—28no—
270#117—28no—
271#117—24no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—42no—
276#117—42no—
277#117—42no—
278#117—42no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2414yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334#107—28no—
335#107—44no—
336#106—44no—
337C4⋊C8—28no—
338C4⋊C8—28no—
339#103—28no—
340#103—44no—
341#103—44no—
342#102—28no—
343#102—44no—
344#102—44no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8C2×D418yes—
348C22⋊C8C2×D418yes—
349#97—28no—
350C4×C8—28no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—44no—
368C24—42no—
369C24—82no—
370C24—82no—
371C24#12114yes—
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374C24C22×D411yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—84no—
382C2×Q8—82no—
383C2×Q8—82no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—42no—
394C2×D4—84no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4—82no—
398C2×D4—82no—
399C2×D4—816no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#12114yes—
432C22×C4#12114yes—
433C22×C4#12114yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
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
448SD16—88no—
449M16—48no—
450M16—48no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8C22×D418yes—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
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—44no—
472C4⋊C4—44no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C42—24no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—44no—
502C42—44no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—88no—
524C23—168no—
525C23C2×#12112yes—
526C23C2×#12112yes—
527C23C2×#12112yes—
528C23C23×D411yescharacteristic
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—84no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—84no—
537D4—88no—
538D4—88no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4C2×#12112yes—
585C2×C4C23×D411yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4#58512yes—
588C2×C4#58512yes—
589C8—28no—
590C8—44no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—88no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—84no—
608C22—84no—
609C22—168no—
610C22C22×#12111yescharacteristic
611C22#344912yes—
612C22#358512yes—
613C22#358612yes—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—84no—
626C4#348512yes—
627C4#349111yescharacteristic
628C4#349111yescharacteristic
629C2—42no—
630C2—42no—
631C2—42no—
632C2—162no—
633C2#3508911yescharacteristic
634C2#6045411yescharacteristic
635C2#6045511yescharacteristic
636C2#6100414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2921C2semidirect
3. #35311C2semidirect
5. #33145C2semidirect
6. #29784C2semidirect
7. #24614C2semidirect
8. #24612C2semidirect
9. #24518C2semidirect
10. #24516C2semidirect
11. #21943C2semidirect
12. #21768C2semidirect
25. C2×#433C22semidirect
26. C2×#423C22semidirect
29. C2×C4×M16C22semidirect
30. #2855C22semidirect
31. #2855C22semidirect
49. #2317C22semidirect
50. #2310C22semidirect
54. #2087C22semidirect
55. #2086C22semidirect
56. #2068C22semidirect
57. #2067C22semidirect
58. #2066C22semidirect
59. #2065C22semidirect
60. #2055C22semidirect
61. #2055C22semidirect
62. #1973C22semidirect
63. #1968C22semidirect
64. #1944C22semidirect
65. #1942C22semidirect
66. #1936C22semidirect
67. #1935C22semidirect
68. #1934C22semidirect
69. #1933C22semidirect
71. #1894C22semidirect
75. #1854C22semidirect
76. #1824C22semidirect
78. #1809C22semidirect
79. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#29212
#353112
#352622
#331452
#297842
#246142
#246122
#245182
#245162
#219432
#217682
#217672