Finite Group Databasebeta

← #6508846

Subgroups — #6508846

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3094C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic, complemented
3#35076C211yesmaximal, characteristic
4#34809C211yesmaximal, characteristic, complemented
5#28110C214yesmaximal, complemented
6#27671C214yesmaximal, complemented
7#25260C214yesmaximal, complemented
8#25258C214yesmaximal, complemented
9#24605C214yesmaximal, complemented
10#24560C214yesmaximal, complemented
11#21943C211yesmaximal, characteristic, complemented
12#21767C211yesmaximal, characteristic, complemented
13#20701C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3094—28no—
18C2×#481C2212yescomplemented
19C2×#482C2214yes—
20C2×#480C2212yescomplemented
21#2975—28no—
22C2×#436C2214yes—
23C2×#436C2214yescomplemented
24C2×#431C2212yescomplemented
25C2×#428C2212yescomplemented
26C2×#418C2212yescomplemented
27C2×#407C2211yescharacteristic
28C2×#407C2211yescharacteristic
29C2×#407C2212yescomplemented
30#2854C2212yes—
31#2847C2212yes—
32#2635—22no—
33#2591—21no—
34#2539—21no—
35#2487C2212yes—
36#2487C2212yescomplemented
37#2486C2212yescomplemented
38#2485C2214yes—
39#2484—21no—
40#2483—21no—
41#2482—21no—
42#2414C2214yes—
43#2408C2212yes—
44#2405—21no—
45#2405—21no—
46#2403—21no—
47#2395—22no—
48#2321—22no—
49C4×#123—22no—
50#2296C2212yescomplemented
51#2280C2211yescharacteristic
52#2280C2211yescharacteristic
53#2280C2212yescomplemented
54#2088C2214yescomplemented
55#2087C2214yescomplemented
56#2077C2214yescomplemented
57#2075C2214yescomplemented
58#2067C2214yescomplemented
59#2065C2214yescomplemented
60#2034C2214yescomplemented
61#2033C2214yescomplemented
62#1942C2214yescomplemented
63#1941C2214yescomplemented
64#1934C2214yescomplemented
65#1934C2214yescomplemented
66#1895C2214yescomplemented
67#1895C2214yescomplemented
68#1895C2214yescomplemented
69#1875C2214yes—
70#1854C2214yescomplemented
71#1854C2214yescomplemented
72#1825C2214yescomplemented
73#1825C2214yescomplemented
74#1823C2212yescomplemented
75#1809C2214yescomplemented
76#1809C2214yescomplemented
77#1809C2214yescomplemented
78#1786C2212yescomplemented
79#1736C2212yescomplemented
80#1736C2212yescomplemented
81C23×D4—21no—
82C23×D4D411yescharacteristic
83C23×D4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C22×M16C2314yes—
87C22×M16C2312yes—
88C22×M16C2312yes—
89C2×#123—22no—
90C2×#123—22no—
91C2×#121—21no—
92C2×#121—21no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121C2312yes—
96C2×#121C2312yes—
97C2×#121C2314yes—
98C2×#120—22no—
99C2×#120—22no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4D411yescharacteristic
121C22×C22⋊C4D411yescharacteristic
122C22×C42—21no—
123C22×C42C2311yescharacteristic
124#482—216no—
125#481—28no—
126#480—28no—
127#436—216no—
128#436—216no—
129#431—28no—
130#428—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136#418—28no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141#407—24no—
142#407—28no—
143#407D418yescomplemented
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#400—22no—
152#399—22no—
153#399—22no—
154#398—22no—
155#397—22no—
156#397—24no—
157#394—24no—
158#394—22no—
159#394—24no—
160#392—24no—
161#392—22no—
162#390—24no—
163#390C2314yes—
164#389C2314yes—
165#389C2314yes—
166#389C2314yes—
167#388—24no—
168#388—24no—
169#388—24no—
170#387—24no—
171#386—22no—
172#386—22no—
173#386—24no—
174#386—22no—
175#386—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#381—21no—
181#381—21no—
182#381—22no—
183#381—22no—
184#380C2314yes—
185#380C2312yes—
186#380C2312yes—
187#380C2314yes—
188#380C2314yes—
189#380C2312yes—
190#380C2312yes—
191#379—21no—
192#379—22no—
193#379—21no—
194#379—22no—
195C4×C4⋊C4—24no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—24no—
200C4×C22⋊C4—21no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209C25—21no—
210C25C2×D411yescharacteristic
211C25C2×D411yescharacteristic
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—22no—
219C22×D4—44no—
220C22×D4—42no—
221C22×D4—42no—
222C22×D4—44no—
223C22×D4—84no—
224C22×D4C2×D412yes—
225C22×D4C2×D412yes—
226C22×D4C2412yes—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D414yes—
244C23×C4#4114yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—24no—
250C2×M16—28no—
251C2×M16—42no—
252C2×M16—42no—
253C2×M16—44no—
254C2×M16—44no—
255C2×M16—42no—
256C2×M16—42no—
257C2×M16C2×D418yes—
258C22×C8C2414yes—
259C22×C8C2414yes—
260#123—48no—
261#123—84no—
262#121—24no—
263#121—24no—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—44no—
268#121—44no—
269#121—44no—
270#120—48no—
271#120—84no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—24no—
278#117—24no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—42no—
283#117—42no—
284#117—44no—
285#117—44no—
286#117—42no—
287#117—42no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2414yes—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—44no—
339C2×C42C2×D412yes—
340C2×C42C2412yes—
341#107—28no—
342#107—44no—
343#106—28no—
344#106—44no—
345C4⋊C8—28no—
346#102—28no—
347#102—28no—
348#102—44no—
349#102—44no—
350C22⋊C8—24no—
351C22⋊C8—24no—
352C22⋊C8—28no—
353C22⋊C8C2×D418yes—
354C22⋊C8C2×D418yes—
355C4×C8—28no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366C24—22no—
367C24—22no—
368C24—42no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—44no—
375C24—42no—
376C24—82no—
377C24—82no—
378C24#12114yes—
379C24C22×D412yes—
380C24C22×D411yescharacteristic
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—42no—
401C2×D4—82no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—816no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—84no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4#12114yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2511yescharacteristic
463M16—48no—
464M16—88no—
465M16—84no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8C22×D418yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—48no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—82no—
507C22⋊C4—84no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C42—24no—
511C42—22no—
512C42—22no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—82no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—82no—
541C23—82no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—88no—
546C23—88no—
547C23—168no—
548C23C2×#12112yes—
549C23C2×#12112yes—
550C23C2×#12112yes—
551C23C23×D411yescharacteristic
552D4—48no—
553D4—48no—
554D4—44no—
555D4—42no—
556D4—42no—
557D4—84no—
558D4—88no—
559D4—88no—
560D4—168no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—84no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4C2×#12112yes—
606C2×C4C2×#12112yes—
607C2×C4C2×#12112yes—
608C2×C4#57514yes—
609C2×C4C23×D411yescharacteristic
610C2×C4C22×#4111yescharacteristic
611C2×C4#58512yes—
612C2×C4#58512yes—
613C8—28no—
614C8—44no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—88no—
633C22—88no—
634C22—82no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—82no—
639C22—168no—
640C22C22×#12111yescharacteristic
641C22#348512yes—
642C22C2×#57512yes—
643C22#358512yes—
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—82no—
651C4—84no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4#348512yes—
656C4#349111yescharacteristic
657C4#349111yescharacteristic
658C2—42no—
659C2—42no—
660C2—42no—
661C2—82no—
662C2—162no—
663C2#3544711yescharacteristic
664C2#6044411yescharacteristic
665C2#6045411yescharacteristic
666C2#6172514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35076C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3094C2semidirect
2. C2×#2975C2semidirect
4. #34809C2semidirect
5. #28110C2semidirect
6. #27671C2semidirect
7. #25260C2semidirect
8. #25258C2semidirect
9. #24605C2semidirect
10. #24560C2semidirect
11. #21943C2semidirect
12. #21767C2semidirect
13. #20701C2semidirect
18. C2×#481C22semidirect
20. C2×#480C22semidirect
23. C2×#436C22semidirect
24. C2×#431C22semidirect
25. C2×#428C22semidirect
26. C2×#418C22semidirect
29. C2×#407C22semidirect
36. #2487C22semidirect
37. #2486C22semidirect
50. #2296C22semidirect
53. #2280C22semidirect
54. #2088C22semidirect
55. #2087C22semidirect
56. #2077C22semidirect
57. #2075C22semidirect
58. #2067C22semidirect
59. #2065C22semidirect
60. #2034C22semidirect
61. #2033C22semidirect
62. #1942C22semidirect
63. #1941C22semidirect
64. #1934C22semidirect
65. #1934C22semidirect
66. #1895C22semidirect
67. #1895C22semidirect
68. #1895C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
72. #1825C22semidirect
73. #1825C22semidirect
74. #1823C22semidirect
75. #1809C22semidirect
76. #1809C22semidirect
77. #1809C22semidirect
78. #1786C22semidirect
79. #1736C22semidirect
80. #1736C22semidirect
143. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30942
C2×#29752
#350762
#348092
#281102
#276712
#252602
#252582
#246052
#245602
#219432
#217672
#207012