Finite Group Databasebeta

← #6510728

Subgroups — #6510728

Order 512.

1760 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#29110C212yesmaximal, complemented
4#29103C212yesmaximal, complemented
5#27969C212yesmaximal, complemented
6#27964C212yesmaximal, complemented
7#27725C214yesmaximal, complemented
8#27711C214yesmaximal, complemented
9#27703C214yesmaximal, complemented
10#27701C214yesmaximal, complemented
11#21764C211yesmaximal, characteristic
12#21761C211yesmaximal, characteristic
13#20700C211yesmaximal, characteristic, complemented
14C2×#1823C211yesmaximal, characteristic, complemented
15#20525C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2857C2212yes—
21#2851C2212yes—
22#2667—21no—
23#2665—21no—
24#2569—21no—
25#2549—21no—
26#2504—22no—
27#2504—21no—
28#2504—21no—
29#2504C2212yes—
30#2504C2214yes—
31#2504C2212yes—
32#2442C2212yescomplemented
33#2441C2212yescomplemented
34#2440C2212yescomplemented
35#2439C2212yescomplemented
36#2437—22no—
37C2×#397—21no—
38C2×#397—21no—
39#2329—22no—
40C2×#390C2212yes—
41C2×#386—22no—
42C2×#382C2212yes—
43#2280C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#2101C2214yescomplemented
47#2101C2214yescomplemented
48#2101C2214yescomplemented
49#2101C2214yescomplemented
50#2073C2214yescomplemented
51#2071C2214yescomplemented
52#2025C2214yescomplemented
53#2023C2214yescomplemented
54#1993C2214yescomplemented
55#1991C2214yescomplemented
56#1936C2214yescomplemented
57#1936C2214yescomplemented
58#1934C2214yescomplemented
59#1934C2214yescomplemented
60#1895C2212yes—
61#1895C2212yes—
62#1893C2214yes—
63#1875C2214yes—
64#1874C2214yes—
65#1836C2214yescomplemented
66#1835C2214yescomplemented
67#1824C2211yescharacteristic
68#1824C2211yescharacteristic
69#1824C2212yes—
70#1824C2214yescomplemented
71#1823C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1823C2212yescomplemented
76#1823C2212yescomplemented
77#1786C2211yescharacteristic
78#1786C2211yescharacteristic
79#1786C2212yes—
80#1753C2214yescomplemented
81#1752C2212yescomplemented
82C8×C22⋊C4C2212yescomplemented
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2214yescomplemented
86C2×#340C2214yescomplemented
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#125C2312yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×#121—21no—
98C2×#121—21no—
99C2×#121—24no—
100C2×#121—24no—
101C2×#121C2312yes—
102C2×#121C2312yes—
103C2×#117C2312yes—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—22no—
117C22×C22⋊C4—22no—
118C22×C22⋊C4D412yes—
119C22×C42C2312yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124#407—24no—
125C2×C22⋊C8—24no—
126C2×C22⋊C8C2311yescharacteristic
127C2×C22⋊C8C2311yescharacteristic
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2314yes—
132C2×C4×C8C2314yes—
133#400—22no—
134#399—22no—
135#398—22no—
136#398—22no—
137#397—24no—
138#397—24no—
139#397—22no—
140#397—22no—
141#397—22no—
142#397—22no—
143#396—22no—
144#396—22no—
145#395—22no—
146#393—22no—
147#391—22no—
148#391—22no—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#390C2314yes—
153#389—22no—
154#389—22no—
155#389C2312yes—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#386—24no—
164#386—24no—
165#386—24no—
166#386—24no—
167#385—22no—
168#385—22no—
169#385—22no—
170#385—22no—
171#384C2312yes—
172#382—22no—
173#382—22no—
174#382C2312yes—
175#382C2314yes—
176#382C2314yes—
177#381—21no—
178#380—22no—
179#380—21no—
180#380—21no—
181#380—22no—
182#380C2311yescharacteristic
183#380C2312yes—
184#380C2311yescharacteristic
185#380C2311yescharacteristic
186#380C2311yescharacteristic
187#380C2312yes—
188#379—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4D411yescharacteristic
193C4×C22⋊C4D411yescharacteristic
194C4×C22⋊C4C2312yes—
195C4×C22⋊C4C2311yescharacteristic
196C4×C22⋊C4C2311yescharacteristic
197C2×#95—21no—
198C2×#95—22no—
199C2×#95—22no—
200C2×#95D411yescharacteristic
201C2×#95D411yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25—21no—
217C25C2×D411yescharacteristic
218C22×Q8—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—44no—
225C22×D4—48no—
226C22×D4—44no—
227C22×D4C2412yes—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—82no—
236C23×C4—82no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4#4114yes—
243C23×C4#4114yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×M16—24no—
248C2×M16—44no—
249C22×C8—24no—
250C22×C8—44no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—44no—
255#125—44no—
256C22⋊Q8—48no—
257C22⋊Q8—84no—
258#121—24no—
259#121—24no—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—44no—
264#117—44no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
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—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—84no—
314C2×C22⋊C4—84no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—44no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2412yes—
325#102—44no—
326#102—44no—
327#102—44no—
328#102—44no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334#97—44no—
335C4×C8—44no—
336#95—21no—
337#95—22no—
338#95—21no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—21no—
344#95—21no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D411yescharacteristic
355#95C2×D411yescharacteristic
356#95C2×D411yescharacteristic
357#95C2×D411yescharacteristic
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—42no—
363C24—44no—
364C24—44no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24#12414yes—
369C24C22×D411yescharacteristic
370C24C22×D411yescharacteristic
371C24C22×D411yescharacteristic
372C2×Q8—44no—
373C2×Q8—82no—
374C2×Q8—82no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—816no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—816no—
423C22×C4#12414yes—
424C22×C4#12414yes—
425C22×C4#12414yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2511yescharacteristic
440M16—84no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—84no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—82no—
488C42—24no—
489C42—24no—
490C42—42no—
491C42—42no—
492C42—82no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—28no—
499C23—28no—
500C23—22no—
501C23—22no—
502C23—21no—
503C23—21no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—84no—
513C23—84no—
514C23—88no—
515C23—88no—
516C23—84no—
517C23C2×#12412yes—
518C23C2×#12412yes—
519C23C2×#12412yes—
520C23#57514yes—
521C23#57612yes—
522C23#57712yes—
523C23C23×D411yescharacteristic
524Q8—164no—
525D4—44no—
526D4—42no—
527D4—42no—
528D4—84no—
529D4—88no—
530D4—88no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4—88no—
578C2×C4—164no—
579C2×C4C2×#12412yes—
580C2×C4C2×#12412yes—
581C2×C4C2×#12412yes—
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4#58514yes—
585C8—42no—
586C8—42no—
587C8—44no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—416no—
600C22—42no—
601C22—42no—
602C22—84no—
603C22—84no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—168no—
608C22#313614yes—
609C22#314014yes—
610C22C22×#12411yescharacteristic
611C22#348512yes—
612C22C2×#57512yes—
613C22C2×#57611yescharacteristic
614C22C2×#57711yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4#349112yes—
629C4#349112yes—
630C2—24no—
631C2—24no—
632C2—42no—
633C2—162no—
634C2#3545511yescharacteristic
635C2C2×#313611yescharacteristic
636C2C2×#314011yescharacteristic
637C2#6172712yes—
638C2#6173612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35079C2semidirect
3. #29110C2semidirect
4. #29103C2semidirect
5. #27969C2semidirect
6. #27964C2semidirect
7. #27725C2semidirect
8. #27711C2semidirect
9. #27703C2semidirect
10. #27701C2semidirect
13. #20700C2semidirect
14. C2×#1823C2semidirect
15. #20525C2semidirect
32. #2442C22semidirect
33. #2441C22semidirect
34. #2440C22semidirect
35. #2439C22semidirect
46. #2101C22semidirect
47. #2101C22semidirect
48. #2101C22semidirect
49. #2101C22semidirect
50. #2073C22semidirect
51. #2071C22semidirect
52. #2025C22semidirect
53. #2023C22semidirect
54. #1993C22semidirect
55. #1991C22semidirect
56. #1936C22semidirect
57. #1936C22semidirect
58. #1934C22semidirect
59. #1934C22semidirect
65. #1836C22semidirect
66. #1835C22semidirect
70. #1824C22semidirect
75. #1823C22semidirect
76. #1823C22semidirect
80. #1753C22semidirect
81. #1752C22semidirect
82. C8×C22⋊C4C22semidirect
85. #1736C22semidirect
86. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
#350792
C2×#25042
#291102
#291032
#279692
#279642
#277252
#277112
#277032
#277012
#217642
#217612
#207002
C2×#18232
#205252