Finite Group Databasebeta

← #6436923

Subgroups — #6436923

Order 512.

1741 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#29940C214yesmaximal, complemented
4#29924C214yesmaximal, complemented
5#29913C212yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#29894C212yesmaximal, complemented
8#29891C212yesmaximal
9#23765C214yesmaximal, complemented
10#23764C214yesmaximal, complemented
11#21391C211yesmaximal, characteristic, complemented
12#21391C211yesmaximal, characteristic, complemented
13#21363C211yesmaximal, characteristic, complemented
14#21363C211yesmaximal, characteristic, complemented
15#20715C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2855C2211yescharacteristic
21#2854C2211yescharacteristic
22#2853—21no—
23#2852—22no—
24#2851C2212yes—
25#2850C2212yes—
26#2849—21no—
27#2848C2211yescharacteristic
28#2847C2211yescharacteristic
29C4×C22⋊Q8C2212yes—
30C4×#123—21no—
31C4×#121C2212yes—
32C4×#120—21no—
33#2290—22no—
34#2283—22no—
35#2283—22no—
36#2283C2211yescharacteristic
37#2283C2214yes—
38#2283C2214yes—
39C2×#383—21no—
40C2×#382C2212yes—
41C2×#381—21no—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4—21no—
46C2×#376—21no—
47#2101C2214yescomplemented
48#2101C2214yescomplemented
49#2097C2214yescomplemented
50#2092C2214yescomplemented
51#2085C2214yescomplemented
52#2085C2214yescomplemented
53#2077C2214yes—
54#2077C2214yes—
55#2076C2214yes—
56#2075C2214yes—
57#2074C2214yescomplemented
58#2073C2214yescomplemented
59#2036C2214yes—
60#2035C2214yes—
61#2034C2214yes—
62#2033C2214yes—
63#1989C2212yescomplemented
64#1989C2212yescomplemented
65#1983C2212yescomplemented
66#1983C2212yescomplemented
67#1949C2214yescomplemented
68#1949C2214yescomplemented
69#1874C2212yes—
70#1874C2212yes—
71#1872C2212yes—
72#1872C2212yes—
73#1853C2212yes—
74#1853C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1817C2212yes—
78#1797C2214yes—
79#1779C2212yes—
80#1776C2214yescomplemented
81#1776C2214yescomplemented
82#1763C2212yescomplemented
83#1763C2212yescomplemented
84#1762C2212yescomplemented
85#1762C2212yescomplemented
86#1749C2214yes—
87C24×C4D411yescharacteristic
88C24×C4D411yescharacteristic
89C24×C4C2311yescharacteristic
90#529—24no—
91#529—24no—
92C2×#126—21no—
93C2×#125C2311yescharacteristic
94C2×#124C2311yescharacteristic
95C2×#123—21no—
96C2×#123—21no—
97C2×C22⋊Q8C2311yescharacteristic
98C2×C22⋊Q8C2312yes—
99C2×#121C2311yescharacteristic
100C2×#121C2312yes—
101C2×#120—21no—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4—22no—
105C2×C4×D4C2311yescharacteristic
106C2×#117—21no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—22no—
110C22×C42—24no—
111C22×C42—24no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42D412yes—
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D412yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122#432—24no—
123#432—24no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2312yes—
139C2×#97C2312yes—
140#391C2312yes—
141#390C2312yes—
142#389C2312yes—
143#389C2312yes—
144#388—22no—
145#388—22no—
146#388—22no—
147#388—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#386—22no—
153#386—21no—
154#386—22no—
155#386—22no—
156#386—22no—
157#386—21no—
158#385—22no—
159#385—21no—
160#385—22no—
161#385—22no—
162#385—22no—
163#385—21no—
164#383—21no—
165#383—24no—
166#383—24no—
167#382—22no—
168#382—22no—
169#382C2314yes—
170#382C2312yes—
171#382C2314yes—
172#381—21no—
173#381—21no—
174#381—24no—
175#381—24no—
176#380—22no—
177#380—22no—
178#380C2312yes—
179#380C2314yes—
180#380C2312yes—
181#380C2314yes—
182#380C2311yescharacteristic
183#380C2311yescharacteristic
184#379—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—24no—
196C4×C22⋊C4—24no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200#376—24no—
201#376—24no—
202#376—21no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206C2×#95—21no—
207C2×#95—21no—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C22×Q8C2411yescharacteristic
220C22×D4—21no—
221C22×D4—22no—
222C22×D4C2411yescharacteristic
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—44no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—44no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C22×C8C2414yes—
247C22×C8C2414yes—
248#126—44no—
249#125—22no—
250#125—22no—
251#124—22no—
252#124—22no—
253#123—44no—
254#123—44no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—44no—
258#121—22no—
259#121—22no—
260#121—44no—
261#120—44no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—48no—
267C4×D4—44no—
268#117—44no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2411yescharacteristic
283C2×C4⋊C4C2411yescharacteristic
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—48no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42#4114yes—
328C2×C42#4114yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—44no—
333#107—44no—
334#106—44no—
335#106—44no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338#103—44no—
339#103—44no—
340#102—44no—
341#102—44no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C22⋊C8—44no—
345C22⋊C8—44no—
346#97—24no—
347#97—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—42no—
375C24C22×D412yes—
376C24C22×D411yescharacteristic
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—24no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—24no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—42no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—42no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4#12314yes—
439C22×C4#12314yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2511yescharacteristic
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—42no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—44no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—44no—
503C42—42no—
504C42#12314yes—
505C42#12314yes—
506C42C2×#4112yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—48no—
520C23—48no—
521C23—44no—
522C23—44no—
523C23—41no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23C2×#12312yes—
529C23#57514yes—
530C23#57512yes—
531C23#57512yes—
532C23C23×D411yescharacteristic
533Q8—42no—
534Q8—42no—
535D4—42no—
536D4—42no—
537D4—88no—
538D4—84no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4C2×#12312yes—
592C2×C4C2×#12312yes—
593C2×C4C2×#12312yes—
594C2×C4C2×#12312yes—
595C2×C4C2×#12312yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4#58514yes—
599C8—44no—
600C8—44no—
601C22—24no—
602C22—24no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—416no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22#309314yes—
619C22C22×#12311yescharacteristic
620C22#344912yes—
621C22C2×#57511yescharacteristic
622C22C2×#57512yes—
623C22C2×#57511yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—82no—
633C4—84no—
634C4—84no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4—81no—
639C4#345512yes—
640C4#345512yes—
641C2—24no—
642C2—44no—
643C2—161no—
644C2#3463511yescharacteristic
645C2C2×#309311yescharacteristic
646C2#6051111yescharacteristic
647C2#6095912yes—
648C2#6096412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #29940C2semidirect
4. #29924C2semidirect
5. #29913C2semidirect
6. #29912C2semidirect
7. #29894C2semidirect
9. #23765C2semidirect
10. #23764C2semidirect
11. #21391C2semidirect
12. #21391C2semidirect
13. #21363C2semidirect
14. #21363C2semidirect
15. #20715C2semidirect
47. #2101C22semidirect
48. #2101C22semidirect
49. #2097C22semidirect
50. #2092C22semidirect
51. #2085C22semidirect
52. #2085C22semidirect
57. #2074C22semidirect
58. #2073C22semidirect
63. #1989C22semidirect
64. #1989C22semidirect
65. #1983C22semidirect
66. #1983C22semidirect
67. #1949C22semidirect
68. #1949C22semidirect
80. #1776C22semidirect
81. #1776C22semidirect
82. #1763C22semidirect
83. #1763C22semidirect
84. #1762C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
#366212
C2×#22832
#299402
#299242
#299132
#299122
#298942
#298912
#237652
#237642
#213912
#213632
#207152