Finite Group Databasebeta

← #6475522

Subgroups — #6475522

Order 512.

1782 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35327C211yesmaximal, characteristic
2#35280C211yesmaximal, characteristic, complemented
3#33057C212yesmaximal, complemented
4#33051C212yesmaximal, complemented
5#30191C214yesmaximal, complemented
6#29342C214yesmaximal, complemented
7#29117C212yesmaximal, complemented
8#29115C212yesmaximal, complemented
9#27978C212yesmaximal, complemented
10#27968C212yesmaximal, complemented
11#25909C212yesmaximal, complemented
12#25864C212yesmaximal, complemented
13#21771C212yesmaximal, complemented
14#21770C212yesmaximal
15#20375C211yesmaximal, characteristic, complemented
16C2×#536C2211yescharacteristic
17C2×#535C2211yescharacteristic
18C2×#533C2211yescharacteristic, complemented
19C22×#124C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#489C2212yes—
22C2×#489C2212yes—
23C2×#408C2212yes—
24C2×C4×M16C2211yescharacteristic, complemented
25C22×#97C2211yescharacteristic, complemented
26#2857C2212yes—
27#2857C2212yescomplemented
28#2850C2212yes—
29#2850C2212yes—
30#2629C2212yes—
31#2629C2212yescomplemented
32#2620C2212yes—
33#2620C2212yes—
34#2582—22no—
35#2545—22no—
36#2544—22no—
37#2538—22no—
38#2534—22no—
39#2534—22no—
40#2417—22no—
41#2417—22no—
42#2316C2212yescomplemented
43#2311C2212yes—
44#2310C2211yescharacteristic
45#2309C2211yescharacteristic
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic, complemented
48#2100C2214yescomplemented
49#2099C2214yescomplemented
50#2072C2214yescomplemented
51#2071C2214yes—
52#2042C2214yes—
53#2042C2214yes—
54#2026C2214yescomplemented
55#2025C2214yes—
56#1993C2212yescomplemented
57#1993C2212yescomplemented
58#1959C2212yes—
59#1959C2212yes—
60#1936C2214yes—
61#1935C2214yescomplemented
62#1934C2214yes—
63#1933C2214yescomplemented
64#1895C2214yes—
65#1894C2214yescomplemented
66#1893C2214yes—
67#1892C2214yescomplemented
68#1875C2214yes—
69#1874C2214yes—
70#1873C2214yescomplemented
71#1872C2214yescomplemented
72#1836C2212yescomplemented
73#1836C2212yescomplemented
74#1825C2214yes—
75#1825C2214yes—
76#1787C2214yes—
77#1786C2214yes—
78#1750C2212yes—
79#1750C2212yescomplemented
80#1749C2212yes—
81#1749C2212yescomplemented
82#1729C2212yescomplemented
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C22×M16C2312yes—
87C23×C8C2312yes—
88#536—28no—
89#535—28no—
90#533—42no—
91#533—42no—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8C2312yes—
94C2×#127—21no—
95C2×#125C2312yes—
96C2×#124—21no—
97C2×#124—21no—
98C2×#124—22no—
99C2×#124—42no—
100C2×#124—42no—
101C2×#124C2312yes—
102C2×#124C2312yes—
103C2×#123—21no—
104C2×#123—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#120—21no—
113C2×#120—21no—
114C4×#41—42no—
115C4×#41—42no—
116C2×C4×Q8—22no—
117C2×C4×Q8D412yes—
118C2×C4×D4D412yes—
119C2×C4×D4D412yes—
120C2×C4×D4D412yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—22no—
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—22no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#489—44no—
134#489—44no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143#408—28no—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C4×M16—42no—
147C2×#97—42no—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×#97C2312yes—
151C2×C4×C8C2312yes—
152#398—24no—
153#398—24no—
154#396—22no—
155#396—22no—
156#396—22no—
157#396—22no—
158#396—24no—
159#396—24no—
160#394—22no—
161#394—22no—
162#394—22no—
163#394—22no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391—22no—
168#391C2312yes—
169#391C2312yes—
170#390—22no—
171#390—22no—
172#390C2312yes—
173#390C2312yes—
174#389—24no—
175#389—22no—
176#389—22no—
177#389C2314yes—
178#389C2314yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#384—22no—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#382C2312yes—
194#381—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—24no—
199#380C2314yes—
200#380C2314yes—
201#379—22no—
202C4×C4⋊C4—22no—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25—21no—
214C2×#41—41no—
215C2×#41—41no—
216C2×#41—42no—
217C2×#41#4118yes—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8—24no—
221C22×Q8—41no—
222C22×Q8—41no—
223C22×Q8—42no—
224C22×Q8C2×D411yescharacteristic
225C22×Q8C2412yes—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—41no—
231C22×D4—41no—
232C22×D4—42no—
233C22×D4C2×D411yescharacteristic
234C22×D4C2×D411yescharacteristic
235C22×D4C2×D411yescharacteristic
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×M16—24no—
246C2×M16—42no—
247C2×M16—42no—
248C22×C8—24no—
249C22×C8—42no—
250C22×C8—42no—
251C22×C8C2414yes—
252C22×C8C2414yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—44no—
256#127—44no—
257#125—42no—
258#125—42no—
259#124—24no—
260#124—24no—
261#124—42no—
262#124—42no—
263#124—48no—
264#124—44no—
265#124—44no—
266#123—44no—
267#123—44no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—48no—
271C22⋊Q8—84no—
272C22⋊Q8—84no—
273#121—28no—
274#121—28no—
275#120—44no—
276#120—44no—
277C4×Q8—44no—
278C4×Q8—48no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—44no—
282#117—24no—
283#117—24no—
284#117—41no—
285#117—41no—
286#117—41no—
287#117—41no—
288#117—41no—
289#117—41no—
290#117—42no—
291#117—42no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2414yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#103—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358#102—44no—
359C22⋊C8—28no—
360C22⋊C8—28no—
361#97—24no—
362#97—24no—
363#97—42no—
364C4×C8—42no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—82no—
380C24—82no—
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384#41—48no—
385#41—48no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—24no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—44no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—84no—
399C2×Q8—84no—
400C2×Q8#12414yes—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4#12414yes—
423C2×D4#12414yes—
424C2×D4#12414yes—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—84no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4C2511yescharacteristic
464M16—48no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—48no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C42—24no—
514C42—24no—
515C42—24no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42H(5,𝔽2)12yes—
526C42H(5,𝔽2)12yes—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—84no—
547C23—88no—
548C23—164no—
549C23—162no—
550C23—162no—
551C23C2×#12412yes—
552C23C2×#12412yes—
553C23C2×#12412yes—
554C23C23×D411yescharacteristic
555Q8—44no—
556Q8—44no—
557Q8—44no—
558Q8—88no—
559Q8—84no—
560D4—44no—
561D4—44no—
562D4—44no—
563D4—44no—
564D4—44no—
565D4—84no—
566D4—164no—
567D4—164no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—164no—
611C2×C4C2×#12412yes—
612C2×C4C2×#12412yes—
613C2×C4C2×#12412yes—
614C2×C4#53414yes—
615C2×C4C22×#4112yes—
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4#58512yes—
619C8—44no—
620C8—44no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—168no—
644C22C22×#12411yescharacteristic
645C22C2×#53411yescharacteristic
646C22#344911yescharacteristic
647C22#358512yes—
648C22#358612yes—
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—84no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#350512yes—
662C4#350512yes—
663C2—42no—
664C2—42no—
665C2—42no—
666C2—162no—
667C2#3501911yescharacteristic
668C2#6062512yes—
669C2#6075512yes—
670C2#6101711yescharacteristic
671C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35280C2semidirect
3. #33057C2semidirect
4. #33051C2semidirect
5. #30191C2semidirect
6. #29342C2semidirect
7. #29117C2semidirect
8. #29115C2semidirect
9. #27978C2semidirect
10. #27968C2semidirect
11. #25909C2semidirect
12. #25864C2semidirect
13. #21771C2semidirect
15. #20375C2semidirect
18. C2×#533C22semidirect
24. C2×C4×M16C22semidirect
25. C22×#97C22semidirect
27. #2857C22semidirect
31. #2629C22semidirect
42. #2316C22semidirect
47. C2×#384C22semidirect
48. #2100C22semidirect
49. #2099C22semidirect
50. #2072C22semidirect
54. #2026C22semidirect
56. #1993C22semidirect
57. #1993C22semidirect
61. #1935C22semidirect
63. #1933C22semidirect
65. #1894C22semidirect
67. #1892C22semidirect
70. #1873C22semidirect
71. #1872C22semidirect
72. #1836C22semidirect
73. #1836C22semidirect
79. #1750C22semidirect
81. #1749C22semidirect
82. #1729C22semidirect

Maximal subgroups

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

subgroupindex
#353272
#352802
#330572
#330512
#301912
#293422
#291172
#291152
#279782
#279682
#259092
#258642
#217712
#217702
#203752