Finite Group Databasebeta

← #6489160

Subgroups — #6489160

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic, complemented
3#32918C212yesmaximal, complemented
4#32917C212yesmaximal, complemented
5#30055C214yesmaximal, complemented
6#29148C212yesmaximal, complemented
7#28446C212yesmaximal, complemented
8#28172C214yesmaximal, complemented
9#27981C211yesmaximal, characteristic, complemented
10#27980C211yesmaximal, characteristic, complemented
11#25731C211yesmaximal, characteristic
12#25730C211yesmaximal, characteristic, complemented
13#23946C212yesmaximal, complemented
14#23945C212yesmaximal, complemented
15#22273C212yesmaximal, complemented
16#22272C212yesmaximal, complemented
17#20783C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C22×#124C2211yescharacteristic
20C2×#519C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#497C2212yes—
23C2×#496C2212yes—
24C2×#464C2214yescomplemented
25C2×#463C2214yes—
26C2×#451C2212yes—
27C2×#448C2212yes—
28C2×#450C2212yes—
29C2×#447C2212yes—
30#2868C2211yescharacteristic, complemented
31#2858C2211yescharacteristic, complemented
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2640C2212yescomplemented
35#2620C2212yes—
36#2512—22no—
37#2491—22no—
38#2452—22no—
39#2422—22no—
40#2367—22no—
41#2367—22no—
42#2342—22no—
43#2342—22no—
44#2326C2212yes—
45#2326C2212yes—
46#2326C2212yescomplemented
47#2310C2211yescharacteristic
48#2309C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic, complemented
50C4×#127C2211yescharacteristic, complemented
51C4×#124C2212yes—
52C4×#124C2212yes—
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2072C2214yescomplemented
56#2071C2214yescomplemented
57#2024C2214yescomplemented
58#2023C2214yes—
59#2015C2214yes—
60#2015C2214yescomplemented
61#2012C2214yes—
62#2012C2214yes—
63#1961C2212yescomplemented
64#1960C2212yescomplemented
65#1959C2212yes—
66#1959C2212yes—
67#1880C2214yes—
68#1880C2214yescomplemented
69#1879C2214yes—
70#1879C2214yes—
71#1878C2214yescomplemented
72#1878C2214yescomplemented
73#1877C2214yes—
74#1877C2214yescomplemented
75#1875C2212yes—
76#1875C2212yes—
77#1873C2212yes—
78#1873C2212yes—
79#1840C2212yescomplemented
80#1836C2212yescomplemented
81#1836C2212yescomplemented
82#1832C2212yescomplemented
83#1818C2212yes—
84#1803C2214yescomplemented
85#1802C2214yes—
86#1798C2212yes—
87#1767C2212yescomplemented
88#1749C2211yescharacteristic
89#1749C2211yescharacteristic
90#1745C2212yescomplemented
91C23×Q8C2311yescharacteristic
92C23×D4C2311yescharacteristic
93C22×Q16C2314yes—
94C22×SD16C2314yes—
95C22×SD16C2314yes—
96C22×D8C2314yes—
97#542—44no—
98#542—44no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#126—22no—
104C2×#124—28no—
105C2×#124—22no—
106C2×#124—22no—
107C2×#124C2312yes—
108C2×#124C2312yes—
109C2×#124C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8C2312yes—
114C2×#121—22no—
115C2×#121C2312yes—
116#519—44no—
117#518—44no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135#497—28no—
136#496—28no—
137#464—48no—
138#463—48no—
139#451—44no—
140#450—44no—
141#448—44no—
142#447—44no—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2312yes—
155C2×#97C2312yes—
156#397—24no—
157#396—24no—
158#394—24no—
159#393—24no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#390C2312yes—
176#389—22no—
177#389—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—24no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#384—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382C2312yes—
198#381—21no—
199#381—21no—
200#380—22no—
201#380—22no—
202#379—21no—
203#379—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D412yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214#376C2311yescharacteristic
215#376C2311yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2314yes—
226#336C2314yes—
227C25—21no—
228C22×Q8—24no—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—24no—
232C22×Q8—42no—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—24no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—24no—
240C22×D4—42no—
241C22×D4C2412yes—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2411yescharacteristic
249C2×Q16—28no—
250C2×Q16—44no—
251C2×Q16—44no—
252C2×SD16—28no—
253C2×SD16—28no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×SD16—44no—
258C2×D8—28no—
259C2×D8—44no—
260C2×D8—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—24no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—84no—
267#127—24no—
268#126—48no—
269#124—216no—
270#124—24no—
271#124—24no—
272#124—24no—
273#124—28no—
274#124—42no—
275#124—42no—
276#124—48no—
277#124#4118yes—
278#123—48no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—48no—
282#121—24no—
283#121—24no—
284#121—84no—
285C4×Q8—28no—
286C4×Q8—44no—
287C4×Q8—42no—
288C4×Q8—42no—
289C4×Q8—48no—
290C4×D4—28no—
291C4×D4—44no—
292C4×D4—42no—
293C4×D4—42no—
294C4×D4—48no—
295C4×D4—84no—
296#117—24no—
297#117—24no—
298#117—48no—
299#117—42no—
300#117—42no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—216no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—48no—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D411yescharacteristic
343C2×C22⋊C4C2×D411yescharacteristic
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C4⋊C8—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—44no—
367#102—44no—
368#102—44no—
369#102—44no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372#97—24no—
373#97—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—42no—
390C24—42no—
391C24—82no—
392C24—82no—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—416no—
403C2×Q8—42no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—84no—
407C2×Q8C2×#4114yes—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—416no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—42no—
423C2×D4—84no—
424C2×D4C2×#4114yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—44no—
442C22×C4—84no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4H(5,𝔽2)14yes—
457C22×C4C2511yescharacteristic
458Q16—88no—
459SD16—88no—
460SD16—88no—
461D8—88no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—44no—
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—48no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42C2×#4114yes—
526C42H(5,𝔽2)14yes—
527C23—22no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—24no—
532C23—48no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—88no—
545C23—162no—
546C23—162no—
547C23—164no—
548C23#53512yes—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—48no—
552Q8—44no—
553Q8—44no—
554Q8—44no—
555Q8—164no—
556D4—28no—
557D4—48no—
558D4—44no—
559D4—44no—
560D4—44no—
561D4—88no—
562D4—164no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—28no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—84no—
602C2×C4—164no—
603C2×C4—164no—
604C2×C4—164no—
605C2×C4#52514yes—
606C2×C4#53514yes—
607C2×C4#53512yes—
608C2×C4C22×#4112yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×H(5,𝔽2)12yes—
611C8—44no—
612C8—44no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22—84no—
629C22—82no—
630C22—168no—
631C22C2×#52511yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53511yescharacteristic
634C22#358512yes—
635C22#358612yes—
636C4—24no—
637C4—24no—
638C4—22no—
639C4—42no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4#351914yes—
652C2—42no—
653C2—84no—
654C2—162no—
655C2#3523711yescharacteristic
656C2#6056712yes—
657C2#6082711yescharacteristic
658C2#6082811yescharacteristic
659C2#6082912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#25731C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35277C2semidirect
2. #34211C2semidirect
3. #32918C2semidirect
4. #32917C2semidirect
5. #30055C2semidirect
6. #29148C2semidirect
7. #28446C2semidirect
8. #28172C2semidirect
9. #27981C2semidirect
10. #27980C2semidirect
12. #25730C2semidirect
13. #23946C2semidirect
14. #23945C2semidirect
15. #22273C2semidirect
16. #22272C2semidirect
17. #20783C2semidirect
24. C2×#464C22semidirect
30. #2868C22semidirect
31. #2858C22semidirect
34. #2640C22semidirect
46. #2326C22semidirect
49. C4×C4⋊Q8C22semidirect
50. C4×#127C22semidirect
55. #2072C22semidirect
56. #2071C22semidirect
57. #2024C22semidirect
60. #2015C22semidirect
63. #1961C22semidirect
64. #1960C22semidirect
68. #1880C22semidirect
71. #1878C22semidirect
72. #1878C22semidirect
74. #1877C22semidirect
79. #1840C22semidirect
80. #1836C22semidirect
81. #1836C22semidirect
82. #1832C22semidirect
84. #1803C22semidirect
87. #1767C22semidirect
90. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#352772
#342112
#329182
#329172
#300552
#291482
#284462
#281722
#279812
#279802
#257312
#257302
#239462
#239452
#222732
#222722
#207832