Finite Group Databasebeta

← #6451982

Subgroups — #6451982

Order 512.

3001 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3004C211yesmaximal, characteristic
2C2×#2920C211yesmaximal, characteristic, complemented
3#36626C211yesmaximal, characteristic
4C22×#391C211yesmaximal, characteristic, complemented
5#24509C214yesmaximal, complemented
6#24508C214yesmaximal, complemented
7#24503C214yesmaximal, complemented
8#24502C214yesmaximal, complemented
9#23619C212yesmaximal, complemented
10#23547C212yesmaximal, complemented
11#21237C212yesmaximal, complemented
12#21094C212yesmaximal, complemented
13#20909C211yesmaximal, characteristic
14#20844C211yesmaximal, characteristic, complemented
15#20599C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#137C2212yes—
18C2×#525C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#3004—28no—
22C2×#451C2214yes—
23C2×#448C2214yes—
24#2920—28no—
25C2×#432C2214yes—
26C2×#431C2214yes—
27C2×#428C2212yes—
28C2×#423C2212yes—
29C2×#417C2212yes—
30C2×#419C2212yes—
31C2×#419C2212yes—
32C2×#419C2212yes—
33C2×#407C2211yescharacteristic
34C2×#407C2211yescharacteristic
35C2×C4×M16C2212yes—
36#2855C2212yes—
37#2855C2212yes—
38#2854C2212yes—
39#2854C2212yes—
40#2853—22no—
41#2849—22no—
42C4×#123—22no—
43C4×#123—22no—
44#2297C2211yescharacteristic
45#2297C2211yescharacteristic
46#2296C2211yescharacteristic
47#2296C2211yescharacteristic
48C2×#391—22no—
49C2×#391—22no—
50C2×#391—22no—
51C2×#391C2212yes—
52C2×#391C2212yes—
53C2×#391C2214yes—
54C2×#391C2214yes—
55#2280C2211yescharacteristic
56C22×#95—22no—
57#2068C2218yescomplemented
58#2066C2218yescomplemented
59#2056C2218yescomplemented
60#2055C2218yescomplemented
61#1973C2214yescomplemented
62#1971C2214yescomplemented
63#1969C2214yescomplemented
64#1968C2214yescomplemented
65#1935C2218yescomplemented
66#1933C2218yescomplemented
67#1874C2214yes—
68#1872C2214yes—
69#1853C2214yes—
70#1852C2214yes—
71#1787C2214yes—
72#1787C2214yes—
73#1736C2212yes—
74#1736C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—28no—
77C23×Q8—22no—
78C23×Q8—21no—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2314yes—
84C24×C4—21no—
85C24×C4—21no—
86C24×C4D412yes—
87C24×C4C2311yescharacteristic
88C2×#137—28no—
89C2×#137—28no—
90C2×#137—44no—
91C22×Q16C2314yes—
92C22×SD16C2314yes—
93C22×M16C2312yes—
94C22×M16C2312yes—
95#525—24no—
96#525—24no—
97C2×#123—22no—
98C2×#123—22no—
99C2×#123—21no—
100C2×C22⋊Q8C2312yes—
101C2×#121C2312yes—
102C2×#120—21no—
103C2×C4×D4—22no—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42D412yes—
115C22×C42D412yes—
116C22×C42D412yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#451—216no—
122#448—216no—
123#432—216no—
124#431—216no—
125#428—44no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128#423—28no—
129C2×C4⋊C8C2318yes—
130#419—28no—
131#419—28no—
132#419—44no—
133#417—44no—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140#407—24no—
141#407D418yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2314yes—
144C4×M16—28no—
145C2×#97C2314yes—
146C2×C4×C8C2314yes—
147#391—22no—
148#391—22no—
149#391—24no—
150#391—24no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391C2318yes—
155#391C2312yes—
156#391C2311yescharacteristic
157#391C2311yescharacteristic
158#391C2314yes—
159#391C2314yes—
160#390C2312yes—
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#385—22no—
183#381—22no—
184#381—22no—
185#381—21no—
186#380C2312yes—
187#380C2312yes—
188#379—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C2×#95—24no—
197C2×#95—24no—
198C2×#95—24no—
199C2×#95—24no—
200C2×#95—24no—
201C2×#95—24no—
202#340C2318yes—
203#340C2318yes—
204C25C2×D411yescharacteristic
205C2×#41—22no—
206C2×#41—22no—
207C2×#41—24no—
208C2×#41—24no—
209C22×Q8—28no—
210C22×Q8—216no—
211C22×Q8—216no—
212C22×Q8—24no—
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—28no—
217C22×Q8—24no—
218C22×Q8—28no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—48no—
222C22×Q8—432no—
223C22×Q8—432no—
224C22×Q8—44no—
225C22×Q8—48no—
226C22×Q8—48no—
227C22×Q8—44no—
228C22×Q8#4118yes—
229C22×Q8C2411yescharacteristic
230C22×Q8C2414yes—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4C2411yescharacteristic
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—28no—
237C23×C4—28no—
238C23×C4—24no—
239C23×C4—22no—
240C23×C4—28no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—21no—
244C23×C4—44no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—48no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4C2×D412yes—
258C23×C4#4114yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261#137—432no—
262C2×Q16—28no—
263C2×Q16—28no—
264C2×Q16—44no—
265C2×SD16—28no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16C2×D418yes—
274C22×C8C2414yes—
275C22×C8C2414yes—
276#123—48no—
277#123—44no—
278#123—48no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281#121—24no—
282#121—24no—
283#120—44no—
284C4×D4—48no—
285#117—22no—
286#117—22no—
287#117—24no—
288#117—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—48no—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
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—22no—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341#107—44no—
342#106—44no—
343C4⋊C8—216no—
344#103—216no—
345#103—216no—
346#103—44no—
347#103—44no—
348#103—44no—
349#102—44no—
350C22⋊C8—28no—
351C22⋊C8C2×D4116yes—
352#97—28no—
353C4×C8—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—21no—
373C24#12114yes—
374C24C22×D411yescharacteristic
375C24C22×D412yes—
376#41—48no—
377#41—48no—
378C2×Q8—216no—
379C2×Q8—28no—
380C2×Q8—24no—
381C2×Q8—216no—
382C2×Q8—216no—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—24no—
386C2×Q8—21no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—432no—
392C2×Q8—432no—
393C2×Q8—432no—
394C2×Q8—48no—
395C2×Q8—464no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8C2×#4118yes—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—44no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—44no—
411C2×D4—44no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—216no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—432no—
447C22×C4—432no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—816no—
461C22×C4#12118yes—
462C22×C4#12114yes—
463C22×C4#13718yescomplemented
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4C2511yescharacteristic
477Q16—416no—
478SD16—416no—
479M16—216no—
480M16—48no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8C22×D418yes—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C42—24no—
513C42—22no—
514C42—24no—
515C42—24no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—21no—
523C42—22no—
524C42—21no—
525C42—42no—
526C42—42no—
527C42—42no—
528C23—28no—
529C23—28no—
530C23—216no—
531C23—28no—
532C23—22no—
533C23—22no—
534C23—216no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23C2×#12112yes—
545C23C2×#12114yes—
546C23#52512yes—
547C23#53512yes—
548C23C2×#13714yes—
549C23#57514yes—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—28no—
553Q8—432no—
554Q8—416no—
555Q8—44no—
556Q8—44no—
557Q8—42no—
558Q8—42no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4—88no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—28no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—216no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—216no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—432no—
586C2×C4—44no—
587C2×C4—416no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—88no—
606C2×C4—816no—
607C2×C4—84no—
608C2×C4C2×#12114yes—
609C2×C4C2×#12112yes—
610C2×C4#52512yes—
611C2×C4#53512yes—
612C2×C4#54618yes—
613C2×C4#55118yes—
614C2×C4C2×#13714yes—
615C2×C4C2×#13714yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4112yes—
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×H(5,𝔽2)12yes—
620C8—28no—
621C8—44no—
622C22—232no—
623C22—24no—
624C22—24no—
625C22—22no—
626C22—24no—
627C22—22no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—81no—
638C22#241114yes—
639C22#305618yes—
640C22C22×#12111yescharacteristic
641C22C2×#52511yescharacteristic
642C22C2×#53511yescharacteristic
643C22C22×#13712yes—
644C22C2×#57512yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—42no—
649C4—42no—
650C4—48no—
651C4—82no—
652C4—81no—
653C4—84no—
654C4—81no—
655C4—82no—
656C4—81no—
657C4#241114yes—
658C4C2×#54612yes—
659C4C2×#55112yes—
660C2—24no—
661C2—28no—
662C2—161no—
663C2C2×#241111yescharacteristic
664C2C2×#305612yes—
665C2#6056012yes—
666C2#6079911yescharacteristic
667C2#6080011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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×#2920C2semidirect
4. C22×#391C2semidirect
5. #24509C2semidirect
6. #24508C2semidirect
7. #24503C2semidirect
8. #24502C2semidirect
9. #23619C2semidirect
10. #23547C2semidirect
11. #21237C2semidirect
12. #21094C2semidirect
14. #20844C2semidirect
15. #20599C2semidirect
57. #2068C22semidirect
58. #2066C22semidirect
59. #2056C22semidirect
60. #2055C22semidirect
61. #1973C22semidirect
62. #1971C22semidirect
63. #1969C22semidirect
64. #1968C22semidirect
65. #1935C22semidirect
66. #1933C22semidirect
463. C22×C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30042
C2×#29202
#366262
C22×#3912
#245092
#245082
#245032
#245022
#236192
#235472
#212372
#210942
#209092
#208442
#205992