Finite Group Databasebeta

← #6493461

Subgroups — #6493461

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2D4×#117C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic, complemented
4#33068C211yesmaximal, characteristic, complemented
5#32905C212yesmaximal
6#29998C211yesmaximal, characteristic, complemented
7#29922C212yesmaximal, complemented
8#29798C211yesmaximal, characteristic, complemented
9#27392C214yesmaximal, complemented
10#27192C214yesmaximal, complemented
11#26166C214yesmaximal, complemented
12#25156C214yesmaximal
13#22146C212yesmaximal, complemented
14#21922C211yesmaximal, characteristic, complemented
15C4×#461C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#461C2212yes—
19C2×#461C2212yes—
20C2×#461C2214yes—
21C2×#450C2214yes—
22C2×C4×SD16C2214yes—
23#2868C2211yescharacteristic, complemented
24#2859C2211yescharacteristic, complemented
25#2857C2212yes—
26#2647—21no—
27#2629—21no—
28#2507—22no—
29#2497—22no—
30#2446C2214yescomplemented
31#2429C2214yes—
32#2354C2212yes—
33#2352—22no—
34D4×C4⋊C4C2212yes—
35#2344C2212yes—
36#2342—22no—
37D4×C22⋊C4C2212yes—
38#2330—21no—
39#2314—21no—
40C4×C4⋊Q8C2212yescomplemented
41#2293C2211yescharacteristic
42#2293C2211yescharacteristic
43#2290—21no—
44#2290—21no—
45#2285C2211yescharacteristic
46#2280—21no—
47C42×D4C2211yescharacteristic
48C42×D4C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2096C2214yescomplemented
51#2088C2214yescomplemented
52#2072C2214yescomplemented
53#2065C2214yes—
54#2043C2214yescomplemented
55#2035C2214yes—
56#2035C2214yes—
57#2013C2214yes—
58#1981C2214yescomplemented
59#1981C2212yescomplemented
60#1981C2212yescomplemented
61#1968C2214yes—
62#1948C2212yescomplemented
63#1948C2212yescomplemented
64#1933C2214yescomplemented
65#1927C2214yescomplemented
66#1923C2214yes—
67#1920C2214yescomplemented
68#1916C2214yescomplemented
69#1880C2214yes—
70#1877C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1872C2214yes—
74#1865C2214yescomplemented
75#1864C2214yes—
76#1842C2211yescharacteristic, complemented
77#1831C2211yescharacteristic, complemented
78#1802C2212yes—
79#1802C2212yes—
80#1797C2212yes—
81C4×C4⋊C8C2212yescomplemented
82C4×#103C2214yescomplemented
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C22×SD16C2314yes—
86C22×SD16C2314yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2314yes—
89C2×#125—21no—
90C2×C22⋊Q8C2314yes—
91C2×C4×Q8C2314yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2311yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2312yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×#117—21no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117—21no—
106C2×#117—21no—
107C2×#117—48no—
108C2×#117D411yescharacteristic
109C2×#117D411yescharacteristic
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—22no—
114C22×C42—21no—
115C22×C42—21no—
116C22×C42C2311yescharacteristic
117#461—28no—
118#461—28no—
119#461—48no—
120#450—48no—
121C4×SD16—48no—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#398—24no—
139#398—24no—
140#396—24no—
141#391C2314yes—
142#391C2314yes—
143#390C2312yes—
144#390C2312yes—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389—22no—
150#389—24no—
151#389C2314yes—
152#389C2312yes—
153#389C2312yes—
154#388—24no—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#387—24no—
160#387—24no—
161#387—22no—
162#387—22no—
163#386—22no—
164#386—22no—
165#385—22no—
166#385—22no—
167#384—21no—
168#384—21no—
169#384—22no—
170#384C2312yes—
171#384C2312yes—
172#382—21no—
173#382—21no—
174#381—22no—
175#380—22no—
176#380—21no—
177#380—22no—
178#380—22no—
179#380—22no—
180#380C2312yes—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2311yescharacteristic
189C4×C4⋊C4C2311yescharacteristic
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4C2312yes—
199#376D411yescharacteristic
200#376D411yescharacteristic
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#336C2314yes—
208#336C2314yes—
209#336C2314yes—
210C25—21no—
211C22×Q8C2414yes—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—42no—
217C22×D4C2411yescharacteristic
218C22×D4C2411yescharacteristic
219C23×C4—28no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2411yescharacteristic
232C2×SD16—28no—
233C2×SD16—28no—
234C2×SD16—28no—
235C2×SD16—28no—
236C2×SD16—44no—
237C2×SD16—44no—
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—44no—
243C4⋊Q8—44no—
244#125—44no—
245C22⋊Q8—44no—
246C22⋊Q8—44no—
247C4×Q8—44no—
248C4×Q8—44no—
249C4×D4—28no—
250C4×D4—28no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4#4114yes—
262C4×D4#4114yes—
263#117—42no—
264#117—44no—
265#117—48no—
266#117—48no—
267#117—44no—
268#117—44no—
269#117SD1618yescomplemented
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C22⋊C4—28no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331#106—44no—
332#106—44no—
333C4⋊C8—24no—
334C4⋊C8—24no—
335C4⋊C8—44no—
336#103—28no—
337#103—28no—
338#103—44no—
339#103—44no—
340#103—44no—
341#103—44no—
342#102—28no—
343#102—44no—
344C22⋊C8—44no—
345C4×C8—44no—
346#95—22no—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362C24—21no—
363C24—24no—
364C24—21no—
365C24—44no—
366C24—42no—
367C24—42no—
368C24—82no—
369C24C22×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×D4—24no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—44no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4C2×#4112yes—
387C2×D4C2×#4112yes—
388C2×D4H(5,𝔽2)14yes—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—432no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—82no—
424C22×C4—82no—
425C22×C4C2×SD1614yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436SD16—416no—
437SD16—88no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C4⋊C4—24no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4C2×SD1618yes—
468C4⋊C4C2×#4112yes—
469C4⋊C4C2×#4112yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—84no—
483C22⋊C4C2×SD1618yes—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—42no—
495C42—42no—
496C42—44no—
497C42—44no—
498C42—42no—
499C42—42no—
500C42—84no—
501C42C2×SD1614yes—
502C42C2×SD1614yes—
503C42C2×#4112yes—
504C42C2×#4112yes—
505C23—22no—
506C23—22no—
507C23—21no—
508C23—24no—
509C23—22no—
510C23—21no—
511C23—22no—
512C23—48no—
513C23—48no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—88no—
522C23—84no—
523C23—88no—
524C23C22×SD1612yes—
525C23C23×D411yescharacteristic
526Q8—44no—
527Q8—44no—
528Q8—44no—
529Q8—44no—
530D4—22no—
531D4—24no—
532D4—24no—
533D4—48no—
534D4—41no—
535D4—41no—
536D4#54014yes—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—82no—
582C2×C4—88no—
583C2×C4—88no—
584C2×C4#52512yes—
585C2×C4#52512yes—
586C2×C4C22×SD1614yes—
587C2×C4C22×SD1614yes—
588C2×C4C22×SD1612yes—
589C2×C4C22×SD1612yes—
590C2×C4#57512yes—
591C2×C4C22×#4111yescharacteristic
592C2×C4C22×#4111yescharacteristic
593C2×C4C2×H(5,𝔽2)12yes—
594C2×C4#58512yes—
595C8—44no—
596C8—44no—
597C22—22no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—44no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—816no—
611C22—82no—
612C22—82no—
613C22C2×#52511yescharacteristic
614C22#348812yes—
615C22C23×SD1611yescharacteristic
616C22C2×#57511yescharacteristic
617C22#358612yes—
618C4—22no—
619C4—22no—
620C4—24no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4—82no—
632C4#310114yes—
633C4C2×#54011yescharacteristic
634C4#345511yescharacteristic
635C4#349412yes—
636C2—42no—
637C2—44no—
638C2—82no—
639C2#3526111yescharacteristic
640C2C2×#310111yescharacteristic
641C2#6056911yescharacteristic
642C2#6194812yes—
643C2#6196712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #35284C2semidirect
3. #33162C2semidirect
4. #33068C2semidirect
6. #29998C2semidirect
7. #29922C2semidirect
8. #29798C2semidirect
9. #27392C2semidirect
10. #27192C2semidirect
11. #26166C2semidirect
13. #22146C2semidirect
14. #21922C2semidirect
15. C4×#461C2semidirect
23. #2868C22semidirect
24. #2859C22semidirect
30. #2446C22semidirect
40. C4×C4⋊Q8C22semidirect
50. #2096C22semidirect
51. #2088C22semidirect
52. #2072C22semidirect
54. #2043C22semidirect
58. #1981C22semidirect
59. #1981C22semidirect
60. #1981C22semidirect
62. #1948C22semidirect
63. #1948C22semidirect
64. #1933C22semidirect
65. #1927C22semidirect
67. #1920C22semidirect
68. #1916C22semidirect
74. #1865C22semidirect
76. #1842C22semidirect
77. #1831C22semidirect
81. C4×C4⋊C8C22semidirect
82. C4×#103C22semidirect
269. #117SD16semidirect

Maximal subgroups

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

subgroupindex
#352842
D4×#1172
#331622
#330682
#329052
#299982
#299222
#297982
#273922
#271922
#261662
#251562
#221462
#219222
C4×#4612