Finite Group Databasebeta

← #6513525

Subgroups — #6513525

Order 512.

2474 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic, complemented
3#35172C211yesmaximal, characteristic, complemented
4C2×#2537C211yesmaximal, characteristic
5#28230C214yesmaximal, complemented
6#28150C214yesmaximal, complemented
7#25393C214yesmaximal, complemented
8#25346C214yesmaximal, complemented
9#24559C214yesmaximal, complemented
10#24385C214yesmaximal, complemented
11#21982C211yesmaximal, characteristic, complemented
12#21860C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C22×#121—21no—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3101—28no—
20C2×#483C2212yes—
21C2×#476C2212yescomplemented
22C2×#468C2218yescomplemented
23C2×#461C2212yescomplemented
24C2×#450C2212yes—
25#2975—28no—
26C2×C4×SD16C2214yescomplemented
27C2×#436C2214yescomplemented
28C2×#436C2214yescomplemented
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#407C2214yes—
32#2848C2214yes—
33#2626—22no—
34#2580—22no—
35#2566—22no—
36#2563C2212yescomplemented
37#2542C2212yescomplemented
38#2538C2212yescomplemented
39#2538C2212yescomplemented
40#2537—22no—
41#2537—22no—
42#2537C2214yes—
43#2537C2214yes—
44C2×#394—22no—
45#2317—21no—
46#2309—21no—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380—21no—
50C2×#380C2212yes—
51#2077C2218yescomplemented
52#2036C2218yescomplemented
53#2033C2218yescomplemented
54#2000C2218yescomplemented
55#1990C2212yescomplemented
56#1956C2212yescomplemented
57#1947C2214yescomplemented
58#1895C2218yescomplemented
59#1894C2218yescomplemented
60#1872C2214yes—
61#1852C2214yes—
62#1845C2212yescomplemented
63#1834C2212yescomplemented
64#1823C2212yes—
65#1809C2218yescomplemented
66#1803C2214yescomplemented
67#1802C2212yescomplemented
68#1802C2212yescomplemented
69#1786C2212yes—
70C4×#102C2214yescomplemented
71C23×D4—22no—
72C23×D4D411yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4—21no—
75C24×C4C2312yes—
76C22×SD16C2314yes—
77C22×M16C2318yes—
78#533—24no—
79#533—24no—
80C2×C4⋊Q8C2311yescharacteristic
81C2×C4⋊Q8C2311yescharacteristic
82C2×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2312yes—
85C2×C22⋊Q8C2314yes—
86C2×C22⋊Q8C2314yes—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121—48no—
91C2×#121—48no—
92C2×#121C2318yes—
93C2×C4×Q8—21no—
94C2×C4×Q8C2312yes—
95C2×C4×D4—24no—
96C2×C4×D4—21no—
97C2×C4×D4—21no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4D412yes—
102C2×C4×D4D412yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—22no—
117C22×C22⋊C4D412yes—
118C22×C42C2311yescharacteristic
119#483—28no—
120#476—28no—
121#468—416no—
122#461—28no—
123#450—28no—
124C4×SD16—216no—
125#436—216no—
126#436—216no—
127#431—24no—
128#431D418yescomplemented
129C2×#107C2318yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2318yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138#407—216no—
139C2×C22⋊C8C2318yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#399—24no—
145#396—24no—
146#396—24no—
147#394—28no—
148#394—28no—
149#393—24no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#389—22no—
155#389—22no—
156#385—24no—
157#385—24no—
158#385—24no—
159#384—22no—
160#384—21no—
161#384—21no—
162#384C2311yescharacteristic
163#384C2311yescharacteristic
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—21no—
168#380—21no—
169#380—24no—
170#380—24no—
171#380—24no—
172#380C2314yes—
173#380C2314yes—
174#380C2318yes—
175#380C2312yes—
176#379—24no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4C2312yes—
179C2×#95—22no—
180#340C2314yes—
181#340C2314yes—
182#340C2318yes—
183#340C2318yes—
184C25—21no—
185C25C2×D412yes—
186C22×Q8—21no—
187C22×Q8C2412yes—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—42no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4—84no—
200C22×D4C2×D412yes—
201C22×D4C2×D412yes—
202C22×D4C2412yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D412yes—
215C23×C4C2×D414yes—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C2×SD16—28no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×M16—216no—
224C2×M16—48no—
225C2×M16—48no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232#125—44no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8—44no—
236C22⋊Q8—48no—
237C22⋊Q8—48no—
238#121—216no—
239#121—48no—
240#121—416no—
241#121—416no—
242#121—416no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—44no—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—44no—
257C4×D4—44no—
258#117—44no—
259#117—44no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D414yes—
311C2×C42#4114yes—
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—48no—
315#106—28no—
316C4⋊C8—24no—
317C4⋊C8C2×D418yescomplemented
318#103—28no—
319#103—48no—
320#102—28no—
321#102—28no—
322#102—44no—
323C22⋊C8—24no—
324C22⋊C8—216no—
325C22⋊C8C2×D418yes—
326C4×C8—24no—
327C4×C8C2×D418yescomplemented
328#95—28no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24C22×D414yes—
349C24C22×D411yescharacteristic
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—42no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—42no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—42no—
375C2×D4—416no—
376C2×D4—88no—
377C2×D4—832no—
378C2×D4H(5,𝔽2)14yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—816no—
414C22×C4#12114yes—
415C22×C4#12114yes—
416C22×C4AGL(1,ℤ/8)14yes—
417C22×C4#13714yes—
418C22×C4C22×D414yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427SD16—416no—
428M16—816no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8C22×D418yes—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C22⋊C4—28no—
462C22⋊C4—28no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—44no—
474C42—44no—
475C42#12114yes—
476C42#12114yes—
477C42C22×D412yes—
478C42C22×D412yes—
479C42C2×#4112yes—
480C42C2×#4112yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—48no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—48no—
503C23—42no—
504C23—42no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23C2×#12112yes—
514C23C2×AGL(1,ℤ/8)12yes—
515C23C2×#13712yes—
516C23C23×D411yescharacteristic
517Q8—44no—
518Q8—44no—
519Q8—84no—
520D4—24no—
521D4—24no—
522D4—48no—
523D4—42no—
524D4—42no—
525D4—48no—
526D4—48no—
527D4—832no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—88no—
561C2×C4—84no—
562C2×C4—88no—
563C2×C4—84no—
564C2×C4—88no—
565C2×C4—88no—
566C2×C4#468116yescomplemented
567C2×C4C2×#12112yes—
568C2×C4C2×#12112yes—
569C2×C4C2×#12112yes—
570C2×C4C2×#12112yes—
571C2×C4C2×#12112yes—
572C2×C4C2×AGL(1,ℤ/8)12yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C23×D411yescharacteristic
577C2×C4C22×#4111yescharacteristic
578C2×C4C2×H(5,𝔽2)12yes—
579C2×C4#58512yes—
580C8—28no—
581C8—44no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—816no—
599C22—816no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C2×#46814yes—
605C22C22×#12111yescharacteristic
606C22#348411yescharacteristic
607C22#348511yescharacteristic
608C22C22×AGL(1,ℤ/8)11yescharacteristic
609C22C22×#13711yescharacteristic
610C22#358511yescharacteristic
611C22#358611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—44no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4—82no—
622C4C2×#46814yes—
623C4C2×#46814yes—
624C4#348812yes—
625C4#349412yes—
626C2—24no—
627C2—84no—
628C2—84no—
629C2#3551411yescharacteristic
630C2C22×#46811yescharacteristic
631C2#6045611yescharacteristic
632C2#6158211yescharacteristic
633C2#6158611yescharacteristic
634C2#6171711yescharacteristic
635C2#6172111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3101C2semidirect
2. C2×#2975C2semidirect
3. #35172C2semidirect
5. #28230C2semidirect
6. #28150C2semidirect
7. #25393C2semidirect
8. #25346C2semidirect
9. #24559C2semidirect
10. #24385C2semidirect
11. #21982C2semidirect
12. #21860C2semidirect
13. #20732C2semidirect
21. C2×#476C22semidirect
22. C2×#468C22semidirect
23. C2×#461C22semidirect
26. C2×C4×SD16C22semidirect
27. C2×#436C22semidirect
28. C2×#436C22semidirect
36. #2563C22semidirect
37. #2542C22semidirect
38. #2538C22semidirect
39. #2538C22semidirect
51. #2077C22semidirect
52. #2036C22semidirect
53. #2033C22semidirect
54. #2000C22semidirect
55. #1990C22semidirect
56. #1956C22semidirect
57. #1947C22semidirect
58. #1895C22semidirect
59. #1894C22semidirect
62. #1845C22semidirect
63. #1834C22semidirect
65. #1809C22semidirect
66. #1803C22semidirect
67. #1802C22semidirect
68. #1802C22semidirect
70. C4×#102C22semidirect
128. #431D4semidirect
317. C4⋊C8C2×D4semidirect
327. C4×C8C2×D4semidirect
566. C2×C4#468semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#29752
#351722
C2×#25372
#282302
#281502
#253932
#253462
#245592
#243852
#219822
#218602
#207322