Finite Group Databasebeta

← #6500404

Subgroups — #6500404

Order 512.

2834 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic, complemented
2C2×#2973C211yesmaximal, characteristic
3#35010C211yesmaximal, characteristic
4C2×#2409C211yesmaximal, characteristic
5#28108C214yesmaximal, complemented
6#25126C218yesmaximal, complemented
7#24508C214yesmaximal, complemented
8#24386C214yesmaximal, complemented
9#24273C214yesmaximal, complemented
10#21938C211yesmaximal, characteristic, complemented
11#21759C211yesmaximal, characteristic, complemented
12C2×#1787C211yesmaximal, characteristic, complemented
13C24×Q8C2211yescharacteristic
14C23×Q16C2212yescomplemented
15C22×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C4×D4C2211yescharacteristic
18C2×#470C2218yescomplemented
19C2×#451—28no—
20C2×#451C2214yes—
21C2×#451C2214yescomplemented
22#2973—28no—
23C2×#435C2214yes—
24C2×#435C2214yes—
25C2×#432C2212yes—
26C22×#103C2212yescomplemented
27C22×#103C2214yescomplemented
28C2×#407C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2855C2212yes—
32#2848C2212yes—
33#2627—21no—
34#2626—21no—
35#2579—22no—
36#2564—22no—
37#2487C2212yes—
38#2417C2214yes—
39#2409—22no—
40#2409—22no—
41#2409C2214yes—
42#2409C2214yes—
43#2409C2212yes—
44C2×#395—21no—
45C2×#393—21no—
46#2315—21no—
47#2308—21no—
48C2×#391C2212yes—
49C2×#386—21no—
50C2×#385—21no—
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53#2066C2218yescomplemented
54#2056C2218yescomplemented
55#2035C2218yescomplemented
56#2001C2218yescomplemented
57#1971C2218yescomplemented
58#1935C2218yescomplemented
59#1894C2218yescomplemented
60#1892C2214yescomplemented
61#1872C2214yes—
62#1853C2214yes—
63#1853C2214yescomplemented
64#1824C2212yescomplemented
65#1807C2214yescomplemented
66#1787C2214yes—
67#1787C2214yescomplemented
68#1787C2212yescomplemented
69C23×Q8—21no—
70C23×Q8—21no—
71C23×Q8—21no—
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—48no—
75C23×Q8C2312yes—
76C23×Q8C2312yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4D411yescharacteristic
80C24×C4—21no—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×Q16—28no—
84C22×Q16—48no—
85C22×Q16C2318yes—
86C22×M16C2314yes—
87C23×C8C2312yes—
88C23×C8C2312yes—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—48no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2318yes—
98C2×C22⋊Q8C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4D412yes—
102C2×C4×D4D412yes—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—22no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42C2311yescharacteristic
118#470—416no—
119#451—216no—
120#451—216no—
121#435—216no—
122#435—216no—
123#432—28no—
124C2×#106C2318yes—
125C2×C4⋊C8C2314yes—
126C2×#103—28no—
127C2×#103—48no—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×#103C2318yes—
132#407—28no—
133C2×C22⋊C8—24no—
134C2×C22⋊C8D418yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141#400—22no—
142#399—22no—
143#398—24no—
144#396—24no—
145#395—22no—
146#395—24no—
147#395—24no—
148#393—22no—
149#393—24no—
150#393—24no—
151#391—24no—
152#391C2314yes—
153#391C2318yes—
154#389C2314yes—
155#387—24no—
156#386—24no—
157#386—22no—
158#386—22no—
159#386—24no—
160#386—24no—
161#385—24no—
162#385—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#385—24no—
167#385—24no—
168#381—22no—
169#380—21no—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—24no—
174#380C2312yes—
175#380C2312yes—
176#380C2312yes—
177#380C2314yes—
178#380C2312yes—
179#379—22no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—22no—
183C2×#95—22no—
184#340C2318yes—
185#340C2318yes—
186C25C2×D411yescharacteristic
187C25C2×D411yescharacteristic
188C22×Q8—28no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—21no—
193C22×Q8—28no—
194C22×Q8—28no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—48no—
207C22×Q8—44no—
208C22×Q8—432no—
209C22×Q8—432no—
210C22×Q8—48no—
211C22×Q8—48no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—44no—
215C22×Q8C2412yes—
216C22×Q8C2414yes—
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—24no—
220C22×D4C2×D412yes—
221C22×D4C2×D412yes—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—48no—
234C23×C4—44no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4#4114yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×Q16—216no—
249C2×Q16—216no—
250C2×Q16—48no—
251C2×Q16—432no—
252C2×M16—28no—
253C2×M16—48no—
254C22×C8—24no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8C2×D418yes—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C22⋊Q8—216no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—48no—
266C22⋊Q8—416no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—48no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2414yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C2×D412yes—
318C2×C42C2412yes—
319#106—48no—
320C4⋊C8—28no—
321#103—216no—
322#103—216no—
323#103—48no—
324#103—48no—
325C22⋊C8—24no—
326C22⋊C8—24no—
327C22⋊C8—28no—
328C22⋊C8C2×D418yes—
329C22⋊C8C2×D418yes—
330#97—28no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—42no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24#12114yes—
349C24#13714yes—
350C24#13714yes—
351C24C22×D412yes—
352C24C22×D411yescharacteristic
353C24C22×D411yescharacteristic
354C24C22×D411yescharacteristic
355C2×Q8—28no—
356C2×Q8—24no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—28no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—432no—
377C2×Q8—432no—
378C2×Q8—432no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—82no—
384C2×Q8—84no—
385C2×Q8H(5,𝔽2)18yes—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—24no—
392C2×D4—416no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—432no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4#12114yes—
439C22×C4#12114yes—
440C22×C4#12114yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2511yescharacteristic
454Q16—432no—
455M16—88no—
456C2×C8—28no—
457C2×C8—24no—
458C2×C8—28no—
459C2×C8—48no—
460C2×C8—84no—
461C2×C8C22×D418yes—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—416no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—84no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—416no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C42—24no—
490C42—22no—
491C42—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—44no—
504C23—44no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—44no—
509C23—82no—
510C23—82no—
511C23—84no—
512C23#470116yescomplemented
513C23C2×#12112yes—
514C23C2×#12112yes—
515C23C2×#12112yes—
516C23C2×#13712yes—
517C23C2×#13712yes—
518C23C2×#13712yes—
519C23C2×#13712yes—
520C23C2×#13712yes—
521C23C2×#13712yes—
522C23C23×D411yescharacteristic
523Q8—28no—
524Q8—28no—
525Q8—416no—
526Q8—416no—
527Q8—416no—
528Q8—44no—
529Q8—44no—
530Q8—44no—
531Q8—44no—
532Q8—84no—
533D4—48no—
534D4—48no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—416no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—84no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—88no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—164no—
580C2×C4C2×#12112yes—
581C2×C4C2×#12112yes—
582C2×C4C2×#12112yes—
583C2×C4C23×D411yescharacteristic
584C2×C4C22×#4111yescharacteristic
585C2×C4C2×H(5,𝔽2)14yes—
586C8—28no—
587C8—44no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—416no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—88no—
607C22C2×#47014yes—
608C22C2×#47014yes—
609C22C2×#47014yes—
610C22C22×#12111yescharacteristic
611C22#348412yes—
612C22C22×#13711yescharacteristic
613C22C22×#13711yescharacteristic
614C22#358612yes—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—48no—
620C4—42no—
621C4—42no—
622C4—82no—
623C4—82no—
624C4—84no—
625C4—84no—
626C4#348412yes—
627C4#349012yes—
628C2—24no—
629C2—24no—
630C2—24no—
631C2—82no—
632C2#3534811yescharacteristic
633C2C22×#47011yescharacteristic
634C2#6045511yescharacteristic
635C2#6157312yes—
636C2#6157412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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. C22×#451C2semidirect
5. #28108C2semidirect
6. #25126C2semidirect
7. #24508C2semidirect
8. #24386C2semidirect
9. #24273C2semidirect
10. #21938C2semidirect
11. #21759C2semidirect
12. C2×#1787C2semidirect
14. C23×Q16C22semidirect
18. C2×#470C22semidirect
21. C2×#451C22semidirect
26. C22×#103C22semidirect
27. C22×#103C22semidirect
53. #2066C22semidirect
54. #2056C22semidirect
55. #2035C22semidirect
56. #2001C22semidirect
57. #1971C22semidirect
58. #1935C22semidirect
59. #1894C22semidirect
60. #1892C22semidirect
63. #1853C22semidirect
64. #1824C22semidirect
65. #1807C22semidirect
67. #1787C22semidirect
68. #1787C22semidirect
512. C23#470semidirect

Maximal subgroups

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

subgroupindex
C22×#4512
C2×#29732
#350102
C2×#24092
#281082
#251262
#245082
#243862
#242732
#219382
#217592
C2×#17872