Finite Group Databasebeta

← #4308944

Subgroups — #4308944

Order 512.

2810 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#16588C214yesmaximal, complemented
4#16581C214yesmaximal, complemented
5#16543C214yesmaximal, complemented
6#16536C214yesmaximal, complemented
7#15019C211yesmaximal, characteristic, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14918C212yesmaximal
10#14820C211yesmaximal, characteristic
11#14786C212yesmaximal, complemented
12#14734C211yesmaximal, characteristic
13#14705C212yesmaximal, complemented
14#14349C211yesmaximal, characteristic, complemented
15#14262C212yesmaximal
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2646—28no—
22C2×#400C2212yescomplemented
23C2×#400C2212yescomplemented
24C2×#399C2212yes—
25C2×#398C2212yescomplemented
26C2×#398C2212yescomplemented
27C2×#398C2214yescomplemented
28C2×#397C2212yes—
29C2×#388—28no—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#379C2212yes—
41C2×#379C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1445C2214yescomplemented
45#1445C2212yescomplemented
46#1445C2212yescomplemented
47#1445C2214yescomplemented
48#1444C2212yescomplemented
49#1444C2214yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1443C2214yescomplemented
53#1442C2212yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—24no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42—24no—
117C22×C42—44no—
118C22×C42C2314yes—
119#400—28no—
120#400—28no—
121#399—44no—
122#398—28no—
123#398—28no—
124#398—48no—
125#397—44no—
126#388—216no—
127#388—216no—
128#385—28no—
129#385—28no—
130#385—44no—
131#385—44no—
132#381—28no—
133#381—44no—
134#380—24no—
135#380—24no—
136#379—28no—
137#379—44no—
138C2×#95—24no—
139C2×#95D418yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25—22no—
169C25—22no—
170C25—24no—
171C25—24no—
172C25—42no—
173C25—42no—
174C25—42no—
175C25—42no—
176C25#4114yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4#4114yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C22⋊C4—232no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—416no—
271C2×C22⋊C4—416no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C42—28no—
283C2×C42—48no—
284C2×C42—416no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—42no—
311#95C2×D418yes—
312#95C2×D418yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—28no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—22no—
327C24—22no—
328C24—216no—
329C24—48no—
330C24—48no—
331C24—44no—
332C24—48no—
333C24—48no—
334C24—44no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24—44no—
339C24—48no—
340C24—88no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24C22×D412yes—
345C24C2×#4114yes—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24H(5,𝔽2)14yes—
349C242−1+414yes—
350C24C2511yescharacteristic
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D414yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)14yes—
443C4⋊C4—416no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—84no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—88no—
453C4⋊C4—88no—
454C22⋊C4—232no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—88no—
466C42—416no—
467C23—24no—
468C23—24no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—24no—
483C23—24no—
484C23—416no—
485C23—416no—
486C23—416no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—832no—
506C23—88no—
507C23—88no—
508C23—88no—
509C23—88no—
510C23—832no—
511C23#52514yes—
512C23#53514yes—
513C23#53614yes—
514C23#53918yes—
515C23#53918yes—
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×2−1+412yes—
522C23#58512yes—
523C23#58512yes—
524C23#58512yes—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—88no—
596C2×C4—88no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—216no—
604C22—216no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—832no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—84no—
617C22#263318yes—
618C22C2×#52511yescharacteristic
619C22C2×#53511yescharacteristic
620C22C2×#53611yescharacteristic
621C22C2×#53912yes—
622C22C2×#53912yes—
623C22#345512yes—
624C22#346012yes—
625C22#348411yescharacteristic
626C22#348511yescharacteristic
627C22#348511yescharacteristic
628C22#348711yescharacteristic
629C22#349112yes—
630C22#350512yes—
631C22#353314yes—
632C22#353414yes—
633C22#356114yes—
634C22#356414yes—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—84no—
647C4—84no—
648C2—28no—
649C2—84no—
650C2C2×#263311yescharacteristic
651C2#3509111yescharacteristic
652C2#3509411yescharacteristic
653C2#3695611yescharacteristic
654C2#3696211yescharacteristic
655C2#3696611yescharacteristic
656C2#3698514yes—
657C2#3718511yescharacteristic
658C2#3750614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2646C2semidirect
3. #16588C2semidirect
4. #16581C2semidirect
5. #16543C2semidirect
6. #16536C2semidirect
7. #15019C2semidirect
8. #14936C2semidirect
11. #14786C2semidirect
13. #14705C2semidirect
14. #14349C2semidirect
22. C2×#400C22semidirect
23. C2×#400C22semidirect
25. C2×#398C22semidirect
26. C2×#398C22semidirect
27. C2×#398C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3882
#165882
#165812
#165432
#165362
#150192
#149362
#149182
#148202
#147862
#147342
#147052
#143492
#142622