Finite Group Databasebeta

← #946560

Subgroups — #946560

Order 512.

4058 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2405C212yesmaximal
2#17193C212yesmaximal, complemented
3#17118C212yesmaximal, complemented
4#16657C214yesmaximal, complemented
5#16308C214yesmaximal, complemented
6#15966C211yesmaximal, characteristic, complemented
7#15949C211yesmaximal, characteristic, complemented
8#15463C212yesmaximal, complemented
9#15014C212yesmaximal, complemented
10#14820C214yesmaximal, complemented
11#14275C212yesmaximal, complemented
12#13877C214yesmaximal, complemented
13C2×#1438C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×#123C2212yes—
16C22×#123C2212yes—
17C23×C22⋊C4C2214yes—
18C23×C22⋊C4C2212yes—
19#2405—216no—
20C2×#397C2214yescomplemented
21C2×#397C2214yescomplemented
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#386C2212yes—
25C2×#386C2212yes—
26C2×#385C2214yes—
27C2×#385C2214yescomplemented
28C2×#385C2214yescomplemented
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#379C2214yes—
33C2×#379C2212yes—
34C22×#95C2211yescharacteristic
35C22×#95C2211yescharacteristic
36#1445C2212yescomplemented
37#1445C2212yescomplemented
38#1444C2212yescomplemented
39#1444C2212yescomplemented
40#1444C2212yescomplemented
41#1444C2212yescomplemented
42#1442C2214yescomplemented
43#1442C2212yescomplemented
44#1442C2214yescomplemented
45#1442C2212yescomplemented
46#1441C2214yescomplemented
47#1441C2214yescomplemented
48#1440C2214yescomplemented
49#1440C2214yescomplemented
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1439C2214yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1438C2214yes—
59#1438C2211yescharacteristic
60#1438C2211yescharacteristic
61#1438C2211yescharacteristic
62#1438C2211yescharacteristic
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C26C2312yes—
69C23×D4—42no—
70C23×D4—42no—
71C23×D4—48no—
72C23×D4D418yes—
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—48no—
84C2×#123—48no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—28no—
93C22×C22⋊C4—216no—
94C22×C22⋊C4—48no—
95C22×C22⋊C4—48no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4—44no—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112#397—48no—
113#397—48no—
114#394—216no—
115#394—216no—
116#386—28no—
117#386—28no—
118#385—216no—
119#385—48no—
120#385—48no—
121#380—28no—
122#380—44no—
123#380—44no—
124#379—216no—
125#379—28no—
126C2×#95—42no—
127C2×#95—42no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C25—28no—
152C25—28no—
153C25—24no—
154C25—24no—
155C25—28no—
156C25—44no—
157C25—44no—
158C25—44no—
159C25C2×D418yes—
160C25C2412yes—
161C25C2412yes—
162C25C2411yescharacteristic
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—416no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—432no—
175C22×D4—432no—
176C22×D4—432no—
177C22×D4C2×D414yes—
178C22×D4C2×D418yes—
179C22×D4C2×D414yes—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—28no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×D414yes—
203C23×C4#4118yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214#123—432no—
215#123—432no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—464no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—41no—
283#95—42no—
284#95—41no—
285#95—41no—
286#95—42no—
287#95—41no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299C24—22no—
300C24—24no—
301C24—22no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—216no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—216no—
319C24—232no—
320C24—48no—
321C24—416no—
322C24—416no—
323C24—416no—
324C24—416no—
325C24—416no—
326C24—416no—
327C24—416no—
328C24—416no—
329C24—48no—
330C24—48no—
331C24—88no—
332C24—88no—
333C24—88no—
334C24—88no—
335C24—88no—
336C24—84no—
337C24—84no—
338C24C22×D414yes—
339C24C22×D414yes—
340C24C22×D412yes—
341C24C2×#4114yes—
342C24H(5,𝔽2)18yes—
343C24H(5,𝔽2)12yes—
344C24H(5,𝔽2)12yes—
345C242−1+414yes—
346C24C2511yescharacteristic
347C2×D4—216no—
348C2×D4—216no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—416no—
357C2×D4—416no—
358C2×D4—4128no—
359C2×D4—48no—
360C2×D4—48no—
361C22×C4—28no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—216no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—48no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—88no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—88no—
429C22×C4—84no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)14yes—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C22⋊C4—232no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—432no—
489C23—464no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—432no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—42no—
516C23—42no—
517C23—816no—
518C23—832no—
519C23—832no—
520C23—832no—
521C23—832no—
522C23—832no—
523C23#53418yes—
524C23#53414yes—
525C23#53918yes—
526C23#54718yes—
527C23#55914yes—
528C23#56314yes—
529C23C23×D411yescharacteristic
530C23C22×#4112yes—
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)14yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+412yes—
536C23#58512yes—
537D4—416no—
538D4—432no—
539D4—416no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—416no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
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—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—816no—
598C2×C4—816no—
599C2×C4—816no—
600C2×C4—816no—
601C22—24no—
602C22—28no—
603C22—24no—
604C22—28no—
605C22—28no—
606C22—28no—
607C22—216no—
608C22—28no—
609C22—216no—
610C22—44no—
611C22—48no—
612C22—48no—
613C22—44no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—864no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22#272418yes—
628C22C2×#53411yescharacteristic
629C22C2×#53412yes—
630C22C2×#53912yes—
631C22C2×#54712yes—
632C22C2×#55911yescharacteristic
633C22#349014yes—
634C22#349414yes—
635C22#349612yes—
636C22C2×#56311yescharacteristic
637C22#353212yes—
638C22#353212yes—
639C22#353412yes—
640C22#353514yes—
641C22#353614yes—
642C22#354311yescharacteristic
643C22#354511yescharacteristic
644C4—42no—
645C4—42no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—84no—
650C4—84no—
651C4—82no—
652C4—84no—
653C4—84no—
654C2—28no—
655C2—48no—
656C2—84no—
657C2#3712911yescharacteristic
658C2#3743611yescharacteristic
659C2#3786014yes—
660C2#4063812yes—
661C2#4200912yes—
662C2#4317512yes—
663C2C2×#272411yescharacteristic
664C2#5241112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17193C2semidirect
3. #17118C2semidirect
4. #16657C2semidirect
5. #16308C2semidirect
6. #15966C2semidirect
7. #15949C2semidirect
8. #15463C2semidirect
9. #15014C2semidirect
10. #14820C2semidirect
11. #14275C2semidirect
12. #13877C2semidirect
20. C2×#397C22semidirect
21. C2×#397C22semidirect
27. C2×#385C22semidirect
28. C2×#385C22semidirect
36. #1445C22semidirect
37. #1445C22semidirect
38. #1444C22semidirect
39. #1444C22semidirect
40. #1444C22semidirect
41. #1444C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1441C22semidirect
47. #1441C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24052
#171932
#171182
#166572
#163082
#159662
#159492
#154632
#150142
#148202
#142752
#138772
C2×#14382