Finite Group Databasebeta

← #6629271

Subgroups — #6629271

Order 512.

6464 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic, complemented
2C2×#3100C212yesmaximal, complemented
3#36624C211yesmaximal, characteristic, complemented
4#33002C214yesmaximal, complemented
5#32978C212yesmaximal, complemented
6#32953C214yesmaximal, complemented
7#32902C214yesmaximal, complemented
8#32878C212yesmaximal, complemented
9C4×#447C218yesmaximal, complemented
10#20732C212yesmaximal, complemented
11#20714C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C25×C4—21no—
14C2×#532C2211yescharacteristic, complemented
15C2×#525—21no—
16C22×C4×D4—22no—
17C22×C4×D4—21no—
18C22×C4×D4—21no—
19C22×C4×D4—22no—
20C22×C4×D4—216no—
21C22×C4×D4C2214yes—
22C22×C4×D4C2214yes—
23C22×C4×D4C2212yes—
24C22×C4×D4C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4—21no—
27C23×C42C2211yescharacteristic
28#3100—216no—
29C2×#480C2214yescomplemented
30C2×#474C2214yescomplemented
31C2×#459C2214yescomplemented
32C2×#447C2214yes—
33C2×#447C2218yes—
34C2×C4×D8C2218yescomplemented
35C2×#431C2212yescomplemented
36#2857C2212yescomplemented
37#2856C2212yescomplemented
38#2854—21no—
39#2854C2212yescomplemented
40#2850—22no—
41#2847—21no—
42#2847C2212yes—
43C4×#121—22no—
44C4×#121C2214yescomplemented
45#2297—21no—
46#2285C2211yescharacteristic, complemented
47#1879C22116yescomplemented
48#1874C2218yescomplemented
49#1862C22116yescomplemented
50#1852C2218yescomplemented
51#1823C2214yescomplemented
52#1817C2212yescomplemented
53#1802C2218yescomplemented
54#1797C2214yescomplemented
55#1786C2218yes—
56#1786C2214yes—
57#1779C2212yes—
58C4×#102C2218yescomplemented
59C4×C22⋊C8C2214yescomplemented
60C26—21no—
61C22×#41—21no—
62C23×D4—28no—
63C23×D4—216no—
64C23×D4—21no—
65C23×D4—22no—
66C23×D4C2312yes—
67C23×D4C2314yes—
68C24×C4—216no—
69C24×C4—21no—
70C24×C4—24no—
71C24×C4—22no—
72C24×C4—24no—
73C24×C4—22no—
74C24×C4—48no—
75C24×C4—48no—
76C24×C4D411yescharacteristic
77C24×C4D411yescharacteristic
78C24×C4C2311yescharacteristic
79C22×D8C2318yes—
80#532—28no—
81C2×C4⋊Q8C2311yescharacteristic, complemented
82C2×#127C2311yescharacteristic, complemented
83C2×#124—21no—
84#525—48no—
85C2×#123—22no—
86C2×C22⋊Q8—22no—
87C2×#121—21no—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121C2312yes—
91C2×#121C2314yes—
92C2×#120—22no—
93C2×C4×D4—216no—
94C2×C4×D4—216no—
95C2×C4×D4—232no—
96C2×C4×D4—232no—
97C2×C4×D4—28no—
98C2×C4×D4—232no—
99C2×C4×D4—232no—
100C2×C4×D4—24no—
101C2×C4×D4—28no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—28no—
110C2×C4×D4—21no—
111C2×C4×D4D4116yescomplemented
112C2×C4×D4C2318yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2314yes—
115C2×C4×D4C2312yescomplemented
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—28no—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4—216no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4—24no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C42—28no—
128C22×C42D412yes—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D412yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#480—216no—
137#474—216no—
138#459—216no—
139#447—216no—
140#447—232no—
141C4×D8—232no—
142#431—28no—
143C2×#107C2318yescomplemented
144C2×C4⋊C8C2314yescomplemented
145C2×#102C23116yes—
146C2×#102C2318yes—
147C2×#102C2318yescomplemented
148C2×C22⋊C8C2318yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2314yescomplemented
151#391—22no—
152#390—22no—
153#390—22no—
154#390C2314yescomplemented
155#389—22no—
156#389C2314yescomplemented
157#388—24no—
158#388—24no—
159#386—24no—
160#386—22no—
161#386—24no—
162#386—22no—
163#384C2312yescomplemented
164#382—21no—
165#380—21no—
166#380—22no—
167#380C2312yes—
168#380C2314yes—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2312yescomplemented
171C4×C22⋊C4—24no—
172#340C23116yescomplemented
173#336C2318yescomplemented
174C25—28no—
175C25—216no—
176C25—24no—
177C25—22no—
178C25—44no—
179C25—48no—
180C25—44no—
181C25C2×D411yescharacteristic
182C2×#41—44no—
183C2×#41—44no—
184C2×#41—44no—
185C22×Q8—21no—
186C22×D4—28no—
187C22×D4—216no—
188C22×D4—2128no—
189C22×D4—216no—
190C22×D4—216no—
191C22×D4—216no—
192C22×D4—216no—
193C22×D4—216no—
194C22×D4—216no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—28no—
203C22×D4C2×D418yes—
204C22×D4#41116yescomplemented
205C22×D4C2412yes—
206C22×D4C2414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—216no—
213C23×C4—232no—
214C23×C4—232no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—216no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—44no—
224C23×C4—48no—
225C23×C4—416no—
226C23×C4—416no—
227C23×C4—4128no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—416no—
231C23×C4—416no—
232C23×C4—48no—
233C23×C4—48no—
234C23×C4—48no—
235C23×C4D818yescomplemented
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C2×D8—216no—
249C2×D8—232no—
250C22×C8C2418yes—
251C4⋊Q8—24no—
252#127—24no—
253#124—44no—
254#123—48no—
255C22⋊Q8—48no—
256#121—28no—
257#121—216no—
258#121—48no—
259#121—44no—
260#121—48no—
261#120—48no—
262C4×D4—232no—
263C4×D4—232no—
264C4×D4—232no—
265C4×D4—232no—
266C4×D4—232no—
267C4×D4—216no—
268C4×D4—216no—
269C4×D4—216no—
270C4×D4—216no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—416no—
275C4×D4—44no—
276C4×D4C2×D4116yes—
277C4×D4C2×D4116yes—
278#117—44no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—216no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2414yes—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—232no—
295C2×C22⋊C4—232no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—22no—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D418yes—
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326#107—216no—
327C4⋊C8—28no—
328#102—216no—
329#102—232no—
330#102—216no—
331C22⋊C8—28no—
332C22⋊C8—216no—
333C4×C8—28no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—24no—
342C24—216no—
343C24—264no—
344C24—216no—
345C24—216no—
346C24—28no—
347C24—216no—
348C24—24no—
349C24—22no—
350C24—432no—
351C24—48no—
352C24—432no—
353C24—464no—
354C24—464no—
355C24—48no—
356C24—48no—
357C24—48no—
358C24—44no—
359C24—44no—
360C24—48no—
361C24C2×D814yes—
362C24C22×D411yescharacteristic
363C24C22×D412yes—
364#41—816no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×D4—216no—
369C2×D4—2128no—
370C2×D4—264no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—232no—
374C2×D4—264no—
375C2×D4—232no—
376C2×D4—264no—
377C2×D4—28no—
378C2×D4—216no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4C22×D418yes—
395C2×D4C2×#41116yes—
396C22×C4—232no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—232no—
412C22×C4—216no—
413C22×C4—232no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—464no—
430C22×C4—416no—
431C22×C4—4128no—
432C22×C4—4128no—
433C22×C4—416no—
434C22×C4—432no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—464no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4#12014yes—
449C22×C4#12014yes—
450C22×C4C2×D818yes—
451C22×C4C2×D818yes—
452C22×C4C2×D814yes—
453C22×C4C2×D814yes—
454C22×C4AGL(1,ℤ/8)14yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4118yes—
465C22×C4C2511yescharacteristic
466D8—432no—
467C2×C8—28no—
468C2×C8—216no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4C22×D418yes—
479C22⋊C4—232no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—44no—
496C42#12014yes—
497C42#12014yes—
498C42C22×D414yes—
499C42C22×D414yes—
500C23—28no—
501C23—28no—
502C23—216no—
503C23—216no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—232no—
512C23—264no—
513C23—28no—
514C23—216no—
515C23—432no—
516C23—4256no—
517C23—432no—
518C23—432no—
519C23—432no—
520C23—432no—
521C23—432no—
522C23—416no—
523C23—432no—
524C23—432no—
525C23—416no—
526C23—44no—
527C23—44no—
528C23—48no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23C2×#12012yes—
535C23#52514yes—
536C23C22×D814yes—
537C23C22×D812yes—
538C23C2×AGL(1,ℤ/8)12yes—
539C23#57514yes—
540C23C23×D411yescharacteristic
541Q8—84no—
542D4—232no—
543D4—264no—
544D4—28no—
545D4—216no—
546D4—48no—
547D4—84no—
548D4—816no—
549D4—88no—
550D4—84no—
551D4#525116yescomplemented
552C2×C4—216no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—216no—
564C2×C4—216no—
565C2×C4—216no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—464no—
574C2×C4—44no—
575C2×C4—432no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—432no—
580C2×C4—432no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—88no—
588C2×C4#447116yescomplemented
589C2×C4C2×#12012yes—
590C2×C4C2×#12012yes—
591C2×C4C2×#12012yes—
592C2×C4C2×#12012yes—
593C2×C4C2×#12012yes—
594C2×C4#52514yes—
595C2×C4C22×D814yes—
596C2×C4C22×D812yes—
597C2×C4C22×D812yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C23×D412yes—
601C2×C4C22×#4114yes—
602C8—48no—
603C22—232no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—24no—
614C22—216no—
615C22—4128no—
616C22—464no—
617C22—416no—
618C22—416no—
619C22—48no—
620C22—48no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22C2×#44714yes—
627C22#310018yescomplemented
628C22C22×#12011yescharacteristic
629C22C2×#52512yes—
630C22C23×D811yescharacteristic
631C22C22×AGL(1,ℤ/8)11yescharacteristic
632C22C2×#57512yes—
633C4—22no—
634C4—24no—
635C4—24no—
636C4—22no—
637C4—22no—
638C4—48no—
639C4—42no—
640C4—84no—
641C4—82no—
642C4C2×#44714yes—
643C4C2×#44714yes—
644C4C2×#52514yes—
645C2—28no—
646C2—24no—
647C2—416no—
648C2—162no—
649C2#3661411yescharacteristic
650C2C22×#44711yescharacteristic
651C2#6035911yescharacteristic
652C2C2×#310012yes—
653C2#6055812yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×D4C2semidirect
2. C2×#3100C2semidirect
3. #36624C2semidirect
4. #33002C2semidirect
5. #32978C2semidirect
6. #32953C2semidirect
7. #32902C2semidirect
8. #32878C2semidirect
9. C4×#447C2semidirect
10. #20732C2semidirect
11. #20714C2semidirect
14. C2×#532C22semidirect
29. C2×#480C22semidirect
30. C2×#474C22semidirect
31. C2×#459C22semidirect
34. C2×C4×D8C22semidirect
35. C2×#431C22semidirect
36. #2857C22semidirect
37. #2856C22semidirect
39. #2854C22semidirect
44. C4×#121C22semidirect
46. #2285C22semidirect
47. #1879C22semidirect
48. #1874C22semidirect
49. #1862C22semidirect
50. #1852C22semidirect
51. #1823C22semidirect
52. #1817C22semidirect
53. #1802C22semidirect
54. #1797C22semidirect
58. C4×#102C22semidirect
59. C4×C22⋊C8C22semidirect
81. C2×C4⋊Q8C23semidirect
82. C2×#127C23semidirect
111. C2×C4×D4D4semidirect
115. C2×C4×D4C23semidirect
143. C2×#107C23semidirect
144. C2×C4⋊C8C23semidirect
147. C2×#102C23semidirect
150. C2×C4×C8C23semidirect
154. #390C23semidirect
156. #389C23semidirect
163. #384C23semidirect
170. C4×C4⋊C4C23semidirect
172. #340C23semidirect
173. #336C23semidirect
204. C22×D4#41semidirect
235. C23×C4D8semidirect
551. D4#525semidirect
588. C2×C4#447semidirect
627. C22#3100semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×#31002
#366242
#330022
#329782
#329532
#329022
#328782
C4×#4472
#207322
#207142