Finite Group Databasebeta

← #6510725

Subgroups — #6510725

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35079C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3#29109C212yesmaximal, complemented
4#29104C212yesmaximal, complemented
5#27965C212yesmaximal, complemented
6#27961C212yesmaximal, complemented
7#27719C214yesmaximal, complemented
8#27717C214yesmaximal, complemented
9#27709C214yesmaximal, complemented
10#27695C214yesmaximal, complemented
11#21764C211yesmaximal, characteristic
12#21761C211yesmaximal, characteristic
13#20700C211yesmaximal, characteristic, complemented
14C2×#1824C211yesmaximal, characteristic, complemented
15#20525C211yesmaximal, characteristic, complemented
16C22×#123—22no—
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#407C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2857C2212yes—
22#2851C2212yes—
23#2667—21no—
24#2665—21no—
25#2569—21no—
26#2549—21no—
27#2501—22no—
28#2501—21no—
29#2501—21no—
30#2501C2212yes—
31#2501C2214yes—
32#2501C2212yes—
33#2442C2212yescomplemented
34#2441C2212yescomplemented
35#2440C2212yescomplemented
36#2439C2212yescomplemented
37#2437—22no—
38C2×#399—21no—
39C2×#399—21no—
40#2329—22no—
41C2×#390C2212yes—
42C2×#384C2212yes—
43#2280C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#2100C2214yescomplemented
47#2100C2214yescomplemented
48#2099C2214yescomplemented
49#2099C2214yescomplemented
50#2073C2214yescomplemented
51#2071C2214yescomplemented
52#2025C2214yescomplemented
53#2023C2214yescomplemented
54#1992C2214yescomplemented
55#1990C2214yescomplemented
56#1936C2214yescomplemented
57#1936C2214yescomplemented
58#1934C2214yescomplemented
59#1934C2214yescomplemented
60#1895C2212yes—
61#1895C2212yes—
62#1893C2214yes—
63#1875C2214yes—
64#1874C2214yes—
65#1836C2214yescomplemented
66#1834C2214yescomplemented
67#1824C2211yescharacteristic
68#1824C2211yescharacteristic
69#1824C2212yes—
70#1824C2212yes—
71#1824C2212yes—
72#1824C2212yes—
73#1824C2212yes—
74#1824C2212yescomplemented
75#1824C2212yescomplemented
76#1823C2214yescomplemented
77#1786C2211yescharacteristic
78#1786C2211yescharacteristic
79#1786C2212yes—
80#1753C2214yescomplemented
81#1752C2212yescomplemented
82C8×C22⋊C4C2212yescomplemented
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2214yescomplemented
86C2×#340C2214yescomplemented
87C23×D4—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C23×C8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×#125C2312yes—
95C2×#123—24no—
96C2×#123—24no—
97C2×#123—24no—
98C2×#123—24no—
99C2×#123—416no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121—24no—
105C2×#121—24no—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—24no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—22no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4D412yes—
126C22×C42C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#407—24no—
132C2×C22⋊C8—24no—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2311yescharacteristic
135C2×C22⋊C8C2311yescharacteristic
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2314yes—
140#400—22no—
141#399—22no—
142#399—24no—
143#399—24no—
144#399—22no—
145#399—22no—
146#399—22no—
147#399—22no—
148#398—22no—
149#398—22no—
150#396—22no—
151#396—22no—
152#395—22no—
153#393—22no—
154#391—22no—
155#391—22no—
156#390—22no—
157#390—22no—
158#390C2314yes—
159#390C2314yes—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#384—22no—
175#384—22no—
176#384C2312yes—
177#384C2314yes—
178#384C2314yes—
179#382C2312yes—
180#381—21no—
181#380—22no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380C2312yes—
186#380C2311yescharacteristic
187#380C2312yes—
188#380C2311yescharacteristic
189#380C2311yescharacteristic
190#380C2311yescharacteristic
191#379—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4D412yes—
195C4×C22⋊C4D411yescharacteristic
196C4×C22⋊C4D411yescharacteristic
197C4×C22⋊C4C2312yes—
198C4×C22⋊C4C2311yescharacteristic
199C4×C22⋊C4C2311yescharacteristic
200C2×#95—21no—
201C2×#95D411yescharacteristic
202C2×#95D411yescharacteristic
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25—21no—
218C25C2×D411yescharacteristic
219C22×Q8—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—44no—
226C22×D4—48no—
227C22×D4—44no—
228C22×D4C2412yes—
229C23×C4—22no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—44no—
235C23×C4—48no—
236C23×C4—44no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4#4114yes—
243C23×C4#4114yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×M16—24no—
248C2×M16—44no—
249C22×C8—24no—
250C22×C8—44no—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C4⋊Q8—44no—
255#125—44no—
256#123—416no—
257#123—416no—
258#123—416no—
259#123—416no—
260C22⋊Q8—48no—
261C22⋊Q8—84no—
262#121—24no—
263#121—24no—
264#121—24no—
265#121—24no—
266#121—44no—
267#121—44no—
268#117—44no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—48no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2412yes—
335#102—44no—
336#102—44no—
337#102—44no—
338#102—44no—
339C22⋊C8—22no—
340C22⋊C8—22no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344#97—44no—
345C4×C8—44no—
346#95—21no—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D411yescharacteristic
361#95C2×D411yescharacteristic
362#95C2×D411yescharacteristic
363#95C2×D411yescharacteristic
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—42no—
369C24—44no—
370C24—44no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24#12414yes—
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378C2×Q8—44no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—816no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—816no—
427C22×C4#12414yes—
428C22×C4#12414yes—
429C22×C4#12414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2511yescharacteristic
444M16—84no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—84no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—82no—
495C22⋊C4AGL(1,ℤ/8)18yes—
496C42—24no—
497C42—24no—
498C42—42no—
499C42—42no—
500C42—82no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—21no—
505C23—21no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—44no—
513C23—48no—
514C23—48no—
515C23—42no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—84no—
521C23—84no—
522C23—88no—
523C23—88no—
524C23—84no—
525C23C2×#12412yes—
526C23C2×#12412yes—
527C23C2×#12412yes—
528C23C2×AGL(1,ℤ/8)14yes—
529C23#57712yes—
530C23#57812yes—
531C23C23×D411yescharacteristic
532Q8—164no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—84no—
537D4—88no—
538D4—88no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—88no—
581C2×C4—88no—
582C2×C4—84no—
583C2×C4—88no—
584C2×C4—164no—
585C2×C4C2×#12412yes—
586C2×C4C2×#12412yes—
587C2×C4C2×#12412yes—
588C2×C4C2×AGL(1,ℤ/8)14yes—
589C2×C4C2×AGL(1,ℤ/8)14yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4#58514yes—
593C8—42no—
594C8—42no—
595C8—44no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—42no—
609C22—42no—
610C22—84no—
611C22—84no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—168no—
616C22#313714yes—
617C22#313814yes—
618C22C22×#12411yescharacteristic
619C22#348512yes—
620C22C22×AGL(1,ℤ/8)12yes—
621C22C2×#57711yescharacteristic
622C22C2×#57811yescharacteristic
623C4—24no—
624C4—24no—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—84no—
635C4—82no—
636C4#349112yes—
637C4#349112yes—
638C2—24no—
639C2—24no—
640C2—42no—
641C2—162no—
642C2#3545511yescharacteristic
643C2C2×#313711yescharacteristic
644C2C2×#313811yescharacteristic
645C2#6172812yes—
646C2#6173712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35079C2semidirect
3. #29109C2semidirect
4. #29104C2semidirect
5. #27965C2semidirect
6. #27961C2semidirect
7. #27719C2semidirect
8. #27717C2semidirect
9. #27709C2semidirect
10. #27695C2semidirect
13. #20700C2semidirect
14. C2×#1824C2semidirect
15. #20525C2semidirect
33. #2442C22semidirect
34. #2441C22semidirect
35. #2440C22semidirect
36. #2439C22semidirect
46. #2100C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2099C22semidirect
50. #2073C22semidirect
51. #2071C22semidirect
52. #2025C22semidirect
53. #2023C22semidirect
54. #1992C22semidirect
55. #1990C22semidirect
56. #1936C22semidirect
57. #1936C22semidirect
58. #1934C22semidirect
59. #1934C22semidirect
65. #1836C22semidirect
66. #1834C22semidirect
74. #1824C22semidirect
75. #1824C22semidirect
76. #1823C22semidirect
80. #1753C22semidirect
81. #1752C22semidirect
82. C8×C22⋊C4C22semidirect
85. #1736C22semidirect
86. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
#350792
C2×#25012
#291092
#291042
#279652
#279612
#277192
#277172
#277092
#276952
#217642
#217612
#207002
C2×#18242
#205252