Finite Group Databasebeta

← #6345402

Subgroups — #6345402

Order 512.

1703 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60561C211yesmaximal, characteristic
2#60282C211yesmaximal, characteristic, complemented
3#60232C211yesmaximal, characteristic, complemented
4#18046C218yesmaximal, complemented
5#18022C214yesmaximal, complemented
6#18015C214yesmaximal, complemented
7#17951C214yesmaximal, complemented
8#17922C214yesmaximal, complemented
9#17823C211yesmaximal, characteristic, complemented
10#17500C211yesmaximal, characteristic, complemented
11#17447C211yesmaximal, characteristic, complemented
12#17334C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#3115C414yes—
16#3112C412yes—
17#3109C412yes—
18#3106C2214yes—
19#3103C2212yes—
20#3100C2212yes—
21#3094C412yes—
22#3079C412yes—
23#3038C412yes—
24#3006C412yes—
25#2990—22no—
26#2990C2214yescomplemented
27#2983C2212yescomplemented
28#2981—22no—
29#2981C2214yes—
30#2980—22no—
31#2979C2214yescomplemented
32#2973—22no—
33#2972C2212yescomplemented
34#2970—22no—
35#2970C2214yes—
36C4×AGL(1,ℤ/8)C2214yes—
37D4×M16—22no—
38D4×M16C2214yescomplemented
39#2920C2211yescharacteristic
40#2920C2211yescharacteristic
41#2920C2211yescharacteristic
42#2920C2211yescharacteristic
43C2×#432—22no—
44#2879—22no—
45C2×C4×M16C2211yescharacteristic
46#1592C2218yescomplemented
47#1590C2218yescomplemented
48#1586C2218yescomplemented
49#1585C2218yescomplemented
50#1582C2218yescomplemented
51#1581C2218yescomplemented
52#1580C2218yescomplemented
53C8×D8C2218yescomplemented
54#1574C2214yescomplemented
55#1572C2214yescomplemented
56#1569C2214yescomplemented
57#1492C2212yescomplemented
58#1492C2212yescomplemented
59#1491C2212yescomplemented
60#1490C2212yescomplemented
61#1490C2214yescomplemented
62#1489C2214yes—
63#1484C2214yes—
64#1476C2212yescomplemented
65#1476C2212yescomplemented
66#1455C2214yescomplemented
67#1453C2214yescomplemented
68#1452C2212yescomplemented
69#1450C2212yescomplemented
70C22×#41—21no—
71C22×#41C2×C411yescharacteristic
72C2×AGL(1,ℤ/8)C2×C414yes—
73C22×M16—21no—
74C22×M16—24no—
75#532—21no—
76#532—21no—
77#532—21no—
78#532—21no—
79#532C2×C412yes—
80#532C2×C412yes—
81#532C2312yes—
82#532C2312yes—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#127C2×C411yescharacteristic
85C2×#124—21no—
86C2×#121—21no—
87C2×#121C2×C412yes—
88C4×#41—21no—
89C4×#41—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41C2×C412yes—
93C4×#41C2×C412yes—
94C4×#41C2312yes—
95C4×#41C2312yes—
96C2×C4×Q8C2×C411yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4C2×C411yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×C4×D4C2312yes—
104C2×#117—21no—
105C2×#117C2×C411yescharacteristic
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#482C2×C418yes—
109#481C2×C414yes—
110#480C2×C414yes—
111#477C2×C414yes—
112#476C2×C414yes—
113#475C2×C414yes—
114#474C2×C414yes—
115#460C2×C414yes—
116#460C2×C414yes—
117#459C2×C414yes—
118#459C2×C414yes—
119#449C2×C418yes—
120#448C2×C414yes—
121#447C2×C414yes—
122#446—24no—
123#446C2318yes—
124#442C2×C418yes—
125#440C2×C418yes—
126C4×SD16C2318yes—
127C4×D8C2318yes—
128#436—24no—
129#436C2318yes—
130#435—28no—
131#435—28no—
132#435—28no—
133#435—28no—
134C8×D4—24no—
135C8×D4—24no—
136C8×D4C2318yes—
137C8×D4C2318yes—
138#432—21no—
139#432—21no—
140#432—24no—
141#432—24no—
142#432—24no—
143#432C2×C412yes—
144#432C2×C412yes—
145#432D412yes—
146#432D412yes—
147#432C2312yes—
148#432C2312yes—
149#431—21no—
150#431—21no—
151#431C2312yes—
152#431C2312yes—
153#431C2312yes—
154#431C2312yes—
155#431C2312yes—
156#431C2312yes—
157#428C2×C414yes—
158#423C2×C412yes—
159#423D412yes—
160#423C2312yes—
161#423C2312yes—
162C2×C4⋊C8—24no—
163#418C2×C414yes—
164#417C2×C414yes—
165#408—28no—
166#408—28no—
167#407—21no—
168#407—21no—
169#407—24no—
170#407C2×C412yes—
171#407D412yes—
172C2×C22⋊C8—24no—
173C4×M16—21no—
174C4×M16—21no—
175C4×M16—21no—
176C4×M16—21no—
177C4×M16C2312yes—
178C4×M16C2312yes—
179C4×M16C2312yes—
180C4×M16C2312yes—
181C2×#97—21no—
182C2×#97—24no—
183C2×C4×C8C2311yescharacteristic
184C2×C4×C8C2311yescharacteristic
185#335C2318yes—
186#334C2318yes—
187Q8⋊C8C2314yes—
188Q8⋊C8C2314yes—
189D4⋊C8C2314yes—
190D4⋊C8C2318yes—
191D4⋊C8C2314yes—
192D4⋊C8C2318yes—
193#322C2314yes—
194#322C2314yes—
195#322C2314yes—
196C82C2314yes—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—22no—
200C2×#41—22no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41—44no—
204C2×#41—84no—
205C2×#41C22×C412yes—
206C2×#41C22×C412yes—
207C22×Q8—21no—
208C22×Q8C22×C411yescharacteristic
209C22×D4—21no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C22×C411yescharacteristic
214C22×D4C22×C412yes—
215C23×C4—21no—
216C23×C4—22no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4#4111yescharacteristic
222AGL(1,ℤ/8)—48no—
223C2×SD16C22×C418yes—
224C2×D8C22×C418yes—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—22no—
229C2×M16—28no—
230C2×M16—44no—
231C2×M16—88no—
232C2×M16C22×C414yes—
233C2×M16C2×D414yes—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—28no—
237C22×C8—28no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8C22×C412yes—
241C4⋊Q8C22×C412yes—
242#127—21no—
243#127—21no—
244#127C22×C412yes—
245#127C22×C412yes—
246#124—22no—
247#124—22no—
248#124—22no—
249#124—22no—
250#121—22no—
251#121—22no—
252#121—22no—
253#121—22no—
254#121—22no—
255#121—22no—
256#121C22×C414yes—
257#121C22×C414yes—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×Q8C22×C412yes—
263C4×Q8C22×C412yes—
264C4×Q8C22×C412yes—
265C4×Q8C2412yes—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—21no—
281C4×D4—22no—
282C4×D4—21no—
283C4×D4—22no—
284C4×D4C22×C412yes—
285C4×D4C22×C412yes—
286C4×D4C22×C412yes—
287C4×D4C22×C414yes—
288C4×D4C2412yes—
289C4×D4C2414yes—
290#117—21no—
291#117—21no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117C22×C412yes—
297#117C22×C412yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C22×C411yescharacteristic
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C412yes—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42#4111yescharacteristic
323C2×C42#4111yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107C22×C418yes—
329#106C22×C418yes—
330C4⋊C8—28no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8C22×C414yes—
334C4⋊C8C22×C414yes—
335C4⋊C8C2×D414yes—
336C4⋊C8C2×D414yes—
337C4⋊C8C2414yes—
338C4⋊C8C2414yes—
339#103C22×C418yes—
340#102C22×C418yes—
341#102C22×C418yes—
342#102C22×C418yes—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346C22⋊C8—22no—
347C22⋊C8—28no—
348C22⋊C8—28no—
349C22⋊C8—28no—
350C22⋊C8C22×C414yes—
351C22⋊C8C22×C414yes—
352C22⋊C8C2×D414yes—
353C22⋊C8C2×D414yes—
354#97—28no—
355#97—22no—
356#97—22no—
357#97—22no—
358#97—22no—
359#97C22×C414yes—
360#97C2×D414yes—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8C2412yes—
366C4×C8C2412yes—
367C4×C8C2414yes—
368C4×C8C2414yes—
369C24—21no—
370C24—22no—
371C24C4×D411yescharacteristic
372#41—24no—
373#41—24no—
374#41—42no—
375#41—42no—
376#41—44no—
377#41—44no—
378#41—82no—
379#41—82no—
380#41—84no—
381#41—84no—
382#41—84no—
383C2×Q8—21no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—82no—
388C2×Q8C23×C412yes—
389C2×D4—24no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—82no—
403C2×D4—82no—
404C2×D4—84no—
405C2×D4C23×C414yes—
406C2×D4C23×C412yes—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4C4×D411yescharacteristic
431C22×C4C4×D411yescharacteristic
432C22×C4C4×D411yescharacteristic
433C22×C4C4×D411yescharacteristic
434C22×C4C4×D411yescharacteristic
435C22×C4C23×C411yescharacteristic
436C22×C4C23×C411yescharacteristic
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444SD16—48no—
445D8—216no—
446M16—44no—
447M16—44no—
448M16—44no—
449M16—44no—
450M16—44no—
451M16—88no—
452M16—82no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—42no—
459C2×C8—88no—
460C2×C8—88no—
461C2×C8C23×C418yes—
462C2×C8C22×D418yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4C23×C412yes—
473C4⋊C4C23×C412yes—
474C4⋊C4C23×C414yes—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—22no—
484C22⋊C4—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42C4×D411yescharacteristic
492C42C4×D411yescharacteristic
493C42C4×D411yescharacteristic
494C42C4×D411yescharacteristic
495C42C4×D411yescharacteristic
496C42C4×D411yescharacteristic
497C42C23×C411yescharacteristic
498C42C23×C411yescharacteristic
499C42C23×C411yescharacteristic
500C42C22×D411yescharacteristic
501C42C22×D411yescharacteristic
502C42C22×D411yescharacteristic
503C42C2×#4111yescharacteristic
504C42C2×#4111yescharacteristic
505C42C2×#4111yescharacteristic
506C42C2511yescharacteristic
507C23—24no—
508C23—22no—
509C23—24no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—44no—
514C23—81no—
515C23—81no—
516C23—82no—
517C23—82no—
518C23C2×C4×D411yescharacteristic
519C23C2×C4×D411yescharacteristic
520C23C2×C4×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—42no—
524Q8—82no—
525Q8—82no—
526D4—24no—
527D4—22no—
528D4—28no—
529D4—44no—
530D4—44no—
531D4—44no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—84no—
537D4—84no—
538D4#56814yes—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4C2×C4×D411yescharacteristic
575C2×C4C2×C4×D411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×C4×D411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C4×D411yescharacteristic
581C2×C4C2×C4×D411yescharacteristic
582C2×C4C2×C4×D411yescharacteristic
583C2×C4C24×C411yescharacteristic
584C2×C4C23×D411yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C8—24no—
587C8—28no—
588C8—28no—
589C8—42no—
590C8—44no—
591C8—44no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—24no—
602C22—42no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—84no—
608C22—82no—
609C22—164no—
610C22C22×C4×D411yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4#295612yes—
625C4C2×#56812yes—
626C4#358512yes—
627C2—41no—
628C2—41no—
629C2—41no—
630C2—44no—
631C2—84no—
632C2#6014111yescharacteristic
633C2#6016811yescharacteristic
634C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60561C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60282C2semidirect
3. #60232C2semidirect
4. #18046C2semidirect
5. #18022C2semidirect
6. #18015C2semidirect
7. #17951C2semidirect
8. #17922C2semidirect
9. #17823C2semidirect
10. #17500C2semidirect
11. #17447C2semidirect
12. #17334C2semidirect
26. #2990C22semidirect
27. #2983C22semidirect
31. #2979C22semidirect
33. #2972C22semidirect
38. D4×M16C22semidirect
46. #1592C22semidirect
47. #1590C22semidirect
48. #1586C22semidirect
49. #1585C22semidirect
50. #1582C22semidirect
51. #1581C22semidirect
52. #1580C22semidirect
53. C8×D8C22semidirect
54. #1574C22semidirect
55. #1572C22semidirect
56. #1569C22semidirect
57. #1492C22semidirect
58. #1492C22semidirect
59. #1491C22semidirect
60. #1490C22semidirect
61. #1490C22semidirect
64. #1476C22semidirect
65. #1476C22semidirect
66. #1455C22semidirect
67. #1453C22semidirect
68. #1452C22semidirect
69. #1450C22semidirect

Maximal subgroups

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

subgroupindex
#605612
#602822
#602322
#180462
#180222
#180152
#179512
#179222
#178232
#175002
#174472
#173342