Finite Group Databasebeta

← #6346071

Subgroups — #6346071

Order 512.

2017 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D8×#41C211yesmaximal, characteristic
2#60147C211yesmaximal, characteristic, complemented
3D4×#131C211yesmaximal, characteristic, complemented
4#18073C213yesmaximal, complemented
5#18044C213yesmaximal, complemented
6#18001C213yesmaximal, complemented
7#18001C213yesmaximal, complemented
8#17981C213yesmaximal, complemented
9#17935C213yesmaximal, complemented
10#17907C213yesmaximal, complemented
11C2×#1586C213yesmaximal, complemented
12#17826C211yesmaximal, characteristic, complemented
13#17507C211yesmaximal, characteristic, complemented
14#17507C211yesmaximal, characteristic, complemented
15#17320C211yesmaximal, characteristic, complemented
16C2×#568—21no—
17C22×#131—21no—
18D4×#41C2211yescharacteristic
19D4×#41C2211yescharacteristic
20#3391C411yescharacteristic
21Q8×D8C411yescharacteristic
22#3358C411yescharacteristic
23#3358C411yescharacteristic
24#3334C413yes—
25#3294C413yes—
26#3294C413yes—
27D4×D8C413yes—
28#3169C2213yes—
29#3100C2213yes—
30#3100C2213yes—
31C8×#41C2211yescharacteristic
32C8×#41C2211yescharacteristic
33C2×C4×D8C2213yes—
34#2935C2213yescomplemented
35#2935C2213yescomplemented
36D4×M16C2213yescomplemented
37D4×M16C2213yescomplemented
38#2933—23no—
39#2933—23no—
40#2931C2213yescomplemented
41#2931C2213yescomplemented
42#2930—23no—
43#2929—23no—
44C2×#436C2213yescomplemented
45C2×C8×D4C2213yescomplemented
46#2906C2211yescharacteristic
47#2906C2211yescharacteristic
48#2880—21no—
49#2880—21no—
50C4×#131C2211yescharacteristic
51#1594C2216yescomplemented
52#1594C2216yescomplemented
53#1586C2213yes—
54#1586C2211yescharacteristic
55#1586C2213yescomplemented
56#1586C2216yescomplemented
57#1586C2213yescomplemented
58#1586C2216yescomplemented
59#1586C2216yescomplemented
60#1586C2213yescomplemented
61#1586C2213yescomplemented
62#1586C2213yescomplemented
63#1586C2213yescomplemented
64#1580C2216yescomplemented
65#1580C2216yescomplemented
66#1570C2213yescomplemented
67#1570C2213yescomplemented
68#1566C2213yescomplemented
69C2×#334C2213yescomplemented
70#1492C2213yescomplemented
71#1492C2213yescomplemented
72#1486C2213yescomplemented
73#1486C2213yescomplemented
74#1480C2213yescomplemented
75#1480C2213yescomplemented
76C2×D4⋊C8C2213yescomplemented
77C2×D4⋊C8C2213yescomplemented
78#1452C2213yescomplemented
79#1451C2213yescomplemented
80#1451C2213yescomplemented
81C2×#322C2213yescomplemented
82C22×#41—21no—
83C22×#41—21no—
84C22×D8C2×C413yes—
85#568—48no—
86C2×#131—21no—
87C2×#131—21no—
88C2×#131—23no—
89C2×#131—23no—
90C2×#131—23no—
91C2×#131—23no—
92C2×#131—48no—
93C2×#131D411yescharacteristic
94C2×#131D411yescharacteristic
95C22×M16—23no—
96C22×M16—23no—
97C22×M16—23no—
98C23×C8—23no—
99#550C2×C411yescharacteristic
100#550C2×C411yescharacteristic
101D4×Q8C2×C411yescharacteristic
102D4×Q8C2×C411yescharacteristic
103#548—21no—
104#548—21no—
105#547C2×C413yes—
106#547C2×C413yes—
107#546—23no—
108#546—23no—
109D42C2×C413yes—
110D42C2×C413yes—
111#532C2313yes—
112#532C2313yes—
113#525—23no—
114#525—23no—
115C4×#41C2311yescharacteristic
116C2×C4×D4C2313yes—
117C2×C4×D4C2313yes—
118#500C2×C413yes—
119#493C2×C413yes—
120#486C2×C413yes—
121#486C2×C413yes—
122#480C2×C416yes—
123#480C2×C416yes—
124#474C2×C413yes—
125#474C2×C413yes—
126#474C2×C413yes—
127#474C2×C413yes—
128#466C2×C416yes—
129#466C2×C416yes—
130#459C2×C413yes—
131#459C2×C413yes—
132#459C2×C413yes—
133#459C2×C413yes—
134#447C2×C416yes—
135#447C2×C416yes—
136C8×Q8—21no—
137C8×Q8C2×C411yescharacteristic
138C8×Q8C2×C411yescharacteristic
139C4×D8C2×C413yes—
140C4×D8C2×C413yes—
141C4×D8C2×C413yes—
142C4×D8C2×C413yes—
143C4×D8C2313yes—
144C4×D8C2316yes—
145C4×D8C2313yes—
146C4×D8C2311yescharacteristic
147C4×D8C2313yes—
148#436—23no—
149#436—23no—
150#436—23no—
151#436—23no—
152#436C2311yescharacteristic
153#436C2313yes—
154#436C2313yes—
155#436C2313yes—
156#436C2316yes—
157#435—26no—
158#435—26no—
159#435—26no—
160#435—26no—
161C8×D4—23no—
162C8×D4—23no—
163C8×D4—23no—
164C8×D4—23no—
165C8×D4—23no—
166C8×D4C2×C413yes—
167C8×D4C2×C413yes—
168C8×D4C2313yes—
169C8×D4C2313yes—
170C8×D4C2316yes—
171C8×D4C2311yescharacteristic
172C8×D4C2313yes—
173#431C2313yes—
174#431C2313yes—
175C2×#107C2×C413yes—
176#424—23no—
177#424—23no—
178#423C2313yes—
179#423C2313yes—
180C2×C4⋊C8C2313yes—
181C2×C4⋊C8C2313yes—
182C2×#102C2×C413yes—
183C2×#102C2×C413yes—
184#408—23no—
185#408—23no—
186#408—23no—
187#408—23no—
188#407—23no—
189#407—23no—
190C2×C22⋊C8—23no—
191C2×C22⋊C8—23no—
192#405D413yes—
193#405D413yes—
194C4×M16C2313yes—
195C2×C4×C8C2313yes—
196C2×C4×C8C2313yes—
197C2×C4×C8C2313yes—
198#334C2311yescharacteristic
199#334C2313yes—
200#334C2313yes—
201#334C2313yes—
202#334C2316yes—
203D4⋊C8C2313yes—
204D4⋊C8C2313yes—
205D4⋊C8C2316yes—
206D4⋊C8C2316yes—
207D4⋊C8C2311yescharacteristic
208D4⋊C8C2311yescharacteristic
209D4⋊C8C2313yes—
210D4⋊C8C2313yes—
211D4⋊C8C2313yes—
212D4⋊C8C2313yes—
213#322C2316yes—
214#322C2311yescharacteristic
215#322C2313yes—
216#322C2313yes—
217#322C2313yes—
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—23no—
221C2×#41—23no—
222C2×#41—23no—
223C2×#41—23no—
224C2×#41—44no—
225C2×#41—44no—
226C2×#41—44no—
227C2×#41—44no—
228C2×#41C2×D411yescharacteristic
229C22×Q8—21no—
230C22×Q8—21no—
231C22×D4—23no—
232C22×D4—23no—
233C22×D4C22×C413yes—
234C22×D4C22×C413yes—
235C23×C4—23no—
236C23×C4—23no—
237C2×D8—46no—
238C2×D8C22×C411yescharacteristic
239C2×D8C22×C413yes—
240C2×D8C22×C413yes—
241C2×D8C22×C413yes—
242C2×D8C22×C416yes—
243#131—24no—
244#131—24no—
245#131—412no—
246#131—412no—
247#131—412no—
248#131—412no—
249#131—44no—
250#131—44no—
251C2×M16—23no—
252C2×M16—23no—
253C2×M16—23no—
254C2×M16—21no—
255C2×M16—26no—
256C2×M16—23no—
257C2×M16—23no—
258C2×M16—412no—
259C2×M16—412no—
260C2×M16—412no—
261C2×M16C2×D413yes—
262C2×M16C2×D413yes—
263C22×C8—23no—
264C22×C8—23no—
265C22×C8—23no—
266C22×C8—21no—
267C22×C8—23no—
268C22×C8—23no—
269C22×C8—23no—
270C22×C8—26no—
271C22×C8—412no—
272C22×C8C22×C413yes—
273C22×C8C22×C413yes—
274C22×C8C2×D413yes—
275C22×C8C2×D413yes—
276C4⋊Q8C22×C413yes—
277C4⋊Q8C22×C413yes—
278#127C22×C413yes—
279#127C22×C413yes—
280#124—23no—
281#124—23no—
282#123—26no—
283#123—26no—
284C22⋊Q8—23no—
285C22⋊Q8—23no—
286C22⋊Q8—23no—
287C22⋊Q8—23no—
288#121—23no—
289#121—23no—
290#121—23no—
291#121—23no—
292#121—23no—
293#121—23no—
294#121C22×C416yes—
295#121C22×C416yes—
296#120—26no—
297#120—26no—
298C4×Q8C22×C411yescharacteristic
299C4×Q8#4111yescharacteristic
300C4×D4—23no—
301C4×D4—23no—
302C4×D4—23no—
303C4×D4—23no—
304C4×D4—23no—
305C4×D4—23no—
306C4×D4C22×C413yes—
307C4×D4C22×C413yes—
308C4×D4C22×C413yes—
309C4×D4C22×C413yes—
310C4×D4C22×C413yes—
311C4×D4#4113yes—
312C4×D4C2411yescharacteristic
313C4×D4C2411yescharacteristic
314C4×D4C2413yes—
315C4×D4C2413yes—
316#117C2×D413yes—
317C2×C4⋊C4C22×C413yes—
318C2×C4⋊C4C22×C413yes—
319C2×C22⋊C4—23no—
320C2×C22⋊C4—23no—
321C2×C42C2413yes—
322#107C22×C411yescharacteristic
323#107C22×C413yes—
324#107C22×C413yes—
325#107C22×C413yes—
326#107C22×C416yes—
327C4⋊C8—23no—
328C4⋊C8—23no—
329C4⋊C8—23no—
330C4⋊C8—23no—
331C4⋊C8—23no—
332C4⋊C8C22×C413yes—
333C4⋊C8C22×C413yes—
334C4⋊C8C2411yescharacteristic
335C4⋊C8C2411yescharacteristic
336C4⋊C8C2413yes—
337C4⋊C8C2413yes—
338#102C22×C411yescharacteristic
339#102C22×C411yescharacteristic
340#102C22×C413yes—
341#102C22×C413yes—
342#102C22×C413yes—
343#102C22×C413yes—
344#102C22×C413yes—
345#102C22×C413yes—
346#102C22×C416yes—
347#102C22×C416yes—
348C22⋊C8—23no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—23no—
352C22⋊C8—23no—
353C22⋊C8—23no—
354C22⋊C8—23no—
355C22⋊C8—23no—
356C22⋊C8—23no—
357C22⋊C8—26no—
358C22⋊C8—26no—
359C22⋊C8C22×C416yes—
360#97C2×D416yes—
361C4×C8—23no—
362C4×C8C22×C413yes—
363C4×C8C22×C413yes—
364C4×C8C2×D413yes—
365C4×C8C2×D413yes—
366C4×C8C2411yescharacteristic
367C4×C8C2413yes—
368C4×C8C2411yescharacteristic
369C4×C8C2413yes—
370C4×C8C2411yescharacteristic
371C4×C8C2413yes—
372C24—23no—
373C24—23no—
374#41—44no—
375#41—44no—
376#41—412no—
377#41—412no—
378#41—46no—
379#41—46no—
380#41—46no—
381#41—46no—
382#41AGL(1,ℤ/8)14yes—
383#41AGL(1,ℤ/8)14yes—
384C2×Q8—23no—
385C2×Q8—23no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8C4×D411yescharacteristic
389C2×D4—26no—
390C2×D4—26no—
391C2×D4—26no—
392C2×D4—23no—
393C2×D4—23no—
394C2×D4—23no—
395C2×D4—23no—
396C2×D4—23no—
397C2×D4—23no—
398C2×D4—43no—
399C2×D4—43no—
400C2×D4—43no—
401C2×D4—412no—
402C2×D4—412no—
403C2×D4C4×D413yes—
404C2×D4C23×C411yescharacteristic
405C2×D4C23×C413yes—
406C2×D4C23×C413yes—
407C2×D4C23×C411yescharacteristic
408C22×C4—23no—
409C22×C4—23no—
410C22×C4—23no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—23no—
414C22×C4—21no—
415C22×C4—23no—
416C22×C4—23no—
417C22×C4—21no—
418C22×C4—46no—
419C22×C4—46no—
420C22×C4—46no—
421C22×C4—46no—
422C22×C4C23×C413yes—
423C22×C4C22×D413yes—
424C22×C4C2×#4113yes—
425D8—22no—
426D8—26no—
427D8—43no—
428D8—43no—
429D8—46no—
430M16—26no—
431M16—26no—
432M16—42no—
433M16—46no—
434M16—46no—
435M16—46no—
436M16—412no—
437M16—46no—
438M16—46no—
439C2×C8—26no—
440C2×C8—26no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—23no—
445C2×C8—23no—
446C2×C8—23no—
447C2×C8—43no—
448C2×C8—43no—
449C2×C8—46no—
450C2×C8—42no—
451C2×C8—46no—
452C2×C8—46no—
453C2×C8—46no—
454C2×C8—412no—
455C2×C8—46no—
456C2×C8—46no—
457C2×C8C23×C411yescharacteristic
458C2×C8C23×C411yescharacteristic
459C2×C8C23×C413yes—
460C2×C8C23×C413yes—
461C2×C8C22×D411yescharacteristic
462C2×C8C22×D411yescharacteristic
463C2×C8C22×D413yes—
464C2×C8C22×D413yes—
465C4⋊C4—23no—
466C4⋊C4—23no—
467C4⋊C4—23no—
468C4⋊C4—23no—
469C4⋊C4C4×D413yes—
470C4⋊C4C4×D413yes—
471C4⋊C4C23×C411yescharacteristic
472C4⋊C4C23×C411yescharacteristic
473C4⋊C4C23×C413yes—
474C4⋊C4C23×C413yes—
475C22⋊C4—23no—
476C22⋊C4—23no—
477C22⋊C4—23no—
478C22⋊C4—23no—
479C22⋊C4—26no—
480C22⋊C4—26no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—23no—
484C22⋊C4—23no—
485C22⋊C4C4×D416yes—
486C42C23×C413yes—
487C42C22×D413yes—
488C42C2×#4113yes—
489C42C2511yescharacteristic
490C23—21no—
491C23—21no—
492C23—23no—
493C23—23no—
494C23—46no—
495C23—46no—
496C23—46no—
497C23—83no—
498C23—83no—
499C23—83no—
500C23C2×C4×D413yes—
501Q8—46no—
502Q8—46no—
503Q8#44214yes—
504D4—21no—
505D4—23no—
506D4—23no—
507D4—21no—
508D4—23no—
509D4—23no—
510D4—21no—
511D4—46no—
512D4—412no—
513D4—412no—
514D4—46no—
515D4—46no—
516D4—83no—
517D4—83no—
518D4—83no—
519D4#442112yes—
520D4#56812yescomplemented
521C2×C4—23no—
522C2×C4—23no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—23no—
526C2×C4—23no—
527C2×C4—43no—
528C2×C4—46no—
529C2×C4—46no—
530C2×C4—46no—
531C2×C4—46no—
532C2×C4—46no—
533C2×C4—46no—
534C2×C4—41no—
535C2×C4—43no—
536C2×C4—43no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—43no—
541C2×C4—43no—
542C2×C4C2×C4×D413yes—
543C2×C4C2×C4×D413yes—
544C2×C4C2×C4×D413yes—
545C2×C4C2×AGL(1,ℤ/8)16yes—
546C2×C4C2×AGL(1,ℤ/8)16yes—
547C2×C4C24×C411yescharacteristic
548C2×C4C23×D411yescharacteristic
549C2×C4C22×#4111yescharacteristic
550C8—22no—
551C8—21no—
552C8—23no—
553C8—21no—
554C8—23no—
555C8—21no—
556C8—21no—
557C8—23no—
558C8—23no—
559C8—43no—
560C8—41no—
561C8—43no—
562C8—43no—
563C8—41no—
564C22—23no—
565C22—23no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—43no—
570C22—43no—
571C22—41no—
572C22—43no—
573C22—41no—
574C22—43no—
575C22—41no—
576C22—43no—
577C22—43no—
578C22—812no—
579C22—166no—
580C22C2×#44216yes—
581C22C22×C4×D411yescharacteristic
582C4—21no—
583C4—23no—
584C4—23no—
585C4—41no—
586C4—41no—
587C4—43no—
588C4—43no—
589C4C2×#44216yes—
590C4C2×#56811yescharacteristic
591C4#357311yescharacteristic
592C4C22×AGL(1,ℤ/8)11yescharacteristic
593C4C22×AGL(1,ℤ/8)11yescharacteristic
594C2—26no—
595C2—42no—
596C2—83no—
597C2—83no—
598C2—81no—
599C2#6014611yescharacteristic
600C2C22×#44211yescharacteristic
601C2#6018311yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalD8×#41C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#131 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60147C2semidirect
3. D4×#131C2semidirect
4. #18073C2semidirect
5. #18044C2semidirect
6. #18001C2semidirect
7. #18001C2semidirect
8. #17981C2semidirect
9. #17935C2semidirect
10. #17907C2semidirect
11. C2×#1586C2semidirect
12. #17826C2semidirect
13. #17507C2semidirect
14. #17507C2semidirect
15. #17320C2semidirect
34. #2935C22semidirect
35. #2935C22semidirect
36. D4×M16C22semidirect
37. D4×M16C22semidirect
40. #2931C22semidirect
41. #2931C22semidirect
44. C2×#436C22semidirect
45. C2×C8×D4C22semidirect
51. #1594C22semidirect
52. #1594C22semidirect
55. #1586C22semidirect
56. #1586C22semidirect
57. #1586C22semidirect
58. #1586C22semidirect
59. #1586C22semidirect
60. #1586C22semidirect
61. #1586C22semidirect
62. #1586C22semidirect
63. #1586C22semidirect
64. #1580C22semidirect
65. #1580C22semidirect
66. #1570C22semidirect
67. #1570C22semidirect
68. #1566C22semidirect
69. C2×#334C22semidirect
70. #1492C22semidirect
71. #1492C22semidirect
72. #1486C22semidirect
73. #1486C22semidirect
74. #1480C22semidirect
75. #1480C22semidirect
76. C2×D4⋊C8C22semidirect
77. C2×D4⋊C8C22semidirect
78. #1452C22semidirect
79. #1451C22semidirect
80. #1451C22semidirect
81. C2×#322C22semidirect
520. D4#568semidirect

Maximal subgroups

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

subgroupindex
D8×#412
#601472
D4×#1312
#180732
#180442
#180012
#179812
#179352
#179072
C2×#15862
#178262
#175072
#173202