Finite Group Databasebeta

← #6344662

Subgroups — #6344662

Order 512.

1926 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2#60236C211yesmaximal, characteristic, complemented
3C2×#2972C211yesmaximal, characteristic, complemented
4#17922C218yesmaximal, complemented
5#17907C214yesmaximal, complemented
6#17900C214yesmaximal, complemented
7#17877C214yesmaximal, complemented
8#17854C214yesmaximal, complemented
9#17803C211yesmaximal, characteristic, complemented
10#17463C211yesmaximal, characteristic, complemented
11#17448C211yesmaximal, characteristic, complemented
12#17325C211yesmaximal, characteristic, complemented
13C2×#532C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15#3100C414yes—
16#3100C414yes—
17#3100C2214yes—
18#3100C2212yes—
19#3100C2212yes—
20C2×#480C412yes—
21C2×#474C412yes—
22C2×#459C412yes—
23C2×#447C412yes—
24#2981C2214yescomplemented
25#2979—22no—
26#2975C2212yescomplemented
27#2974—22no—
28#2972—22no—
29#2972—22no—
30#2972C2212yes—
31#2972C2212yes—
32#2972C2214yescomplemented
33#2972C2212yescomplemented
34C2×C4×D8C2214yes—
35#2935—22no—
36#2931C2214yescomplemented
37C2×C8×D4—22no—
38C2×C8×D4C2214yescomplemented
39#2920C2211yescharacteristic
40#2920C2211yescharacteristic
41C2×#431C2211yescharacteristic
42C2×#431C2211yescharacteristic
43C2×#431C2211yescharacteristic
44#2880—22no—
45C22×C22⋊C8—22no—
46#1586C22116yescomplemented
47#1581C22116yescomplemented
48#1580C22116yescomplemented
49C8×D8C22116yescomplemented
50#1572C2214yescomplemented
51#1571C2212yescomplemented
52#1570C2212yescomplemented
53#1564C2214yescomplemented
54#1490C2214yescomplemented
55#1486C2214yescomplemented
56#1484C2214yescomplemented
57#1480C2214yescomplemented
58#1478C2214yes—
59#1478C2214yescomplemented
60#1453C2214yescomplemented
61#1452C2212yescomplemented
62#1452C2212yescomplemented
63#1450C2212yescomplemented
64#1450C2212yescomplemented
65C22×#41—21no—
66C23×D4C2×C411yescharacteristic
67C24×C4—21no—
68C22×D8C2×C414yes—
69C2×#131—24no—
70C23×C8—24no—
71#532—21no—
72#532—21no—
73#532—21no—
74#532—21no—
75#532C2×C412yes—
76#532C2×C412yes—
77#532C2312yes—
78#532C2312yes—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#127C2×C411yescharacteristic
81C2×#124—21no—
82C2×#121—21no—
83C2×#121C2×C412yes—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4C2×C412yes—
92C2×C4×D4C2×C412yes—
93C2×C4×D4C2×C412yes—
94C2×C4×D4C2312yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C22⋊C4—21no—
100C22×C42C2311yescharacteristic
101#480C2×C418yes—
102#480C2×C414yes—
103#480C2×C414yes—
104#474C2×C418yes—
105#474C2×C414yes—
106#474C2×C414yes—
107#459C2×C418yes—
108#459C2×C414yes—
109#459C2×C414yes—
110#447C2×C418yes—
111#447C2×C414yes—
112#447C2×C414yes—
113C4×D8C2×C418yes—
114C4×D8C2×C418yes—
115C4×D8C2318yes—
116C4×D8C2318yes—
117#436—24no—
118#436—24no—
119#436C2318yescomplemented
120#435—28no—
121C8×D4—24no—
122C8×D4—24no—
123C8×D4—28no—
124C8×D4—24no—
125C8×D4—24no—
126C8×D4—24no—
127C8×D4—24no—
128C8×D4C2318yes—
129C8×D4C2318yes—
130C8×D4C2318yescomplemented
131#432—21no—
132#432—21no—
133#432—21no—
134#432C2312yes—
135#431—21no—
136#431—21no—
137#431—21no—
138#431C2×C412yes—
139#431C2×C412yes—
140#431D412yes—
141#431D412yes—
142#431C2312yes—
143#431C2312yes—
144#431C2312yes—
145#431C2312yes—
146#431C2312yes—
147#431C2311yescharacteristic
148#431C2311yescharacteristic
149#431C2312yes—
150#431C2311yescharacteristic
151#431C2311yescharacteristic
152#431C2312yes—
153C2×#107C2×C414yes—
154#423—21no—
155#423C2312yes—
156C2×C4⋊C8C2×C412yes—
157C2×C4⋊C8D412yes—
158C2×C4⋊C8C2312yes—
159C2×#102C2×C414yes—
160C2×#102C2×C414yes—
161#408—24no—
162#408—28no—
163#407—21no—
164#407—21no—
165#407—24no—
166C2×C22⋊C8—21no—
167C2×C22⋊C8—24no—
168C2×C22⋊C8—24no—
169C2×C22⋊C8—24no—
170C2×C22⋊C8—28no—
171C2×C22⋊C8—24no—
172C2×C22⋊C8C2×C412yes—
173C2×C22⋊C8D412yes—
174C4×M16—21no—
175C4×M16C2312yes—
176C2×C4×C8—21no—
177C2×C4×C8C2312yes—
178C2×C4×C8C2312yes—
179#335C2318yescomplemented
180#334C2318yescomplemented
181D4⋊C8C23116yes—
182D4⋊C8C23116yescomplemented
183#322C2314yescomplemented
184#322C2314yescomplemented
185#322C2314yescomplemented
186C82C2314yescomplemented
187C25—21no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41—22no—
192C2×#41—84no—
193C22×Q8—21no—
194C22×D4—28no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×D4C22×C412yes—
202C22×D4C22×C412yes—
203C22×D4C22×C412yes—
204C22×D4C22×C412yes—
205C23×C4—21no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—84no—
214C23×C4C22×C411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4#4111yescharacteristic
217C2×D8—216no—
218C2×D8C22×C418yes—
219C2×D8C22×C418yes—
220#131—88no—
221C2×M16—22no—
222C2×M16—22no—
223C2×M16—28no—
224C22×C8—22no—
225C22×C8—28no—
226C22×C8—28no—
227C22×C8—28no—
228C22×C8—88no—
229C22×C8C22×C414yes—
230C22×C8C2×D414yes—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22×C412yes—
234C4⋊Q8C22×C412yes—
235#127—21no—
236#127—21no—
237#127C22×C412yes—
238#127C22×C412yes—
239#124—22no—
240#124—22no—
241#124—22no—
242#124—22no—
243#121—22no—
244#121—22no—
245#121—22no—
246#121—22no—
247#121—22no—
248#121—22no—
249#121C22×C414yes—
250#121C22×C414yes—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4C22×C414yes—
270C4×D4C22×C414yes—
271C4×D4C22×C414yes—
272C4×D4C22×C414yes—
273C4×D4C2414yes—
274C4×D4C2414yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107C22×C418yes—
304#107C22×C418yes—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8C22×C412yes—
310C4⋊C8C22×C412yes—
311C4⋊C8C22×C414yes—
312C4⋊C8C2×D412yes—
313C4⋊C8C2×D414yes—
314C4⋊C8C2×D412yes—
315C4⋊C8C2414yes—
316C4⋊C8C2414yes—
317#102C22×C418yes—
318#102C22×C418yes—
319#102C22×C418yes—
320#102C22×C418yes—
321C22⋊C8—22no—
322C22⋊C8—22no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—28no—
328C22⋊C8—28no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331C22⋊C8C22×C412yes—
332C22⋊C8C22×C412yes—
333C22⋊C8C22×C414yes—
334C22⋊C8C2×D412yes—
335C22⋊C8C2×D412yes—
336C22⋊C8C2×D414yes—
337#97—22no—
338#97—22no—
339C4×C8—22no—
340C4×C8—22no—
341C4×C8—22no—
342C4×C8C22×C414yes—
343C4×C8C2×D414yes—
344C4×C8C2414yes—
345C4×C8C2412yes—
346C4×C8C2412yes—
347C4×C8C2412yes—
348C4×C8C2412yes—
349C24—28no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—82no—
355C24—82no—
356C24C4×D411yescharacteristic
357#41—48no—
358#41—84no—
359#41—84no—
360#41—84no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—82no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—84no—
384C2×D4#13118yescomplemented
385C2×D4C23×C414yes—
386C2×D4C23×C414yes—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—48no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—81no—
407C22×C4—82no—
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C23×C411yescharacteristic
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426D8—216no—
427D8—216no—
428M16—44no—
429M16—88no—
430M16—82no—
431C2×C8—28no—
432C2×C8—28no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—44no—
437C2×C8—88no—
438C2×C8—88no—
439C2×C8—88no—
440C2×C8C23×C414yes—
441C2×C8C23×C414yes—
442C2×C8C22×D414yes—
443C2×C8C22×D414yes—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4C23×C414yes—
451C4⋊C4C23×C414yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42C4×D411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C23×C411yescharacteristic
467C42C23×C411yescharacteristic
468C42C23×C411yescharacteristic
469C42C22×D411yescharacteristic
470C42C22×D411yescharacteristic
471C42C22×D411yescharacteristic
472C42C2×#4111yescharacteristic
473C42C2×#4111yescharacteristic
474C42C2×#4111yescharacteristic
475C42C2511yescharacteristic
476C23—28no—
477C23—28no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—28no—
482C23—24no—
483C23—41no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—816no—
494C23C2×C4×D411yescharacteristic
495C23C2×C4×D411yescharacteristic
496C23C2×C4×D411yescharacteristic
497Q8—44no—
498Q8—82no—
499D4—28no—
500D4—28no—
501D4—28no—
502D4—24no—
503D4—28no—
504D4—44no—
505D4—48no—
506D4—44no—
507D4—44no—
508D4—82no—
509D4—82no—
510D4—84no—
511D4C2×#13118yes—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×C4×D411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×D411yescharacteristic
548C2×C4C2×C4×D411yescharacteristic
549C2×C4C2×#13114yes—
550C2×C4#57614yes—
551C2×C4C24×C411yescharacteristic
552C2×C4C23×D411yescharacteristic
553C2×C4C22×#4111yescharacteristic
554C8—24no—
555C8—24no—
556C8—24no—
557C8—24no—
558C8—44no—
559C8—44no—
560C8—44no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—28no—
569C22—44no—
570C22—416no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—88no—
577C22—88no—
578C22—164no—
579C22#292912yes—
580C22#294812yes—
581C22C22×C4×D411yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—44no—
591C4—44no—
592C4#292912yes—
593C4#294812yes—
594C4C22×#13112yes—
595C4C2×#57612yes—
596C2—22no—
597C2—22no—
598C2—41no—
599C2—48no—
600C2—84no—
601C2C2×#292911yescharacteristic
602C2C2×#294811yescharacteristic
603C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC2×#3100C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60236C2semidirect
3. C2×#2972C2semidirect
4. #17922C2semidirect
5. #17907C2semidirect
6. #17900C2semidirect
7. #17877C2semidirect
8. #17854C2semidirect
9. #17803C2semidirect
10. #17463C2semidirect
11. #17448C2semidirect
12. #17325C2semidirect
24. #2981C22semidirect
26. #2975C22semidirect
32. #2972C22semidirect
33. #2972C22semidirect
36. #2931C22semidirect
38. C2×C8×D4C22semidirect
46. #1586C22semidirect
47. #1581C22semidirect
48. #1580C22semidirect
49. C8×D8C22semidirect
50. #1572C22semidirect
51. #1571C22semidirect
52. #1570C22semidirect
53. #1564C22semidirect
54. #1490C22semidirect
55. #1486C22semidirect
56. #1484C22semidirect
57. #1480C22semidirect
59. #1478C22semidirect
60. #1453C22semidirect
61. #1452C22semidirect
62. #1452C22semidirect
63. #1450C22semidirect
64. #1450C22semidirect
119. #436C23semidirect
130. C8×D4C23semidirect
179. #335C23semidirect
180. #334C23semidirect
182. D4⋊C8C23semidirect
183. #322C23semidirect
184. #322C23semidirect
185. #322C23semidirect
186. C82C23semidirect
384. C2×D4#131semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
#602362
C2×#29722
#179222
#179072
#179002
#178772
#178542
#178032
#174632
#174482
#173252