Finite Group Databasebeta

← #6358470

Subgroups — #6358470

Order 512.

1803 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61697C212yesmaximal, complemented
2C2×#3380C211yesmaximal, characteristic, complemented
3#61084C211yesmaximal, characteristic
4#19980C214yesmaximal, complemented
5#19976C214yesmaximal, complemented
6#19806C218yesmaximal, complemented
7#19655C211yesmaximal, characteristic, complemented
8#19391C212yesmaximal, complemented
9#19193C211yesmaximal, characteristic, complemented
10#19114C211yesmaximal, characteristic, complemented
11#18893C212yesmaximal, complemented
12#18748C212yesmaximal, complemented
13#17901C211yesmaximal, characteristic, complemented
14#17854C211yesmaximal, characteristic, complemented
15C2×#576—21no—
16#3496C2212yes—
17C2×#554C2212yescomplemented
18#3380—22no—
19#3380—22no—
20#3380C2214yescomplemented
21#3380C2214yescomplemented
22#3335—24no—
23#3335C2218yescomplemented
24#3295—24no—
25#3295C2218yescomplemented
26#3283C2212yes—
27#3280—22no—
28D4×D8C2212yes—
29#3273—22no—
30#3272C2214yes—
31#3227C2212yes—
32#3227C2212yescomplemented
33#3206C2212yescomplemented
34#3187—24no—
35C2×#500C2211yescharacteristic, complemented
36C2×#499—21no—
37#3147C2211yescharacteristic
38#3103C2211yescharacteristic
39#3103C2212yescomplemented
40C2×#481—24no—
41C2×#478—22no—
42C2×#476C2212yescomplemented
43#3061—21no—
44#3036C2212yes—
45#3012—22no—
46#2972C2212yescomplemented
47C2×C8×Q8C2211yescharacteristic, complemented
48C2×C4×D8—21no—
49C2×C4×D8C2211yescharacteristic
50#2931C2211yescharacteristic
51#1714C2214yescomplemented
52#1707C2212yescomplemented
53#1706C2212yescomplemented
54#1695C2214yescomplemented
55#1677C2218yescomplemented
56#1675C2214yescomplemented
57#1671C2218yescomplemented
58#1670C2212yescomplemented
59#1668C2212yescomplemented
60#1666C2214yescomplemented
61#1643C2218yescomplemented
62#1635C2218yescomplemented
63#1629C2218yescomplemented
64#1622C2218yescomplemented
65#1583C2214yescomplemented
66#1580C2214yescomplemented
67C8×SD16C2214yescomplemented
68C8×D8C2214yescomplemented
69#1570C2211yescharacteristic, complemented
70#1564C2211yescharacteristic, complemented
71#1548C2212yescomplemented
72#1540C2212yescomplemented
73#1487C2211yescharacteristic, complemented
74#1480C2211yescharacteristic
75#1478C2212yescomplemented
76#1450C2211yescharacteristic, complemented
77C2×H(5,𝔽2)—22no—
78#576—48no—
79C2×AGL(1,ℤ/8)—22no—
80C2×AGL(1,ℤ/8)—24no—
81C2×#135—22no—
82C2×#135—21no—
83C2×#135—21no—
84C22×D8—22no—
85C22×D8D411yescharacteristic
86C2×#131—21no—
87#554—24no—
88#554—24no—
89#554—42no—
90#554C2318yes—
91#547—22no—
92#547C2314yes—
93#546—24no—
94#546—42no—
95D42—22no—
96D42C2314yes—
97#537—22no—
98#534—42no—
99#532C2312yes—
100#532C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—42no—
105C2×#121—24no—
106C2×C4×Q8C2311yescharacteristic
107C2×C4×D4—22no—
108C2×C4×D4C2312yes—
109C22×C4⋊C4—22no—
110#500—21no—
111#500—21no—
112#500C2312yes—
113#500C2312yes—
114#499—24no—
115#499—24no—
116#498—22no—
117#498C2314yes—
118#495—21no—
119#495—21no—
120#494C2311yescharacteristic
121#493C2311yescharacteristic
122#492C2312yes—
123#492C2314yes—
124#487C2314yes—
125#486—22no—
126#486C2314yes—
127#483—28no—
128#481—24no—
129#481—22no—
130#481—28no—
131#481—28no—
132#481—24no—
133#481D412yes—
134#478—28no—
135#478—28no—
136#476—24no—
137#476—24no—
138#476C2318yes—
139#476C2318yes—
140#475D412yes—
141#475C2314yes—
142#468—24no—
143#467—22no—
144#466—22no—
145#465—28no—
146#465—28no—
147#463—24no—
148#461—42no—
149#461C2314yes—
150#460D412yes—
151#460C2314yes—
152#459C2314yes—
153#452—24no—
154#450—28no—
155#450—24no—
156#449—24no—
157#448—22no—
158#448—42no—
159#448D412yes—
160#447—24no—
161C8×Q8—21no—
162C8×Q8—21no—
163C8×Q8C2312yes—
164C8×Q8C2312yes—
165C4×SD16—24no—
166C4×SD16C2314yes—
167C4×D8—24no—
168C4×D8—24no—
169C4×D8D412yes—
170C4×D8D412yes—
171C4×D8C2312yes—
172C4×D8C2312yes—
173#436—21no—
174#436C2312yes—
175C8×D4—24no—
176C8×D4—21no—
177C8×D4C2318yes—
178C8×D4C2312yes—
179#431C2312yes—
180#431C2311yescharacteristic
181#431C2311yescharacteristic
182C2×#107—21no—
183C2×#107—21no—
184C2×#107D411yescharacteristic
185C2×#106—22no—
186C2×#106—28no—
187#423C2311yescharacteristic
188C2×C4⋊C8—21no—
189C2×C4⋊C8C2311yescharacteristic
190#417—22no—
191C2×#102—22no—
192C2×#102—28no—
193C2×#102D412yes—
194#408—21no—
195C2×C22⋊C8—24no—
196C2×C4×C8—21no—
197C2×C4×C8C2311yescharacteristic
198#335C2312yes—
199#334C2312yes—
200#334C2314yes—
201#333C2318yes—
202#332C2314yes—
203#331C2314yes—
204Q8⋊C8C2314yes—
205D4⋊C8C2318yes—
206D4⋊C8C2314yes—
207#322C2312yes—
208C82C2312yes—
209H(5,𝔽2)—48no—
210H(5,𝔽2)—84no—
211C2×#41—21no—
212C2×#41—24no—
213C2×#41—22no—
214C2×#41—42no—
215C22×Q8—22no—
216C22×Q8C2×D411yescharacteristic
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4C2×D412yes—
222C23×C4—22no—
223C23×C4—42no—
224AGL(1,ℤ/8)—48no—
225AGL(1,ℤ/8)—416no—
226#135—48no—
227#135—44no—
228#135—82no—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×SD16—24no—
232C2×SD16—24no—
233C2×SD16—22no—
234C2×SD16—22no—
235C2×SD16—44no—
236C2×SD16C2×D414yes—
237C2×D8—24no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—48no—
241C2×D8SD1618yescomplemented
242C2×D8C2×D412yes—
243C2×D8C2×D412yes—
244#131—44no—
245C2×M16—21no—
246C2×M16—22no—
247C22×C8—21no—
248C22×C8—22no—
249C22×C8—28no—
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8C2×D411yescharacteristic
256C4⋊Q8C2414yes—
257C4⋊Q8C2412yes—
258#127C2×D411yescharacteristic
259#127C2412yes—
260#125—28no—
261#125—44no—
262#124—22no—
263#124—21no—
264#123—44no—
265#123—44no—
266C22⋊Q8—28no—
267C22⋊Q8—24no—
268C22⋊Q8—48no—
269C22⋊Q8—48no—
270C22⋊Q8—42no—
271C22⋊Q8—44no—
272#121—21no—
273#121—24no—
274#121—22no—
275#121—22no—
276#121—24no—
277#121—24no—
278#121—44no—
279#121—42no—
280#121—44no—
281#121C2×D412yes—
282#120—44no—
283#120—44no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×Q8C2×D412yes—
287C4×Q8C2412yes—
288C4×D4—28no—
289C4×D4—22no—
290C4×D4—21no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—44no—
295C4×D4C2×D414yes—
296C4×D4C2414yes—
297C4×D4C2412yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—28no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C42—21no—
312C2×C42C2411yescharacteristic
313#107—22no—
314#107—22no—
315#107—22no—
316#107—22no—
317#107C2×D412yes—
318#107C2×D412yes—
319#106—28no—
320#106—24no—
321#106—28no—
322#106—28no—
323#106C2×D414yes—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8C2×D412yes—
330C4⋊C8C2412yes—
331C4⋊C8C2414yes—
332C4⋊C8C2412yes—
333#103—24no—
334#103—24no—
335#103—44no—
336#103C2×D414yes—
337#102—28no—
338#102—28no—
339#102—28no—
340#102—24no—
341#102—24no—
342#102C2×D414yes—
343#102C2×D414yes—
344#102C2×D414yes—
345C22⋊C8—22no—
346C22⋊C8—21no—
347C22⋊C8—28no—
348C22⋊C8—22no—
349C22⋊C8—22no—
350C22⋊C8C2×D412yes—
351C4×C8—22no—
352C4×C8—22no—
353C4×C8—22no—
354C4×C8C2×D411yescharacteristic
355C4×C8C2×D411yescharacteristic
356C4×C8C2412yes—
357C4×C8C2411yescharacteristic
358C4×C8C2412yes—
359C4×C8C2411yescharacteristic
360C24—22no—
361C24—44no—
362#41—42no—
363#41—42no—
364#41—48no—
365#41—44no—
366#41—44no—
367#41—84no—
368#41—88no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—162no—
376C2×Q8C22×D412yes—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—42no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—816no—
394C2×D4C2×SD1618yes—
395C2×D4C22×D412yes—
396C2×D4C22×D414yes—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—84no—
408C22×C4—164no—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412Q16—42no—
413Q16—81no—
414SD16—44no—
415SD16—48no—
416SD16—44no—
417SD16—82no—
418SD16—84no—
419SD16—164no—
420D8—48no—
421D8—42no—
422D8—81no—
423D8C2×SD1618yes—
424D8C2×SD1618yes—
425M16—42no—
426M16—44no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—42no—
433C2×C8—44no—
434C2×C8—88no—
435C2×C8C2×SD1614yes—
436C2×C8C22×D411yescharacteristic
437C2×C8C22×D411yescharacteristic
438C2×C8C22×D414yes—
439C2×C8C22×D411yescharacteristic
440C2×C8C22×D411yescharacteristic
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—48no—
451C4⋊C4—44no—
452C4⋊C4—48no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4C22×D412yes—
456C4⋊C4C22×D414yes—
457C4⋊C4C22×D412yes—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C42—22no—
468C42—22no—
469C42C22×D411yescharacteristic
470C42C22×D411yescharacteristic
471C42H(5,𝔽2)11yescharacteristic
472C42C2511yescharacteristic
473C23—24no—
474C23—28no—
475C23—22no—
476C23—42no—
477C23—82no—
478C23—82no—
479C23—88no—
480C23—164no—
481C23—164no—
482C23D4211yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—42no—
489Q8—84no—
490Q8—162no—
491D4—24no—
492D4—28no—
493D4—24no—
494D4—44no—
495D4—44no—
496D4—42no—
497D4—42no—
498D4—42no—
499D4—81no—
500D4—82no—
501D4—88no—
502D4—88no—
503D4—84no—
504D4—164no—
505D4C22×SD1618yes—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—48no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—88no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—168no—
525C2×C4D4211yescharacteristic
526C2×C4D4211yescharacteristic
527C2×C4D4211yescharacteristic
528C2×C4C22×SD1612yes—
529C2×C4#57712yes—
530C2×C4C23×D411yescharacteristic
531C2×C4C23×D411yescharacteristic
532C2×C4C2×H(5,𝔽2)11yescharacteristic
533C8—24no—
534C8—21no—
535C8—21no—
536C8—42no—
537C8—44no—
538C8—42no—
539C8—42no—
540C8C22×SD1612yes—
541C8C22×SD1612yes—
542C22—21no—
543C22—21no—
544C22—28no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—88no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—328no—
555C22D4×SD1612yes—
556C22C2×D4211yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—42no—
561C4—44no—
562C4—42no—
563C4—42no—
564C4—82no—
565C4—84no—
566C4D4×SD1612yes—
567C4C23×SD1611yescharacteristic
568C4C2×#57711yescharacteristic
569C4#358412yes—
570C2—22no—
571C2—48no—
572C2—82no—
573C2—162no—
574C2C2×D4×SD1611yescharacteristic
575C2#6105111yescharacteristic
576C2#6110611yescharacteristic
subgroupG/H
center, socleC22C2×D42
commutator, Frattini, cosocleC42C25
prime radical#61084C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #61697C2semidirect
2. C2×#3380C2semidirect
4. #19980C2semidirect
5. #19976C2semidirect
6. #19806C2semidirect
7. #19655C2semidirect
8. #19391C2semidirect
9. #19193C2semidirect
10. #19114C2semidirect
11. #18893C2semidirect
12. #18748C2semidirect
13. #17901C2semidirect
14. #17854C2semidirect
17. C2×#554C22semidirect
20. #3380C22semidirect
21. #3380C22semidirect
23. #3335C22semidirect
25. #3295C22semidirect
32. #3227C22semidirect
33. #3206C22semidirect
35. C2×#500C22semidirect
39. #3103C22semidirect
42. C2×#476C22semidirect
46. #2972C22semidirect
47. C2×C8×Q8C22semidirect
51. #1714C22semidirect
52. #1707C22semidirect
53. #1706C22semidirect
54. #1695C22semidirect
55. #1677C22semidirect
56. #1675C22semidirect
57. #1671C22semidirect
58. #1670C22semidirect
59. #1668C22semidirect
60. #1666C22semidirect
61. #1643C22semidirect
62. #1635C22semidirect
63. #1629C22semidirect
64. #1622C22semidirect
65. #1583C22semidirect
66. #1580C22semidirect
67. C8×SD16C22semidirect
68. C8×D8C22semidirect
69. #1570C22semidirect
70. #1564C22semidirect
71. #1548C22semidirect
72. #1540C22semidirect
73. #1487C22semidirect
75. #1478C22semidirect
76. #1450C22semidirect
241. C2×D8SD16semidirect

Maximal subgroups

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

subgroupindex
#616972
C2×#33802
#610842
#199802
#199762
#198062
#196552
#193912
#191932
#191142
#188932
#187482
#179012
#178542