Finite Group Databasebeta

← #483257

Subgroups — #483257

Order 512.

756 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62417C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#17829C211yesmaximal, characteristic, complemented
4#11987C211yesmaximal, characteristic, complemented
5#11986C211yesmaximal, characteristic, complemented
6#11982C211yesmaximal, characteristic, complemented
7#11966C211yesmaximal, characteristic, complemented
8#11964C211yesmaximal, characteristic, complemented
9#11963C211yesmaximal, characteristic, complemented
10#11953C211yesmaximal, characteristic, complemented
11#11952C211yesmaximal, characteristic, complemented
12#11948C211yesmaximal, characteristic, complemented
13#11931C211yesmaximal, characteristic, complemented
14#11930C211yesmaximal, characteristic, complemented
15#11929C211yesmaximal, characteristic, complemented
16#3508—21no—
17#3396—21no—
18#3396C412yes—
19#3287—21no—
20#3287C412yes—
21#3280—21no—
22#3280—21no—
23#3280C412yes—
24#3280C412yes—
25#3208—21no—
26#3207—21no—
27#3207—21no—
28#3170C2211yescharacteristic
29#3169C2211yescharacteristic
30#3169C2211yescharacteristic
31#3147C2211yescharacteristic
32#3147C2211yescharacteristic
33C2×#493C2211yescharacteristic
34C8×#41C2211yescharacteristic
35#2925—21no—
36#2183—21no—
37#2169—21no—
38#2168—21no—
39#2168—21no—
40#2163C2211yescharacteristic
41#2163C2211yescharacteristic, complemented
42#2162C2211yescharacteristic, complemented
43#2150C2211yescharacteristic
44#2150C2211yescharacteristic, complemented
45C2×C4⋊C16C2211yescharacteristic, complemented
46#1572—21no—
47#1572—21no—
48#1570—21no—
49#1568C2211yescharacteristic
50#1568C2211yescharacteristic, complemented
51#1567C2211yescharacteristic, complemented
52#1564C2211yescharacteristic
53#1564C2211yescharacteristic, complemented
54C2×#335C2211yescharacteristic, complemented
55#1365C2211yescharacteristic, complemented
56#1365C2211yescharacteristic, complemented
57#1365C2211yescharacteristic, complemented
58#1365C2211yescharacteristic, complemented
59#1362C2211yescharacteristic, complemented
60#1362C2211yescharacteristic, complemented
61#1362C2211yescharacteristic, complemented
62#1362C2211yescharacteristic, complemented
63#1361C2211yescharacteristic
64#1361C2211yescharacteristic
65#1361C2211yescharacteristic
66#1361C2211yescharacteristic
67#1361C2211yescharacteristic
68#1361C2211yescharacteristic
69#1361C2211yescharacteristic
70#1361C2211yescharacteristic
71C2×#135—21no—
72C2×#135—21no—
73C2×#135C2×C412yes—
74C2×#135C2×C412yes—
75C22×D8—21no—
76C22×D8C2×C412yes—
77#559—22no—
78#559—41no—
79#550—22no—
80#550—41no—
81#546—22no—
82#546—22no—
83#546—41no—
84#546—41no—
85#540—41no—
86#540—41no—
87#535—41no—
88#533—21no—
89#532—21no—
90#532—21no—
91#532—21no—
92#532—21no—
93C2×#127—21no—
94#518—41no—
95C4×#41D411yescharacteristic
96C2×M32—21no—
97C2×M32—21no—
98C22×C16—21no—
99#500C2311yescharacteristic
100#495—21no—
101#495—21no—
102#495C2×C412yes—
103#495C2×C412yes—
104#494C2311yescharacteristic
105#493—21no—
106#493—21no—
107#493C2×C412yes—
108#493C2311yescharacteristic
109#493C2311yescharacteristic
110#492—21no—
111#492—21no—
112#487—21no—
113#486—21no—
114#486—21no—
115#467—21no—
116#467C2×C412yes—
117#466—21no—
118#466—21no—
119#466C2×C412yes—
120#466C2×C412yes—
121#463—22no—
122#463—22no—
123#463—22no—
124#463—22no—
125#460—22no—
126#460—22no—
127#450—22no—
128#450—22no—
129#449—22no—
130#449—22no—
131#449—22no—
132#449—22no—
133C8×Q8C2×C411yescharacteristic
134C8×Q8D411yescharacteristic
135C4×Q16—21no—
136C4×Q16C2×C412yes—
137C4×D8—21no—
138C4×D8—21no—
139C4×D8C2×C412yes—
140C4×D8C2×C412yes—
141C8×D4C2×C411yescharacteristic
142C8×D4C2×C411yescharacteristic
143C8×D4C2×C411yescharacteristic
144C8×D4D411yescharacteristic
145C8×D4D411yescharacteristic
146C8×D4D411yescharacteristic
147#433—41no—
148#433—41no—
149#432—21no—
150#432—21no—
151#431—21no—
152#431—21no—
153#431D411yescharacteristic
154#431D411yescharacteristic
155#431D411yescharacteristic
156#423—21no—
157C2×C4⋊C8—21no—
158#408—41no—
159#405—41no—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C4⋊C16—21no—
164C4⋊C16—21no—
165C4⋊C16—21no—
166C4⋊C16C2311yescharacteristic
167C4⋊C16C2311yescharacteristic
168C4⋊C16C2311yescharacteristic
169C4⋊C16C2311yescharacteristic
170C22⋊C16—21no—
171C22⋊C16—21no—
172C22⋊C16—21no—
173#346—21no—
174#346—21no—
175C4×C16—21no—
176#335—21no—
177#335—21no—
178#335C2311yescharacteristic
179#335C2311yescharacteristic
180#335C2311yescharacteristic
181#335C2311yescharacteristic
182#334—21no—
183#334—21no—
184#334—21no—
185#334—21no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—41no—
189C2×#41—41no—
190C2×#41D811yescharacteristic
191C22×D4—22no—
192C22×D4—41no—
193C22×D4—41no—
194#135—81no—
195#135—81no—
196#135—81no—
197#135—81no—
198#135—82no—
199#135—82no—
200C2×Q16—21no—
201C2×Q16C22×C412yes—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×D8—22no—
207C2×D8—22no—
208C2×D8—21no—
209C2×D8—21no—
210C2×D8—81no—
211C2×D8—81no—
212C2×D8—82no—
213C2×D8C22×C412yes—
214C2×D8C22×C412yes—
215C2×M16—41no—
216C22×C8—41no—
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225#127—22no—
226#127—21no—
227#127—21no—
228#127—42no—
229#127—41no—
230#127—41no—
231#125—41no—
232#124—22no—
233#124—22no—
234#124—42no—
235#124—41no—
236#124—41no—
237#123—42no—
238#123—42no—
239#123—42no—
240#123—42no—
241C22⋊Q8—22no—
242C22⋊Q8—41no—
243#121—22no—
244#121—22no—
245#121—41no—
246#121—41no—
247#121—41no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—42no—
252#121—42no—
253#121—42no—
254#121—42no—
255#120—42no—
256#120—42no—
257C4×Q8—22no—
258C4×Q8—41no—
259C4×Q8C22⋊C411yescharacteristic
260C4×Q8SD1611yescharacteristic
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4C22⋊C411yescharacteristic
272C4×D4C22⋊C411yescharacteristic
273C4×D4C22⋊C411yescharacteristic
274C4×D4SD1611yescharacteristic
275C4×D4SD1611yescharacteristic
276C4×D4SD1611yescharacteristic
277#117—41no—
278#117D811yescharacteristic
279#117D811yescharacteristic
280#117D811yescharacteristic
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286M32—81no—
287M32—81no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—81no—
293#107—21no—
294#107C22×C412yes—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8C22⋊C411yescharacteristic
302C4⋊C8C22⋊C411yescharacteristic
303C4⋊C8C22⋊C411yescharacteristic
304C4⋊C8C22⋊C411yescharacteristic
305C4⋊C8C22⋊C411yescharacteristic
306C4⋊C8C22⋊C411yescharacteristic
307#103—22no—
308#103—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312#102—22no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8—41no—
317C22⋊C8C22⋊C411yescharacteristic
318C22⋊C8C22⋊C411yescharacteristic
319C22⋊C8C22⋊C411yescharacteristic
320C22⋊C8C22⋊C411yescharacteristic
321C22⋊C8C22⋊C411yescharacteristic
322C22⋊C8C22⋊C411yescharacteristic
323#97—41no—
324#97—41no—
325C4×C8—41no—
326C4×C8—41no—
327C4×C8C22×C411yescharacteristic
328C4×C8C22×C411yescharacteristic
329C4×C8C22×C411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2×D411yescharacteristic
333C4×C8C2×D411yescharacteristic
334C4×C8C2×D411yescharacteristic
335C4×C8C2×D411yescharacteristic
336C4×C8C2411yescharacteristic
337C24—42no—
338#41—41no—
339#41—41no—
340#41—82no—
341#41—82no—
342#41—82no—
343#41—82no—
344#41—162no—
345#41—162no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8#10211yescharacteristic
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—82no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—82no—
366C2×D4—162no—
367C2×D4#10211yescharacteristic
368C2×D4#10211yescharacteristic
369C2×D4#10211yescharacteristic
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×D811yescharacteristic
378C22×C4C2×D811yescharacteristic
379C22×C4C2×D811yescharacteristic
380C22×C4C2×SD1611yescharacteristic
381C22×C4C2×SD1611yescharacteristic
382C22×C4C2×SD1611yescharacteristic
383Q16—42no—
384Q16—41no—
385Q16—41no—
386SD16—82no—
387SD16—82no—
388SD16—82no—
389SD16—82no—
390D8—42no—
391D8—42no—
392D8—41no—
393D8—41no—
394D8—41no—
395D8—41no—
396D8—82no—
397D8—82no—
398M16—82no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—82no—
413C2×C8C2×C22⋊C411yescharacteristic
414C2×C8C2×C22⋊C411yescharacteristic
415C2×C8C2×C22⋊C411yescharacteristic
416C2×C8C2×C22⋊C411yescharacteristic
417C2×C8C2×C22⋊C411yescharacteristic
418C2×C8C2×C22⋊C411yescharacteristic
419C2×C8C23×C411yescharacteristic
420C2×C8C22×D411yescharacteristic
421C4⋊C4—22no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4#10211yescharacteristic
428C4⋊C4#10211yescharacteristic
429C4⋊C4#10211yescharacteristic
430C4⋊C4#10211yescharacteristic
431C4⋊C4#10211yescharacteristic
432C4⋊C4#10211yescharacteristic
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4#10211yescharacteristic
443C22⋊C4#10211yescharacteristic
444C22⋊C4#10211yescharacteristic
445C22⋊C4#10211yescharacteristic
446C22⋊C4#10211yescharacteristic
447C22⋊C4#10211yescharacteristic
448C42—42no—
449C42C2×C22⋊C411yescharacteristic
450C42C2×C22⋊C411yescharacteristic
451C42C2×C22⋊C411yescharacteristic
452C42C2×D811yescharacteristic
453C42C2×D811yescharacteristic
454C42C2×D811yescharacteristic
455C42C2×SD1611yescharacteristic
456C42C2×SD1611yescharacteristic
457C42C2×SD1611yescharacteristic
458C42C22×D411yescharacteristic
459C16—41no—
460C16—41no—
461C16—41no—
462C16—41no—
463C23—42no—
464C23—42no—
465C23—84no—
466C23—162no—
467C23—162no—
468C23C2×#10211yescharacteristic
469C23C2×#10211yescharacteristic
470C23C2×#10211yescharacteristic
471Q8—41no—
472Q8—42no—
473Q8—42no—
474Q8—82no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—84no—
483D4—82no—
484D4—82no—
485D4—81no—
486D4—81no—
487D4—81no—
488D4—81no—
489D4—162no—
490D4—162no—
491D4—162no—
492D4—162no—
493D4—162no—
494D4—162no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—162no—
513C2×C4—162no—
514C2×C4—162no—
515C2×C4—162no—
516C2×C4C2×#10211yescharacteristic
517C2×C4C2×#10211yescharacteristic
518C2×C4C2×#10211yescharacteristic
519C2×C4C2×#10211yescharacteristic
520C2×C4C2×#10211yescharacteristic
521C2×C4C2×#10211yescharacteristic
522C2×C4C2×#10211yescharacteristic
523C2×C4C2×#10211yescharacteristic
524C2×C4C2×#10211yescharacteristic
525C2×C4C22×C22⋊C411yescharacteristic
526C2×C4C22×D811yescharacteristic
527C2×C4C22×SD1611yescharacteristic
528C8—21no—
529C8—21no—
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—21no—
535C8—81no—
536C8—81no—
537C8—81no—
538C8—81no—
539C8#57611yescharacteristic
540C8#57711yescharacteristic
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—324no—
557C22C22×#10211yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—82no—
569C4—82no—
570C4#288811yescharacteristic
571C4#289711yescharacteristic
572C4#341411yescharacteristic
573C4#341411yescharacteristic
574C2—41no—
575C2—41no—
576C2—41no—
577C2—162no—
578C2—162no—
579C2#1772011yescharacteristic
580C2#3338011yescharacteristic
581C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#62417C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33503C2semidirect
3. #17829C2semidirect
4. #11987C2semidirect
5. #11986C2semidirect
6. #11982C2semidirect
7. #11966C2semidirect
8. #11964C2semidirect
9. #11963C2semidirect
10. #11953C2semidirect
11. #11952C2semidirect
12. #11948C2semidirect
13. #11931C2semidirect
14. #11930C2semidirect
15. #11929C2semidirect
41. #2163C22semidirect
42. #2162C22semidirect
44. #2150C22semidirect
45. C2×C4⋊C16C22semidirect
50. #1568C22semidirect
51. #1567C22semidirect
53. #1564C22semidirect
54. C2×#335C22semidirect
55. #1365C22semidirect
56. #1365C22semidirect
57. #1365C22semidirect
58. #1365C22semidirect
59. #1362C22semidirect
60. #1362C22semidirect
61. #1362C22semidirect
62. #1362C22semidirect

Maximal subgroups

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

subgroupindex
#624172
#335032
#178292
#119872
#119862
#119822
#119662
#119642
#119632
#119532
#119522
#119482
#119312
#119302
#119292