Finite Group Databasebeta

← #6484431

Subgroups — #6484431

Order 512.

1668 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#35294C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#32967C212yesmaximal, complemented
4#32966C212yesmaximal
5#29960C212yesmaximal, complemented
6#29959C212yesmaximal
7#29267C214yesmaximal, complemented
8#29198C212yesmaximal, complemented
9#29197C212yesmaximal, complemented
10#29148C212yesmaximal, complemented
11#28517C212yesmaximal, complemented
12#28293C214yesmaximal, complemented
13#21988C212yesmaximal, complemented
14#21808C212yesmaximal
15#20811C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C2×#519C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#478C2214yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic, complemented
23#2859C2212yescomplemented
24#2859C2211yescharacteristic, complemented
25#2629C2212yes—
26#2628C2212yes—
27#2570—22no—
28#2550—22no—
29#2441—22no—
30#2439—22no—
31#2367—22no—
32#2367—22no—
33#2342—22no—
34#2342—22no—
35#2331C2212yescomplemented
36#2330C2212yescomplemented
37#2326C2212yes—
38#2326C2212yes—
39#2310C2211yescharacteristic
40#2309C2211yescharacteristic
41C4×#124C2212yes—
42C4×#124C2212yes—
43#2285C2212yes—
44#2285C2211yescharacteristic
45C4×#117C2211yescharacteristic
46#2097C2214yescomplemented
47#2092C2214yes—
48#2068C2214yescomplemented
49#2067C2214yes—
50#2066C2214yescomplemented
51#2065C2214yes—
52#2024C2214yescomplemented
53#2023C2214yes—
54#1983C2214yes—
55#1983C2214yes—
56#1959C2212yes—
57#1959C2212yes—
58#1959C2212yes—
59#1959C2212yes—
60#1948C2214yescomplemented
61#1947C2214yescomplemented
62#1865C2214yescomplemented
63#1864C2214yes—
64#1863C2214yescomplemented
65#1862C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1847C2212yescomplemented
69#1847C2212yescomplemented
70#1843C2212yescomplemented
71#1836C2212yescomplemented
72#1836C2212yescomplemented
73#1832C2212yescomplemented
74#1803C2214yescomplemented
75#1802C2214yes—
76#1798C2212yes—
77#1798C2212yes—
78#1787C2214yes—
79#1786C2214yes—
80#1767C2211yescharacteristic, complemented
81#1767C2211yescharacteristic, complemented
82#1767C2212yescomplemented
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×#125—22no—
88C2×#125C2312yes—
89C2×#124—22no—
90C2×#124—24no—
91C2×#124Q818yescomplemented
92C2×#124C2312yes—
93C2×#124C2312yes—
94#519—44no—
95#518—44no—
96C2×C4×Q8—22no—
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×C4×D4C2312yes—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C42C2311yescharacteristic
111#479—48no—
112#478—48no—
113#432—28no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2312yes—
132C2×#97C2312yes—
133#400—22no—
134#399—22no—
135#398—24no—
136#396—24no—
137#395—22no—
138#393—22no—
139#391—21no—
140#391—21no—
141#391—24no—
142#391C2312yes—
143#391C2312yes—
144#390—21no—
145#390—21no—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#389—24no—
153#389C2314yes—
154#389C2314yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#387—24no—
164#387—24no—
165#385—22no—
166#385—22no—
167#385—22no—
168#385—22no—
169#384—22no—
170#384C2312yes—
171#384C2312yes—
172#384C2312yes—
173#384C2312yes—
174#382—22no—
175#382—22no—
176#382C2312yes—
177#382C2312yes—
178#381—21no—
179#381—21no—
180#380—22no—
181#380—22no—
182#380C2314yes—
183#379—21no—
184#379—21no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4D412yes—
188C4×C4⋊C4D412yes—
189C4×C4⋊C4C2314yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4D412yes—
195C4×C22⋊C4D412yes—
196#376C2311yescharacteristic
197#376C2311yescharacteristic
198C43C2311yescharacteristic
199C43C2311yescharacteristic
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2312yes—
206#336C2312yes—
207C25—21no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8C2×Q814yes—
214C22×Q8C2412yes—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4C2×Q814yes—
220C22×D4C2412yes—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2×D412yes—
225C23×C4C2411yescharacteristic
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—84no—
231#125—28no—
232#125—84no—
233#124—216no—
234#124—24no—
235#124—24no—
236#124C2×Q818yes—
237#124C2×Q818yes—
238C4×Q8—28no—
239C4×Q8—44no—
240C4×Q8—48no—
241C4×D4—28no—
242C4×D4—44no—
243C4×D4—48no—
244#117—24no—
245#117—42no—
246#117—48no—
247#117—42no—
248#117—42no—
249#117—84no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—24no—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×Q814yes—
299C2×C42#4114yes—
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306#107—44no—
307#107—44no—
308#106—44no—
309#106—44no—
310C4⋊C8—28no—
311C4⋊C8—28no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314#103—44no—
315#103—44no—
316#103—44no—
317#103—44no—
318#102—44no—
319#102—44no—
320#102—44no—
321#102—44no—
322C22⋊C8—28no—
323#97—24no—
324#97—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D412yes—
340#95C2×D412yes—
341C24—22no—
342C24—24no—
343C24—42no—
344C24—42no—
345C24—82no—
346C24—82no—
347C24C22×D411yescharacteristic
348C2×Q8—24no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—48no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—84no—
358C2×Q8C22×Q814yes—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—48no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—48no—
367C2×D4C22×Q814yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4C22⋊Q818yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×Q812yes—
397C22×C4C2×#4112yes—
398C22×C4C2511yescharacteristic
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—48no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—84no—
427C22⋊C4—28no—
428C22⋊C4—28no—
429C22⋊C4—28no—
430C22⋊C4—28no—
431C22⋊C4—24no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—44no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—82no—
441C22⋊C4—82no—
442C22⋊C4—84no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—44no—
453C42—44no—
454C42—84no—
455C42C22⋊Q818yes—
456C42C22×Q812yes—
457C42C22×Q812yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C42H(5,𝔽2)14yes—
461C23—22no—
462C23—22no—
463C23—21no—
464C23—21no—
465C23—24no—
466C23—48no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—88no—
474C23—162no—
475C23—162no—
476C23—164no—
477C23C2×C22⋊Q812yes—
478C23C23×D411yescharacteristic
479Q8—28no—
480Q8—48no—
481Q8—44no—
482Q8—164no—
483D4—28no—
484D4—48no—
485D4—44no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4C2×C22⋊Q812yes—
527C2×C4C2×C22⋊Q814yes—
528C2×C4C2×C22⋊Q814yes—
529C2×C4#53514yes—
530C2×C4C23×Q811yescharacteristic
531C2×C4C22×#4111yescharacteristic
532C2×C4C2×H(5,𝔽2)12yes—
533C2×C4#58512yes—
534C8—44no—
535C8—44no—
536C22—24no—
537C22—22no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—168no—
549C22C22×C22⋊Q811yescharacteristic
550C22C2×#53511yescharacteristic
551C22#344911yescharacteristic
552C22#358512yes—
553C22#358612yes—
554C4—24no—
555C4—24no—
556C4—22no—
557C4—24no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—44no—
563C4—42no—
564C4—84no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4#350112yes—
569C4#350612yes—
570C2—42no—
571C2—84no—
572C2#3517211yescharacteristic
573C2#6049312yes—
574C2#6082912yes—
575C2#6101711yescharacteristic
576C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35294C2semidirect
3. #32967C2semidirect
5. #29960C2semidirect
7. #29267C2semidirect
8. #29198C2semidirect
9. #29197C2semidirect
10. #29148C2semidirect
11. #28517C2semidirect
12. #28293C2semidirect
13. #21988C2semidirect
15. #20811C2semidirect
19. C2×#479C22semidirect
22. #2868C22semidirect
23. #2859C22semidirect
24. #2859C22semidirect
35. #2331C22semidirect
36. #2330C22semidirect
46. #2097C22semidirect
48. #2068C22semidirect
50. #2066C22semidirect
52. #2024C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1865C22semidirect
64. #1863C22semidirect
68. #1847C22semidirect
69. #1847C22semidirect
70. #1843C22semidirect
71. #1836C22semidirect
72. #1836C22semidirect
73. #1832C22semidirect
74. #1803C22semidirect
80. #1767C22semidirect
81. #1767C22semidirect
82. #1767C22semidirect
91. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
#352942
#342112
#329672
#329662
#299602
#299592
#292672
#291982
#291972
#291482
#285172
#282932
#219882
#218082
#208112