Finite Group Databasebeta

← #6661109

Subgroups — #6661109

Order 512.

1567 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35310C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic
3#32755C214yesmaximal, complemented
4#30303C212yesmaximal, complemented
5#30196C212yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#29949C212yesmaximal, complemented
8#29756C212yesmaximal, complemented
9#29701C212yesmaximal, complemented
10#27647C212yesmaximal, complemented
11#24854C214yesmaximal, complemented
12#24559C212yesmaximal
13#24551C212yesmaximal
14#24307C212yesmaximal, complemented
15#20628C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#483C2212yes—
20C2×#483C2212yes—
21C2×#479C2214yescomplemented
22C2×#424C2212yes—
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2857C2211yescharacteristic
26#2851C2211yescharacteristic
27#2848C2212yes—
28#2629—22no—
29#2629—22no—
30#2629C2212yes—
31#2629C2212yes—
32#2629C2212yes—
33#2629C2212yes—
34#2629C2211yescharacteristic
35#2625C2212yes—
36#2621C2211yescharacteristic
37#2480—22no—
38#2478—22no—
39#2475—22no—
40#2473—22no—
41C2×#398—22no—
42C2×#396—22no—
43#2323C2212yes—
44#2314C2211yescharacteristic
45C4×C22⋊Q8C2212yes—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48C2×#380C2212yes—
49#2101C2214yescomplemented
50#2088C2214yescomplemented
51#2087C2214yescomplemented
52#2084C2214yescomplemented
53#2084C2214yescomplemented
54#2077C2214yes—
55#2077C2214yes—
56#2072C2214yescomplemented
57#2066C2214yescomplemented
58#2066C2214yescomplemented
59#2059C2214yescomplemented
60#2059C2214yescomplemented
61#2043C2214yescomplemented
62#2036C2214yes—
63#2035C2214yes—
64#2026C2214yescomplemented
65#1989C2214yescomplemented
66#1948C2214yescomplemented
67#1894C2214yes—
68#1892C2214yes—
69#1872C2212yes—
70#1872C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1823C2212yes—
74#1823C2212yes—
75#1809C2214yes—
76#1807C2214yes—
77#1803C2214yescomplemented
78#1798C2212yes—
79#1798C2212yes—
80#1797C2212yes—
81#1797C2212yes—
82#1729C2212yes—
83C24×C4C2311yescharacteristic
84C22×M16C2312yes—
85C23×C8C2312yes—
86#541—44no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×#126—22no—
89C2×#125—22no—
90C2×#125—22no—
91C2×#125—44no—
92C2×#125C2312yes—
93C2×#125C2311yescharacteristic
94C2×#125C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×C4×Q8C2312yes—
99C2×#117—44no—
100C2×#117D412yes—
101C2×#117D412yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4D412yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113C22×C22⋊C4D412yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#483—44no—
117#483—44no—
118#479—48no—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123#424—44no—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130#407—42no—
131C2×C22⋊C8—42no—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136#400—22no—
137#398—22no—
138#398—22no—
139#398—24no—
140#398—24no—
141#398—24no—
142#398—24no—
143#398—22no—
144#397—22no—
145#397—22no—
146#397—22no—
147#396—24no—
148#396—24no—
149#396—24no—
150#396—24no—
151#396—22no—
152#391C2312yes—
153#391C2312yes—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2314yes—
160#389C2314yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#384—22no—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2312yes—
178#382C2311yescharacteristic
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2312yes—
183#380C2312yes—
184#380C2314yes—
185#380C2314yes—
186#380C2312yes—
187#379—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C2×#95—22no—
192C2×#95—22no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25C2×D411yescharacteristic
204C22×Q8C2412yes—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—42no—
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C2×M16—24no—
220C2×M16—42no—
221C2×M16—42no—
222C22×C8—24no—
223C22×C8—42no—
224C22×C8—42no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—42no—
228#126—48no—
229#125—28no—
230#125—42no—
231#125—42no—
232#125—42no—
233#125—48no—
234#125—48no—
235C22⋊Q8—24no—
236C22⋊Q8—28no—
237C22⋊Q8—24no—
238C22⋊Q8—44no—
239C4×Q8—28no—
240#117—24no—
241#117—24no—
242#117—42no—
243#117—42no—
244#117—44no—
245#117—44no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4—84no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—84no—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C22⋊C4C2×D412yes—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—42no—
297C2×C42C2×D412yes—
298C2×C42C2×D412yes—
299C2×C42C2411yescharacteristic
300C2×C42C2411yescharacteristic
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—44no—
304#106—44no—
305#106—44no—
306#106—44no—
307C4⋊C8—28no—
308C4⋊C8—44no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—41no—
336C24—42no—
337C24C22×D411yescharacteristic
338C24C22×D411yescharacteristic
339C24C22×D411yescharacteristic
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+414yes—
391C22×C42−1+412yes—
392C22×C4C2511yescharacteristic
393M16—84no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—84no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—82no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C42—24no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—42no—
447C42—44no—
448C42—82no—
449C42H(5,𝔽2)14yes—
450C23—22no—
451C23—21no—
452C23—21no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—44no—
463C23—48no—
464C23—44no—
465C23—84no—
466C23—82no—
467C23#53612yes—
468C23#53712yes—
469C23#57714yes—
470C23C23×D411yescharacteristic
471Q8—44no—
472Q8—44no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—44no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4#53414yes—
523C2×C4#53612yes—
524C2×C4#53712yes—
525C2×C4C2×H(5,𝔽2)11yescharacteristic
526C2×C4C2×H(5,𝔽2)12yes—
527C2×C4C2×2−1+412yes—
528C2×C4C2×2−1+411yescharacteristic
529C8—44no—
530C8—44no—
531C22—22no—
532C22—24no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—88no—
546C22#326014yes—
547C22C2×#53411yescharacteristic
548C22C2×#53611yescharacteristic
549C22C2×#53711yescharacteristic
550C22C2×#57712yes—
551C22#358612yes—
552C4—22no—
553C4—22no—
554C4—24no—
555C4—41no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—84no—
562C4—82no—
563C4—84no—
564C4—82no—
565C4#353612yes—
566C4#353812yes—
567C2—24no—
568C2—42no—
569C2—81no—
570C2#3744211yescharacteristic
571C2#6076312yes—
572C2#6087212yes—
573C2C2×#326011yescharacteristic
574C2#6092111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32755C2semidirect
4. #30303C2semidirect
5. #30196C2semidirect
6. #29972C2semidirect
7. #29949C2semidirect
8. #29756C2semidirect
9. #29701C2semidirect
10. #27647C2semidirect
11. #24854C2semidirect
14. #24307C2semidirect
21. C2×#479C22semidirect
49. #2101C22semidirect
50. #2088C22semidirect
51. #2087C22semidirect
52. #2084C22semidirect
53. #2084C22semidirect
56. #2072C22semidirect
57. #2066C22semidirect
58. #2066C22semidirect
59. #2059C22semidirect
60. #2059C22semidirect
61. #2043C22semidirect
64. #2026C22semidirect
65. #1989C22semidirect
66. #1948C22semidirect
77. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#353102
C2×#26292
#327552
#303032
#301962
#299722
#299492
#297562
#297012
#276472
#248542
#245592
#245512
#243072
#206282