Finite Group Databasebeta

← #6630471

Subgroups — #6630471

Order 512.

1972 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#35310C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3#33157C212yesmaximal, complemented
4#33151C212yesmaximal, complemented
5#33117C212yesmaximal, complemented
6#33108C212yesmaximal, complemented
7#33105C214yesmaximal, complemented
8#32985C212yesmaximal
9#32982C212yesmaximal
10#21591C214yesmaximal, complemented
11#21579C212yesmaximal, complemented
12C4×#483C212yesmaximal, complemented
13#21398C212yesmaximal, complemented
14#21369C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#485C2214yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#433C2212yes—
24#2857C2211yescharacteristic
25#2851C2211yescharacteristic
26#2848C2212yes—
27#2629C2211yescharacteristic
28#2625C2212yes—
29#2621C2211yescharacteristic
30#2480—22no—
31#2478—22no—
32#2475—22no—
33#2473—22no—
34#2323C2212yes—
35#2314C2211yescharacteristic
36C4×C22⋊Q8C2212yes—
37#2274—22no—
38#2274—22no—
39#2274C2212yes—
40#2274C2212yes—
41#2274C2212yes—
42#2274C2212yes—
43C2×C4×C4⋊C4—22no—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4—22no—
47#1989C2214yescomplemented
48#1981C2214yescomplemented
49#1949C2214yescomplemented
50#1948C2214yescomplemented
51#1928C2214yescomplemented
52#1927C2214yescomplemented
53#1927C2214yescomplemented
54#1927C2214yescomplemented
55#1921C2214yescomplemented
56#1920C2214yescomplemented
57#1919C2214yescomplemented
58#1919C2214yescomplemented
59#1873C2214yes—
60#1872C2212yes—
61#1872C2212yes—
62#1872C2214yes—
63#1872C2214yes—
64#1854C2214yes—
65#1853C2212yes—
66#1853C2214yes—
67#1853C2214yes—
68#1852C2212yes—
69#1825C2214yes—
70#1823C2212yes—
71#1823C2212yes—
72#1818C2212yes—
73#1798C2212yes—
74#1797C2212yes—
75#1779C2212yes—
76#1776C2214yescomplemented
77C4×#106C2214yescomplemented
78#1763C2214yescomplemented
79C4×#103C2214yescomplemented
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85#541—28no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#126—22no—
88C2×#125—22no—
89C2×#125C2311yescharacteristic
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117D412yes—
99C2×#117D412yes—
100C2×#117C2312yes—
101C2×#117C2312yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—28no—
110C22×C4⋊C4D412yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D412yes—
117C22×C42—21no—
118C22×C42—22no—
119C22×C42—22no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#485—216no—
123#483—28no—
124#483—28no—
125#433—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2312yes—
140C2×C4×C8C2312yes—
141#400—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#396—22no—
149#391C2312yes—
150#391C2312yes—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#385—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#385—22no—
167#384C2311yescharacteristic
168#382C2311yescharacteristic
169#380—22no—
170#380C2312yes—
171#380C2312yes—
172#380C2312yes—
173#379—22no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—24no—
190C4×C22⋊C4—24no—
191C4×C22⋊C4—24no—
192C4×C22⋊C4—24no—
193C2×#95—22no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206C25C2×D411yescharacteristic
207C22×Q8C2412yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—84no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—24no—
228#126—48no—
229#125—24no—
230#125—48no—
231C22⋊Q8—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—28no—
235C4×Q8—28no—
236#117—28no—
237#117—28no—
238#117—24no—
239#117—24no—
240#117—28no—
241#117—28no—
242#117—28no—
243#117—24no—
244#117—24no—
245#117#4118yes—
246#117#4118yes—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4#4118yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2414yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D412yes—
290C2×C22⋊C4C2×D412yes—
291C2×C22⋊C4C2×D412yes—
292C2×C42—28no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—84no—
304C2×C42—84no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—28no—
312#106—28no—
313#106—28no—
314#106—28no—
315C4⋊C8—28no—
316#103—28no—
317#103—28no—
318#103—28no—
319#103—28no—
320C22⋊C8—28no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—28no—
324#97—24no—
325C4×C8—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—42no—
346C24C22×D411yescharacteristic
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8—24no—
353C22×C4—232no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—816no—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4114yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4C2511yescharacteristic
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4#12318yes—
432C4⋊C4C2×#4118yes—
433C4⋊C4C2×#4118yes—
434C22⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—28no—
437C22⋊C4—24no—
438C22⋊C4—24no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4#12318yes—
446C42—28no—
447C42—24no—
448C42—24no—
449C42—28no—
450C42—22no—
451C42—22no—
452C42—44no—
453C42—82no—
454C42—84no—
455C42—84no—
456C42—84no—
457C42—84no—
458C42C2×#4114yes—
459C42C2×#4114yes—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—28no—
464C23—22no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—48no—
473C23—48no—
474C23—44no—
475C23—44no—
476C23C2×#12312yes—
477C23#52512yes—
478C23#52512yes—
479C23#57714yes—
480C23C23×D411yescharacteristic
481Q8—44no—
482Q8—44no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—416no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—88no—
529C2×C4C2×#12314yes—
530C2×C4C2×#12314yes—
531C2×C4C2×#12312yes—
532C2×C4#52512yes—
533C2×C4#52512yes—
534C2×C4C22×#4112yes—
535C2×C4C22×#4112yes—
536C2×C4C22×#4112yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—24no—
541C22—22no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—416no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22#312314yes—
555C22C22×#12311yescharacteristic
556C22C2×#52511yescharacteristic
557C22C2×#52511yescharacteristic
558C22C2×#57712yes—
559C22#358612yes—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—24no—
564C4—24no—
565C4—42no—
566C4—42no—
567C4—48no—
568C4—48no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4#344414yes—
574C2—24no—
575C2—42no—
576C2—42no—
577C2#3662611yescharacteristic
578C2#6052212yes—
579C2C2×#312311yescharacteristic
580C2#6056611yescharacteristic
581C2#6058512yes—
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. #33157C2semidirect
4. #33151C2semidirect
5. #33117C2semidirect
6. #33108C2semidirect
7. #33105C2semidirect
10. #21591C2semidirect
11. #21579C2semidirect
12. C4×#483C2semidirect
13. #21398C2semidirect
14. #21369C2semidirect
47. #1989C22semidirect
48. #1981C22semidirect
49. #1949C22semidirect
50. #1948C22semidirect
51. #1928C22semidirect
52. #1927C22semidirect
53. #1927C22semidirect
54. #1927C22semidirect
55. #1921C22semidirect
56. #1920C22semidirect
57. #1919C22semidirect
58. #1919C22semidirect
76. #1776C22semidirect
77. C4×#106C22semidirect
78. #1763C22semidirect
79. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#353102
C2×#22742
#331572
#331512
#331172
#331082
#331052
#329852
#329822
#215912
#215792
C4×#4832
#213982
#213692
#207162