Finite Group Databasebeta

← #6479530

Subgroups — #6479530

Order 512.

1787 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#33099C211yesmaximal, characteristic, complemented
4#33098C211yesmaximal, characteristic, complemented
5#33095C211yesmaximal, characteristic, complemented
6#33094C211yesmaximal, characteristic, complemented
7#29913C212yesmaximal, complemented
8#29912C212yesmaximal, complemented
9#27860C214yesmaximal, complemented
10#27859C214yesmaximal, complemented
11#27618C214yesmaximal, complemented
12#27590C214yesmaximal, complemented
13#21941C212yesmaximal, complemented
14#21763C212yesmaximal
15C2×#1817C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19#2855C2211yescharacteristic
20#2855C2212yes—
21#2855C2211yescharacteristic
22#2854C2211yescharacteristic
23#2854C2212yes—
24#2854C2211yescharacteristic
25#2853—22no—
26#2849—21no—
27#2849—21no—
28C4×#123—21no—
29C4×#123—21no—
30C4×#123—22no—
31#2297C2212yes—
32#2296C2212yes—
33#2283—22no—
34#2283—22no—
35#2283C2214yes—
36#2283C2214yes—
37C2×#383—21no—
38C2×#382C2212yes—
39#2280C2211yescharacteristic
40C2×#381—21no—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4—21no—
45C2×#376—21no—
46#2101C2214yescomplemented
47#2101C2214yescomplemented
48#2097C2214yescomplemented
49#2092C2214yescomplemented
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2074C2214yescomplemented
53#2073C2214yescomplemented
54#1928C2214yescomplemented
55#1927C2214yescomplemented
56#1926C2214yescomplemented
57#1925C2214yescomplemented
58#1921C2214yescomplemented
59#1920C2214yescomplemented
60#1919C2214yescomplemented
61#1918C2214yescomplemented
62#1893C2214yes—
63#1892C2214yes—
64#1874C2212yes—
65#1874C2212yes—
66#1872C2212yes—
67#1872C2212yes—
68#1854C2214yes—
69#1854C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1825C2214yes—
75#1817C2211yescharacteristic
76#1817C2211yescharacteristic
77#1817C2214yes—
78#1817C2212yes—
79#1808C2214yes—
80#1807C2214yes—
81#1788C2214yes—
82C2×#336C2214yes—
83C22×#41C2311yescharacteristic
84C24×C4D411yescharacteristic
85C24×C4D411yescharacteristic
86C24×C4C2311yescharacteristic
87C23×C8C2314yes—
88#525—28no—
89C2×#123—21no—
90C2×#123—22no—
91C2×#123—21no—
92C2×#123—21no—
93C2×C22⋊Q8C2311yescharacteristic
94C2×C22⋊Q8C2311yescharacteristic
95C2×#121C2311yescharacteristic
96C2×#121C2311yescharacteristic
97C2×#120—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×#117C2311yescharacteristic
101C2×#117C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—22no—
105C22×C42—28no—
106C22×C42—21no—
107C22×C42—21no—
108C22×C42D412yes—
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D412yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×C22⋊C8—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128#391C2312yes—
129#391C2312yes—
130#390C2312yes—
131#390C2312yes—
132#389C2312yes—
133#389C2312yes—
134#389C2312yes—
135#389C2312yes—
136#388—22no—
137#388—22no—
138#388—22no—
139#388—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#386—22no—
145#386—22no—
146#386—22no—
147#386—21no—
148#386—21no—
149#385—22no—
150#385—22no—
151#385—22no—
152#385—21no—
153#385—21no—
154#383—24no—
155#383—24no—
156#382—22no—
157#382—22no—
158#382C2314yes—
159#382C2314yes—
160#381—21no—
161#381—22no—
162#381—21no—
163#381—21no—
164#381—24no—
165#381—24no—
166#380—22no—
167#380—22no—
168#380C2312yes—
169#380C2314yes—
170#380C2314yes—
171#380C2311yescharacteristic
172#380C2311yescharacteristic
173#379—21no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—24no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186#376—24no—
187#376—24no—
188C2×#95—22no—
189C2×#95—22no—
190C2×#95—22no—
191C2×#95—21no—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203#336C2314yes—
204#336C2314yes—
205C25C2×D411yescharacteristic
206C2×#41—24no—
207C2×#41—44no—
208C22×Q8C2411yescharacteristic
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C2411yescharacteristic
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—44no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4#4118yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C22×C8—28no—
238C22×C8—44no—
239C22×C8—44no—
240C22×C8C2414yes—
241C22×C8C2414yes—
242#123—44no—
243#123—48no—
244#123—44no—
245#123—44no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248#121—24no—
249#121—24no—
250#120—44no—
251C4×D4—44no—
252C4×D4—44no—
253#117—24no—
254#117—44no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C4⋊C4C2411yescharacteristic
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2412yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—48no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2412yes—
308C2×C42C2412yes—
309#107—44no—
310#107—44no—
311#106—44no—
312#106—44no—
313#103—44no—
314#103—44no—
315#102—44no—
316#102—44no—
317C22⋊C8—28no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—21no—
330#95—21no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—21no—
343C24—42no—
344C24#12314yes—
345C24C22×D411yescharacteristic
346C24C22×D411yescharacteristic
347C24C22×D411yescharacteristic
348#41—88no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—44no—
354C2×D4—42no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—44no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—416no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—416no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4#12314yes—
406C22×C4#12314yes—
407C22×C4#12314yes—
408C22×C4#13514yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4114yes—
419C22×C4C2511yescharacteristic
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—88no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—48no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C42—24no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—48no—
467C42—44no—
468C42—42no—
469C42—48no—
470C23—22no—
471C23—22no—
472C23—21no—
473C23—22no—
474C23—21no—
475C23—28no—
476C23—28no—
477C23—22no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—44no—
482C23—48no—
483C23—48no—
484C23—41no—
485C23—42no—
486C23—41no—
487C23—42no—
488C23—41no—
489C23C2×#12312yes—
490C23C2×#12312yes—
491C23C2×#12312yes—
492C23C2×#13512yes—
493C23#57512yes—
494C23#57514yes—
495C23C23×D411yescharacteristic
496Q8—44no—
497D4—44no—
498D4—84no—
499D4—84no—
500D4—84no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—48no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—88no—
550C2×C4—88no—
551C2×C4C2×#12312yes—
552C2×C4C2×#12312yes—
553C2×C4C2×#12312yes—
554C2×C4C2×#13514yes—
555C2×C4C22×#4112yes—
556C2×C4#58514yes—
557C8—44no—
558C8—44no—
559C22—24no—
560C22—24no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—416no—
571C22—42no—
572C22—42no—
573C22—82no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—81no—
578C22#309114yes—
579C22#309314yes—
580C22C22×#12311yescharacteristic
581C22#344912yes—
582C22C22×#13511yescharacteristic
583C22C2×#57512yes—
584C22C2×#57511yescharacteristic
585C4—22no—
586C4—22no—
587C4—22no—
588C4—24no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—82no—
593C4—84no—
594C4—81no—
595C4—82no—
596C4—84no—
597C4—81no—
598C4—81no—
599C4#349114yes—
600C2—24no—
601C2—24no—
602C2—42no—
603C2—161no—
604C2#3509711yescharacteristic
605C2C2×#309111yescharacteristic
606C2C2×#309311yescharacteristic
607C2#6094212yes—
608C2#6095912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #33099C2semidirect
4. #33098C2semidirect
5. #33095C2semidirect
6. #33094C2semidirect
7. #29913C2semidirect
8. #29912C2semidirect
9. #27860C2semidirect
10. #27859C2semidirect
11. #27618C2semidirect
12. #27590C2semidirect
13. #21941C2semidirect
15. C2×#1817C2semidirect
46. #2101C22semidirect
47. #2101C22semidirect
48. #2097C22semidirect
49. #2092C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
52. #2074C22semidirect
53. #2073C22semidirect
54. #1928C22semidirect
55. #1927C22semidirect
56. #1926C22semidirect
57. #1925C22semidirect
58. #1921C22semidirect
59. #1920C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#22832
#330992
#330982
#330952
#330942
#299132
#299122
#278602
#278592
#276182
#275902
#219412
#217632
C2×#18172