Finite Group Databasebeta

← #6532019

Subgroups — #6532019

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic, complemented
3#27622C214yesmaximal, complemented
4#27591C214yesmaximal
5#25069C214yesmaximal, complemented
6#25059C212yesmaximal, complemented
7#25057C212yesmaximal, complemented
8#25002C214yesmaximal, complemented
9#24992C212yesmaximal, complemented
10#24990C212yesmaximal, complemented
11#21988C211yesmaximal, characteristic, complemented
12#21984C211yesmaximal, characteristic, complemented
13#21821C211yesmaximal, characteristic, complemented
14#20738C211yesmaximal, characteristic
15#20732C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2848—22no—
21#2626—22no—
22#2580C2214yes—
23#2566—22no—
24#2563—21no—
25#2542—21no—
26#2538C2212yes—
27#2538C2212yes—
28#2382—22no—
29#2365—21no—
30#2365—21no—
31#2355C2212yescomplemented
32#2355C2212yescomplemented
33Q8×C4⋊C4C2212yescomplemented
34Q8×C4⋊C4C2212yescomplemented
35#2323—22no—
36#2323—22no—
37#2317C2212yes—
38#2314C2214yes—
39#2309C2212yes—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42#2285C2211yescharacteristic
43#2100C2214yescomplemented
44#2099C2214yescomplemented
45#2092C2214yes—
46#2092C2214yes—
47#2085C2214yescomplemented
48#2085C2214yescomplemented
49#2071C2214yes—
50#2071C2214yes—
51#1985C2214yescomplemented
52#1984C2212yescomplemented
53#1984C2212yescomplemented
54#1949C2214yescomplemented
55#1948C2212yescomplemented
56#1947C2212yescomplemented
57#1926C2214yes—
58#1926C2214yes—
59#1925C2214yes—
60#1925C2214yes—
61#1921C2214yescomplemented
62#1920C2214yescomplemented
63#1919C2214yescomplemented
64#1918C2214yescomplemented
65#1854C2214yes—
66#1854C2214yes—
67#1847C2212yescomplemented
68#1845C2211yescharacteristic, complemented
69#1844C2211yescharacteristic, complemented
70#1836C2212yescomplemented
71#1834C2211yescharacteristic, complemented
72#1834C2211yescharacteristic, complemented
73#1824C2212yes—
74#1823C2212yes—
75#1818C2214yes—
76#1802C2212yes—
77#1802C2212yes—
78#1802C2212yes—
79#1802C2212yes—
80#1786C2212yes—
81#1786C2212yes—
82#1762C2214yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87#533—24no—
88#533—24no—
89C2×C4⋊Q8D411yescharacteristic
90C2×C4⋊Q8D411yescharacteristic
91C2×#125D412yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8D412yes—
95C2×C22⋊Q8D412yes—
96C2×C4×Q8—22no—
97C2×C4×Q8—22no—
98C2×C4×Q8C2312yes—
99C2×C4×Q8C2312yes—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C42C2311yescharacteristic
113#432—24no—
114#431—24no—
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2314yes—
118C2×#106C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#102C2312yes—
122C2×#102C2314yes—
123C2×#102C2312yes—
124C2×#102C2312yes—
125C2×#102C2314yes—
126C2×#102C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2312yes—
129C2×#97C2312yes—
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132#399—24no—
133#396—24no—
134#396—24no—
135#393—24no—
136#391—22no—
137#391—22no—
138#390—22no—
139#390—22no—
140#390C2314yes—
141#390C2314yes—
142#389—22no—
143#389—22no—
144#389—24no—
145#389—24no—
146#389C2314yes—
147#389C2314yes—
148#388—24no—
149#388—24no—
150#387—24no—
151#387—24no—
152#387—24no—
153#387—24no—
154#385—24no—
155#385—24no—
156#385—24no—
157#384—22no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#384C2314yes—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#382—22no—
166#382—21no—
167#382—21no—
168#382C2314yes—
169#382C2312yes—
170#380—21no—
171#380—21no—
172#380—21no—
173#380—21no—
174#380—22no—
175#380—22no—
176#380C2312yes—
177#380C2312yes—
178#379—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189#340C2312yes—
190#340C2312yes—
191#340C2312yes—
192#340C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25—21no—
204C22×Q8C2×D412yes—
205C22×Q8C2×D412yes—
206C22×D4—24no—
207C22×D4—21no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—44no—
216C23×C4—82no—
217C23×C4—82no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2411yescharacteristic
221C22×C8C2412yes—
222C22×C8C2412yes—
223C22×C8C2414yes—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226#125—28no—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C22⋊Q8—84no—
230C22⋊Q8—84no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—24no—
234C4×Q8—24no—
235C4×Q8—84no—
236C4×Q8—84no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—44no—
242C4×D4—44no—
243#117—24no—
244#117—24no—
245#117—44no—
246#117—82no—
247#117—82no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4C2×D414yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4114yes—
299C2×C42#4114yes—
300C2×C42C2414yes—
301C2×C42C2411yescharacteristic
302C2×C42C2411yescharacteristic
303#107—44no—
304#107—44no—
305#106—44no—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8—24no—
309#102—24no—
310#102—28no—
311#102—24no—
312#102—42no—
313#102—44no—
314#102—42no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317#97—24no—
318C4×C8—22no—
319C4×C8—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332C24—24no—
333C24—22no—
334C24—42no—
335C24—42no—
336C24—81no—
337C24—81no—
338C24—81no—
339C24—81no—
340C24C22×D411yescharacteristic
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D42−1+414yes—
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—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—816no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4#12314yes—
382C22×C4#12314yes—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2511yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—28no—
405C4⋊C4—28no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C22⋊C4—28no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C42—24no—
446C42—24no—
447C42—44no—
448C42—44no—
449C42—44no—
450C42—84no—
451C42—84no—
452C42#12314yes—
453C42#12314yes—
454C42C2×#4112yes—
455C42C2×#4112yes—
456C42C2×#4112yes—
457C42C2×#4112yes—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—48no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23C2×#12312yes—
479C23C23×D411yescharacteristic
480Q8—48no—
481Q8—48no—
482D4—24no—
483D4—24no—
484D4—48no—
485D4—42no—
486D4—42no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
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—88no—
527C2×C4—84no—
528C2×C4—88no—
529C2×C4C2×#12312yes—
530C2×C4C2×#12312yes—
531C2×C4C2×#12312yes—
532C2×C4C2×#12312yes—
533C2×C4C2×#12312yes—
534C2×C4#57612yes—
535C2×C4#57712yes—
536C2×C4C22×#4111yescharacteristic
537C2×C4C22×#4111yescharacteristic
538C2×C4C2×2−1+412yes—
539C2×C4#58512yes—
540C8—42no—
541C8—42no—
542C8—44no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—816no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C22×#12311yescharacteristic
559C22#348611yescharacteristic
560C22#348711yescharacteristic
561C22C2×#57611yescharacteristic
562C22C2×#57711yescharacteristic
563C22#358511yescharacteristic
564C22#358611yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—44no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#309614yes—
579C4#349512yes—
580C4#349512yes—
581C2—42no—
582C2—84no—
583C2#3567211yescharacteristic
584C2C2×#309611yescharacteristic
585C2#6051711yescharacteristic
586C2#6184711yescharacteristic
587C2#6184811yescharacteristic
588C2#6191211yescharacteristic
589C2#6191311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34374C2semidirect
3. #27622C2semidirect
5. #25069C2semidirect
6. #25059C2semidirect
7. #25057C2semidirect
8. #25002C2semidirect
9. #24992C2semidirect
10. #24990C2semidirect
11. #21988C2semidirect
12. #21984C2semidirect
13. #21821C2semidirect
31. #2355C22semidirect
32. #2355C22semidirect
33. Q8×C4⋊C4C22semidirect
34. Q8×C4⋊C4C22semidirect
43. #2100C22semidirect
44. #2099C22semidirect
47. #2085C22semidirect
48. #2085C22semidirect
51. #1985C22semidirect
52. #1984C22semidirect
53. #1984C22semidirect
54. #1949C22semidirect
55. #1948C22semidirect
56. #1947C22semidirect
61. #1921C22semidirect
62. #1920C22semidirect
63. #1919C22semidirect
64. #1918C22semidirect
67. #1847C22semidirect
68. #1845C22semidirect
69. #1844C22semidirect
70. #1836C22semidirect
71. #1834C22semidirect
72. #1834C22semidirect

Maximal subgroups

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

subgroupindex
#351722
#343742
#276222
#275912
#250692
#250592
#250572
#250022
#249922
#249902
#219882
#219842
#218212
#207382
#207322