Finite Group Databasebeta

← #6493669

Subgroups — #6493669

Order 512.

1759 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35327C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#33115C214yesmaximal, complemented
4#29935C214yesmaximal, complemented
5#27978C212yesmaximal, complemented
6#27915C212yesmaximal, complemented
7#27895C212yesmaximal, complemented
8#27894C212yesmaximal, complemented
9#27592C212yesmaximal, complemented
10#27591C212yesmaximal, complemented
11#24016C212yesmaximal, complemented
12#24015C212yesmaximal
13#21988C212yesmaximal, complemented
14#21808C212yesmaximal, complemented
15#20719C211yesmaximal, characteristic, complemented
16C2×#536C2211yescharacteristic
17C2×#535C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C2×#485C2214yescomplemented
21C2×#482C2214yes—
22C2×#432C2212yes—
23#2850C2212yes—
24#2850C2212yes—
25#2620C2212yes—
26#2620C2212yes—
27#2545—22no—
28#2544—22no—
29#2534—22no—
30#2534—22no—
31#2355—22no—
32#2354—22no—
33#2352—22no—
34#2352—22no—
35#2314C2212yes—
36#2314C2212yescomplemented
37#2311C2212yes—
38#2310C2211yescharacteristic
39#2309C2211yescharacteristic
40#2285C2212yes—
41#2285C2211yescharacteristic
42#2285C2212yescomplemented
43#2285C2211yescharacteristic, complemented
44C2×#384C2211yescharacteristic, complemented
45#2274C2212yescomplemented
46#2097C2214yescomplemented
47#2092C2214yes—
48#2072C2214yescomplemented
49#2071C2214yes—
50#2036C2214yescomplemented
51#2035C2214yescomplemented
52#2034C2214yes—
53#2033C2214yes—
54#1983C2214yes—
55#1983C2214yes—
56#1959C2212yes—
57#1959C2212yes—
58#1959C2212yes—
59#1959C2212yes—
60#1948C2214yescomplemented
61#1947C2214yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1926C2214yes—
65#1925C2214yes—
66#1874C2214yes—
67#1872C2214yescomplemented
68#1854C2214yes—
69#1854C2214yes—
70#1847C2212yescomplemented
71#1847C2212yescomplemented
72#1836C2212yescomplemented
73#1836C2212yescomplemented
74#1818C2212yes—
75#1798C2212yescomplemented
76#1798C2212yescomplemented
77#1787C2214yes—
78#1786C2214yes—
79#1780C2212yes—
80#1749C2212yescomplemented
81#1749C2211yescharacteristic, complemented
82#1749C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86#536—44no—
87#535—44no—
88C2×C4⋊Q8—21no—
89C2×#127—21no—
90C2×#124—21no—
91C2×#124—21no—
92C2×#124—22no—
93C2×#124—28no—
94C2×#124C2312yes—
95C2×#124C2312yes—
96C2×#123—21no—
97C2×#123—21no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×#121C2312yes—
101C2×#121C2312yes—
102C2×#120—21no—
103C2×#120—21no—
104C4×#41—28no—
105C2×C4×Q8D412yes—
106C2×C4×D4D412yes—
107C2×C4×D4D412yes—
108C2×C4×D4D412yes—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117C2312yes—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2311yescharacteristic
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—22no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#485—48no—
124#482—48no—
125#432—28no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145#396—22no—
146#396—22no—
147#396—22no—
148#396—22no—
149#394—22no—
150#394—22no—
151#394—22no—
152#394—22no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#390—22no—
158#390—22no—
159#390C2312yes—
160#390C2312yes—
161#388—24no—
162#388—24no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—24no—
168#387—24no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#386—22no—
175#384—22no—
176#384C2312yes—
177#384C2312yes—
178#384C2312yes—
179#384C2312yes—
180#382—22no—
181#382—22no—
182#382C2312yes—
183#382C2312yes—
184#381—22no—
185#380—22no—
186#380—22no—
187#380—22no—
188#380—22no—
189#380C2314yes—
190#380C2314yes—
191#379—22no—
192C4×C4⋊C4—24no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2314yes—
199C4×C22⋊C4—24no—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2312yes—
206#336C2312yes—
207C25—21no—
208C2×#41—24no—
209C2×#41—24no—
210C2×#41—42no—
211C22×Q8—21no—
212C22×Q8—24no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8C2×D411yescharacteristic
216C22×Q8C2412yes—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—24no—
221C22×D4—42no—
222C22×D4—42no—
223C22×D4C2×D411yescharacteristic
224C22×D4C2×D411yescharacteristic
225C22×D4C2×D411yescharacteristic
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—22no—
229C23×C4C2×D412yes—
230C23×C4C2×D412yes—
231C23×C4C2×D412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C22×C8C2414yes—
236C22×C8C2414yes—
237C4⋊Q8—44no—
238#127—44no—
239#124—28no—
240#124—28no—
241#124—44no—
242#124—44no—
243#124—48no—
244#123—44no—
245#123—44no—
246C22⋊Q8—28no—
247C22⋊Q8—44no—
248#121—28no—
249#121—44no—
250#120—44no—
251#120—44no—
252C4×Q8—28no—
253C4×D4—28no—
254C4×D4—28no—
255C4×D4—28no—
256#117—24no—
257#117—24no—
258#117—24no—
259#117—28no—
260#117—42no—
261#117—42no—
262#117—84no—
263#117—84no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—84no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42#4118yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—44no—
321#107—44no—
322#106—44no—
323#106—44no—
324C4⋊C8—28no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329#102—44no—
330#102—44no—
331#102—44no—
332#102—44no—
333C22⋊C8—28no—
334C22⋊C8—28no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337#97—24no—
338#97—24no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350C24—22no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—82no—
355C24—82no—
356C24C22×D411yescharacteristic
357C24C22×D411yescharacteristic
358C24C22×D411yescharacteristic
359#41—416no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—41no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—88no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—42no—
387C2×D4—88no—
388C2×D4—82no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—44no—
416C22×C4—84no—
417C22×C4#12318yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4114yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C4C2511yescharacteristic
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—44no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—24no—
474C42—24no—
475C42—22no—
476C42—22no—
477C42—42no—
478C42—42no—
479C42—44no—
480C42—84no—
481C42—84no—
482C42#12318yes—
483C42C2×#4114yes—
484C42C2×#4114yes—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—84no—
506C23—88no—
507C23—162no—
508C23—162no—
509C23—164no—
510C23C2×#12312yes—
511C23#53512yes—
512C23C23×D411yescharacteristic
513Q8—44no—
514Q8—48no—
515Q8—84no—
516D4—44no—
517D4—44no—
518D4—44no—
519D4—48no—
520D4—84no—
521D4—164no—
522D4—164no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—84no—
562C2×C4—164no—
563C2×C4C2×#12314yes—
564C2×C4C2×#12312yes—
565C2×C4C2×#12314yes—
566C2×C4#53512yes—
567C2×C4C22×#4112yes—
568C2×C4C2×H(5,𝔽2)12yes—
569C2×C4#58512yes—
570C8—44no—
571C8—44no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—168no—
595C22C22×#12311yescharacteristic
596C22C2×#53511yescharacteristic
597C22#344911yescharacteristic
598C22#358512yes—
599C22#358612yes—
600C4—24no—
601C4—24no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—84no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4#349414yes—
613C2—42no—
614C2—42no—
615C2—81no—
616C2—162no—
617C2#3526211yescharacteristic
618C2#6051712yes—
619C2#6082912yes—
620C2#6101711yescharacteristic
621C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #33115C2semidirect
4. #29935C2semidirect
5. #27978C2semidirect
6. #27915C2semidirect
7. #27895C2semidirect
8. #27894C2semidirect
9. #27592C2semidirect
10. #27591C2semidirect
11. #24016C2semidirect
13. #21988C2semidirect
14. #21808C2semidirect
15. #20719C2semidirect
20. C2×#485C22semidirect
36. #2314C22semidirect
42. #2285C22semidirect
43. #2285C22semidirect
44. C2×#384C22semidirect
45. #2274C22semidirect
46. #2097C22semidirect
48. #2072C22semidirect
50. #2036C22semidirect
51. #2035C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
67. #1872C22semidirect
70. #1847C22semidirect
71. #1847C22semidirect
72. #1836C22semidirect
73. #1836C22semidirect
75. #1798C22semidirect
76. #1798C22semidirect
80. #1749C22semidirect
81. #1749C22semidirect
82. #1749C22semidirect

Maximal subgroups

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

subgroupindex
#353272
#341882
#331152
#299352
#279782
#279152
#278952
#278942
#275922
#275912
#240162
#240152
#219882
#218082
#207192