Finite Group Databasebeta

← #6497133

Subgroups — #6497133

Order 512.

1734 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2D4×#117C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#33066C211yesmaximal, characteristic, complemented
5#32956C212yesmaximal
6#29994C211yesmaximal, characteristic, complemented
7#29939C212yesmaximal, complemented
8#29868C211yesmaximal, characteristic, complemented
9#27541C214yesmaximal
10#27477C214yesmaximal, complemented
11#27381C214yesmaximal, complemented
12#27317C214yesmaximal, complemented
13#22117C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21614C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#476C2212yes—
19C2×#476C2212yes—
20#2868C2211yescharacteristic, complemented
21#2859C2211yescharacteristic, complemented
22#2857C2212yes—
23#2647—21no—
24#2629—21no—
25#2507C2214yescomplemented
26#2497C2214yes—
27#2446—22no—
28#2429—22no—
29#2354C2212yes—
30#2352—22no—
31D4×C4⋊C4C2212yes—
32#2344C2212yes—
33#2342—22no—
34D4×C22⋊C4C2212yes—
35#2330—21no—
36#2314—21no—
37C4×C4⋊Q8C2212yescomplemented
38#2293C2211yescharacteristic
39#2293C2211yescharacteristic
40#2290—21no—
41#2290—21no—
42#2285C2211yescharacteristic
43#2280—21no—
44C42×D4C2211yescharacteristic
45C42×D4C2211yescharacteristic
46C4×#117C2211yescharacteristic
47#2100C2214yescomplemented
48#2086C2214yescomplemented
49#2084C2214yescomplemented
50#2083C2214yescomplemented
51#2076C2214yes—
52#2076C2214yes—
53#2071C2214yes—
54#2067C2214yes—
55#1987C2212yescomplemented
56#1987C2212yescomplemented
57#1948C2214yescomplemented
58#1947C2212yescomplemented
59#1947C2212yescomplemented
60#1944C2214yescomplemented
61#1936C2214yes—
62#1925C2214yes—
63#1924C2214yes—
64#1920C2214yescomplemented
65#1919C2214yescomplemented
66#1918C2214yescomplemented
67#1917C2214yescomplemented
68#1864C2214yes—
69#1864C2214yes—
70#1862C2214yes—
71#1862C2214yes—
72#1853C2214yes—
73#1842C2211yescharacteristic, complemented
74#1831C2211yescharacteristic, complemented
75#1802C2212yes—
76#1802C2212yes—
77#1802C2214yes—
78#1797C2212yes—
79#1786C2214yes—
80#1776C2214yescomplemented
81#1767C2212yescomplemented
82#1762C2214yes—
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×#125—21no—
88C2×C22⋊Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4C2311yescharacteristic
95C2×C4×D4C2311yescharacteristic
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×C4×D4C2311yescharacteristic
99C2×C4×D4C2311yescharacteristic
100C2×#117—21no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—48no—
106C2×#117D411yescharacteristic
107C2×#117D411yescharacteristic
108C22×C4⋊C4—22no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4—22no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115#476—28no—
116#476—28no—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130#400—24no—
131#398—24no—
132#398—24no—
133#396—24no—
134#391—22no—
135#391—22no—
136#390C2312yes—
137#390C2312yes—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—24no—
142#389C2314yes—
143#389C2314yes—
144#389C2314yes—
145#389C2312yes—
146#389C2312yes—
147#388—24no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#387—24no—
153#387—24no—
154#387—22no—
155#387—22no—
156#386—22no—
157#386—22no—
158#385—22no—
159#385—22no—
160#384—21no—
161#384—21no—
162#384C2312yes—
163#384C2314yes—
164#384C2312yes—
165#382—21no—
166#382—21no—
167#381—22no—
168#380—22no—
169#380—21no—
170#380—22no—
171#380—22no—
172#380C2312yes—
173#380C2314yes—
174#379—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2311yescharacteristic
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2312yes—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4D412yes—
191C4×C22⋊C4C2312yes—
192#376D411yescharacteristic
193#376D411yescharacteristic
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2314yes—
205#336C2314yes—
206#336C2314yes—
207C25—21no—
208C22×Q8—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—42no—
214C22×D4C2411yescharacteristic
215C22×D4C2411yescharacteristic
216C23×C4—28no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—84no—
234#125—44no—
235C22⋊Q8—84no—
236C4×Q8—84no—
237C4×D4—28no—
238C4×D4—28no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4#4114yes—
250C4×D4#4114yes—
251#117—24no—
252#117—24no—
253#117—44no—
254#117—48no—
255#117—48no—
256#117—44no—
257#117—44no—
258C2×C4⋊C4—21no—
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—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2411yescharacteristic
281C2×C4⋊C4C2411yescharacteristic
282C2×C22⋊C4—28no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319#107—44no—
320#107—44no—
321#106—28no—
322#106—28no—
323C4⋊C8—24no—
324C4⋊C8—24no—
325C4⋊C8—44no—
326#102—28no—
327#102—28no—
328#102—28no—
329#102—44no—
330C22⋊C8—44no—
331#97—44no—
332#95—22no—
333#95—22no—
334#95—21no—
335#95—21no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347#95C2×D412yes—
348C24—21no—
349C24—24no—
350C24—21no—
351C24—44no—
352C24—42no—
353C24—42no—
354C24—82no—
355C24C22×D411yescharacteristic
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×D4—24no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4C2×#4112yes—
370C2×D4C2×#4112yes—
371C2×D42−1+414yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—432no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4#13714yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2511yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—82no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4C2×#4112yes—
451C4⋊C4C2×#4112yes—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—84no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—21no—
472C42—21no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—42no—
478C42—42no—
479C42—44no—
480C42—44no—
481C42—42no—
482C42—42no—
483C42—84no—
484C42AGL(1,ℤ/8)14yes—
485C42C2×#4112yes—
486C42C2×#4112yes—
487C23—22no—
488C23—22no—
489C23—21no—
490C23—24no—
491C23—22no—
492C23—21no—
493C23—22no—
494C23—48no—
495C23—48no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—88no—
504C23—84no—
505C23—88no—
506C23C2×#13712yes—
507C23C23×D411yescharacteristic
508Q8—88no—
509D4—22no—
510D4—24no—
511D4—24no—
512D4—48no—
513D4—41no—
514D4—41no—
515D4#54114yes—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—82no—
562C2×C4—88no—
563C2×C4—88no—
564C2×C4#52512yes—
565C2×C4#52512yes—
566C2×C4C2×AGL(1,ℤ/8)14yes—
567C2×C4C2×AGL(1,ℤ/8)12yes—
568C2×C4C2×#13712yes—
569C2×C4C2×#13712yes—
570C2×C4C22×#4111yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4C2×2−1+412yes—
573C2×C4#58512yes—
574C8—44no—
575C8—44no—
576C22—22no—
577C22—24no—
578C22—24no—
579C22—42no—
580C22—41no—
581C22—42no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—41no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—816no—
590C22—82no—
591C22—82no—
592C22C2×#52511yescharacteristic
593C22#348912yes—
594C22C22×AGL(1,ℤ/8)11yescharacteristic
595C22C22×#13711yescharacteristic
596C22#358512yes—
597C4—22no—
598C4—22no—
599C4—24no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—84no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4#311014yes—
612C4C2×#54111yescharacteristic
613C4#345511yescharacteristic
614C4#349512yes—
615C2—42no—
616C2—44no—
617C2—82no—
618C2#3531011yescharacteristic
619C2C2×#311011yescharacteristic
620C2#6056811yescharacteristic
621C2#6210812yes—
622C2#6211012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #33197C2semidirect
4. #33066C2semidirect
6. #29994C2semidirect
7. #29939C2semidirect
8. #29868C2semidirect
10. #27477C2semidirect
11. #27381C2semidirect
12. #27317C2semidirect
13. #22117C2semidirect
14. #21926C2semidirect
15. #21614C2semidirect
20. #2868C22semidirect
21. #2859C22semidirect
25. #2507C22semidirect
37. C4×C4⋊Q8C22semidirect
47. #2100C22semidirect
48. #2086C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
55. #1987C22semidirect
56. #1987C22semidirect
57. #1948C22semidirect
58. #1947C22semidirect
59. #1947C22semidirect
60. #1944C22semidirect
64. #1920C22semidirect
65. #1919C22semidirect
66. #1918C22semidirect
67. #1917C22semidirect
73. #1842C22semidirect
74. #1831C22semidirect
80. #1776C22semidirect
81. #1767C22semidirect

Maximal subgroups

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

subgroupindex
#352842
D4×#1172
#331972
#330662
#329562
#299942
#299392
#298682
#275412
#274772
#273812
#273172
#221172
#219262
#216142