Finite Group Databasebeta

← #6481942

Subgroups — #6481942

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#33151C212yesmaximal, complemented
4#33148C211yesmaximal, characteristic, complemented
5#33146C211yesmaximal, characteristic, complemented
6#29956C212yesmaximal
7#29952C211yesmaximal, characteristic
8#29950C211yesmaximal, characteristic
9#26099C214yesmaximal, complemented
10#26098C214yesmaximal, complemented
11#26033C214yesmaximal, complemented
12#26032C214yesmaximal, complemented
13#21987C212yesmaximal, complemented
14#21955C211yesmaximal, characteristic, complemented
15#20739C212yesmaximal
16C22×C4×D4C2211yescharacteristic
17C22×#117—21no—
18C2×#432C2212yes—
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21#2851C2212yes—
22#2627—21no—
23#2627—21no—
24#2606—22no—
25#2605—22no—
26#2570C2212yescomplemented
27#2570C2212yescomplemented
28#2550C2212yescomplemented
29#2550C2212yescomplemented
30#2354—22no—
31#2354—22no—
32#2354C2214yes—
33#2354C2214yes—
34#2329—21no—
35#2329—21no—
36C2×#390C2211yescharacteristic
37C2×#390C2211yescharacteristic
38C2×#388—21no—
39C2×#388—21no—
40C2×#384C2211yescharacteristic
41#2283C2212yes—
42C2×#380—21no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2094C2212yes—
47#2093C2212yes—
48#2092C2214yes—
49#2067C2214yes—
50#2067C2214yes—
51#2065C2214yes—
52#2065C2214yes—
53#2025C2214yes—
54#2023C2212yes—
55#2023C2212yes—
56#1989C2214yescomplemented
57#1988C2212yescomplemented
58#1987C2212yescomplemented
59#1949C2214yescomplemented
60#1948C2212yescomplemented
61#1948C2214yescomplemented
62#1947C2212yescomplemented
63#1947C2214yescomplemented
64#1940C2214yescomplemented
65#1940C2214yescomplemented
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1918C2214yescomplemented
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1847C2214yescomplemented
75#1836C2214yescomplemented
76#1823C2212yes—
77#1823C2212yes—
78#1804C2214yes—
79#1804C2214yes—
80#1797C2211yescharacteristic
81#1797C2212yes—
82#1797C2211yescharacteristic
83#1786C2212yes—
84#1786C2212yes—
85#1762C2214yes—
86#1762C2214yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—21no—
94C2×C4×D4—22no—
95C2×C4×D4—24no—
96C2×C4×D4Q818yescomplemented
97C2×C4×D4C2314yes—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—48no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4D411yescharacteristic
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#432—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2312yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×#97C2314yes—
134#400—22no—
135#400—22no—
136#398—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#395—22no—
141#395—22no—
142#391—21no—
143#391—21no—
144#391—22no—
145#391—22no—
146#390—22no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#390C2314yes—
151#390C2312yes—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389C2312yes—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#388—24no—
161#388—24no—
162#388—24no—
163#388—24no—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—22no—
171#385—22no—
172#384—22no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2314yes—
176#383—22no—
177#382C2312yes—
178#382C2312yes—
179#382C2312yes—
180#381—22no—
181#380—21no—
182#380—21no—
183#380—24no—
184#380—24no—
185#380C2312yes—
186#380C2312yes—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4D412yes—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200#376—22no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95D411yescharacteristic
204C2×#95D411yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216C25—21no—
217C22×Q8—21no—
218C22×Q8—21no—
219C22×D4—21no—
220C22×D4—22no—
221C22×D4—42no—
222C22×D4—42no—
223C22×D4C2×Q814yes—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243#125—44no—
244C22⋊Q8—44no—
245C22⋊Q8—44no—
246C4×D4—216no—
247C4×D4—28no—
248C4×D4C2×Q8116yes—
249#117—416no—
250#117—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—82no—
300C2×C22⋊C4—82no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D412yes—
314C2×C42C2×Q814yes—
315C2×C42#4114yes—
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2412yes—
320#107—44no—
321#107—44no—
322#106—44no—
323#106—44no—
324C4⋊C8—28no—
325C4⋊C8—44no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328#102—44no—
329#102—44no—
330#102—44no—
331#102—44no—
332C22⋊C8—24no—
333C22⋊C8—24no—
334#97—28no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356C24—24no—
357C24—22no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24C22×D411yescharacteristic
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—48no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—48no—
379C2×D4C22×Q814yes—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—216no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—48no—
407C22×C4—432no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C22⋊Q814yes—
418C22×C4C22⋊Q814yes—
419C22×C4AGL(1,ℤ/8)14yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×Q812yes—
430C22×C4C2×#4112yes—
431C22×C4C2511yescharacteristic
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4C22×Q814yes—
468C22⋊C4—216no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—41no—
481C42—41no—
482C42—42no—
483C42—48no—
484C42—44no—
485C42—44no—
486C42—82no—
487C42—82no—
488C42C22⋊Q818yes—
489C42C22×Q814yes—
490C42C2×#4114yes—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—24no—
496C23—48no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—84no—
506C23—84no—
507C23—84no—
508C23—84no—
509C23C2×C22⋊Q812yes—
510C23C2×AGL(1,ℤ/8)12yes—
511C23#57512yes—
512C23C23×D411yescharacteristic
513Q8—162no—
514Q8—162no—
515D4—28no—
516D4—48no—
517D4—44no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—432no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4C2×C22⋊Q812yes—
574C2×C4C2×C22⋊Q814yes—
575C2×C4C2×C22⋊Q814yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C23×Q811yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4#58512yes—
581C2×C4#58512yes—
582C8—44no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—816no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22#307714yes—
599C22C22×C22⋊Q811yescharacteristic
600C22#348511yescharacteristic
601C22#348711yescharacteristic
602C22C22×AGL(1,ℤ/8)11yescharacteristic
603C22C2×#57511yescharacteristic
604C22#358511yescharacteristic
605C22#358611yescharacteristic
606C4—22no—
607C4—22no—
608C4—24no—
609C4—24no—
610C4—22no—
611C4—42no—
612C4—41no—
613C4—42no—
614C4—41no—
615C4—44no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#345812yes—
623C4#349312yes—
624C2—24no—
625C2—84no—
626C2#3515111yescharacteristic
627C2C2×#307711yescharacteristic
628C2#6049211yescharacteristic
629C2#6170611yescharacteristic
630C2#6171011yescharacteristic
631C2#6189011yescharacteristic
632C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35104C2semidirect
3. #33151C2semidirect
4. #33148C2semidirect
5. #33146C2semidirect
9. #26099C2semidirect
10. #26098C2semidirect
11. #26033C2semidirect
12. #26032C2semidirect
13. #21987C2semidirect
14. #21955C2semidirect
26. #2570C22semidirect
27. #2570C22semidirect
28. #2550C22semidirect
29. #2550C22semidirect
56. #1989C22semidirect
57. #1988C22semidirect
58. #1987C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1947C22semidirect
64. #1940C22semidirect
65. #1940C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
69. #1918C22semidirect
74. #1847C22semidirect
75. #1836C22semidirect
96. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23542
#331512
#331482
#331462
#299562
#299522
#299502
#260992
#260982
#260332
#260322
#219872
#219552
#207392