Finite Group Databasebeta

← #6481938

Subgroups — #6481938

Order 512.

1616 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic, complemented
2C2×#2357C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#33149C211yesmaximal, characteristic, complemented
5#33147C211yesmaximal, characteristic, complemented
6#29956C212yesmaximal
7#29952C211yesmaximal, characteristic
8#29950C211yesmaximal, characteristic
9#26096C214yesmaximal, complemented
10#26094C214yesmaximal, complemented
11#26030C214yesmaximal, complemented
12#26028C214yesmaximal, complemented
13#21983C212yesmaximal, complemented
14#21951C211yesmaximal, characteristic, complemented
15#20733C212yesmaximal
16C22×C4×D4C2211yescharacteristic
17C2×#431C2212yes—
18#2857C2211yescharacteristic
19#2857C2211yescharacteristic
20#2851C2212yes—
21#2627—21no—
22#2627—21no—
23#2606—22no—
24#2605—22no—
25#2570C2212yescomplemented
26#2570C2212yescomplemented
27#2550C2212yescomplemented
28#2550C2212yescomplemented
29#2357—22no—
30#2357—22no—
31#2357C2214yes—
32#2357C2214yes—
33#2329—21no—
34#2329—21no—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#388—21no—
38C2×#388—21no—
39#2283C2212yes—
40C2×#382C2211yescharacteristic
41C2×#381—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4—21no—
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#1949C2214yescomplemented
61#1949C2214yescomplemented
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1939C2214yescomplemented
65#1939C2214yescomplemented
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1918C2214yescomplemented
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1846C2214yescomplemented
75#1835C2214yescomplemented
76#1824C2212yes—
77#1824C2212yes—
78#1804C2214yes—
79#1804C2214yes—
80#1797C2211yescharacteristic
81#1797C2212yes—
82#1797C2211yescharacteristic
83#1786C2212yes—
84#1786C2212yes—
85C4×#102C2214yes—
86C4×#102C2214yes—
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—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C42—21no—
109C22×C42—21no—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#431—28no—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2314yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2314yes—
121C2×C4⋊C8C2312yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C4×C8C2314yes—
129#400—22no—
130#400—22no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#395—22no—
136#395—22no—
137#391—21no—
138#391—21no—
139#391—22no—
140#391—22no—
141#390—22no—
142#390—21no—
143#390—21no—
144#390C2312yes—
145#390C2314yes—
146#390C2312yes—
147#389—21no—
148#389—21no—
149#389—22no—
150#389—22no—
151#389C2312yes—
152#389C2312yes—
153#389C2312yes—
154#389C2312yes—
155#388—24no—
156#388—24no—
157#388—24no—
158#388—24no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#385—22no—
166#385—22no—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#383—22no—
170#382—22no—
171#382C2312yes—
172#382C2312yes—
173#382C2312yes—
174#382C2314yes—
175#381—22no—
176#381—24no—
177#381—24no—
178#380—21no—
179#380—21no—
180#380C2312yes—
181#380C2312yes—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4D412yes—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2314yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—24no—
196C4×C22⋊C4—24no—
197#376—22no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95D411yescharacteristic
203C2×#95D411yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2312yes—
214#336C2312yes—
215C25—21no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×D4—21no—
219C22×D4—22no—
220C22×D4—42no—
221C22×D4—42no—
222C22×D4C2×Q814yes—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—28no—
226C23×C4—24no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—82no—
230C23×C4—82no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242#125—44no—
243C22⋊Q8—44no—
244C22⋊Q8—44no—
245C4×D4—216no—
246C4×D4—28no—
247C4×D4C2×Q8116yes—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×Q814yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—82no—
293C2×C22⋊C4—82no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—82no—
302C2×C42—82no—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42C2×Q814yes—
308C2×C42#4114yes—
309C2×C42C2412yes—
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2412yes—
313#107—44no—
314#107—44no—
315#106—44no—
316#106—44no—
317C4⋊C8—28no—
318C4⋊C8—44no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321#102—44no—
322#102—44no—
323#102—44no—
324#102—44no—
325C22⋊C8—24no—
326C22⋊C8—24no—
327C4×C8—28no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
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#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351C24—24no—
352C24—22no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24C22×D411yescharacteristic
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—81no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—81no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—48no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—48no—
374C2×D4C22×Q814yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—216no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—816no—
412C22×C4C22⋊Q814yes—
413C22×C4C22⋊Q814yes—
414C22×C4#13514yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q812yes—
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—81no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4C22×Q814yes—
461C22⋊C4—216no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—84no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—41no—
474C42—41no—
475C42—42no—
476C42—82no—
477C42—82no—
478C42—82no—
479C42—82no—
480C42—84no—
481C42C22⋊Q818yes—
482C42C22×Q814yes—
483C42C2×#4114yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—24no—
489C23—48no—
490C23—48no—
491C23—42no—
492C23—42no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23C2×C22⋊Q812yes—
503C23C2×#13512yes—
504C23#57512yes—
505C23C23×D411yescharacteristic
506Q8—162no—
507Q8—162no—
508D4—28no—
509D4—48no—
510D4—44no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
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—816no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4C2×C22⋊Q812yes—
567C2×C4C2×C22⋊Q814yes—
568C2×C4C2×C22⋊Q814yes—
569C2×C4C2×#13512yes—
570C2×C4C2×#13512yes—
571C2×C4C23×Q811yescharacteristic
572C2×C4C22×#4111yescharacteristic
573C2×C4#58512yes—
574C2×C4#58512yes—
575C8—44no—
576C8—44no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—816no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22#307614yes—
592C22C22×C22⋊Q811yescharacteristic
593C22#348511yescharacteristic
594C22#348711yescharacteristic
595C22C22×#13511yescharacteristic
596C22C2×#57511yescharacteristic
597C22#358511yescharacteristic
598C22#358511yescharacteristic
599C4—22no—
600C4—22no—
601C4—24no—
602C4—24no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—44no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#345812yes—
616C4#349312yes—
617C2—24no—
618C2—84no—
619C2#3515111yescharacteristic
620C2C2×#307611yescharacteristic
621C2#6049011yescharacteristic
622C2#6169911yescharacteristic
623C2#6171811yescharacteristic
624C2#6188511yescharacteristic
625C2#6189511yescharacteristic
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. #33150C2semidirect
4. #33149C2semidirect
5. #33147C2semidirect
9. #26096C2semidirect
10. #26094C2semidirect
11. #26030C2semidirect
12. #26028C2semidirect
13. #21983C2semidirect
14. #21951C2semidirect
25. #2570C22semidirect
26. #2570C22semidirect
27. #2550C22semidirect
28. #2550C22semidirect
56. #1989C22semidirect
57. #1988C22semidirect
58. #1987C22semidirect
59. #1949C22semidirect
60. #1949C22semidirect
61. #1949C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1939C22semidirect
65. #1939C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
69. #1918C22semidirect
74. #1846C22semidirect
75. #1835C22semidirect
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×#23572
#331502
#331492
#331472
#299562
#299522
#299502
#260962
#260942
#260302
#260282
#219832
#219512
#207332