Finite Group Databasebeta

← #6481959

Subgroups — #6481959

Order 512.

1554 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37642C211yesmaximal, characteristic, complemented
2#34186C211yesmaximal, characteristic
3#33159C214yesmaximal, complemented
4#29959C214yesmaximal
5#26103C214yesmaximal, complemented
6#26091C212yesmaximal, complemented
7#26087C212yesmaximal, complemented
8#26036C214yesmaximal, complemented
9#26024C212yesmaximal, complemented
10#26020C212yesmaximal, complemented
11#21988C211yesmaximal, characteristic, complemented
12#21984C211yesmaximal, characteristic, complemented
13#21954C211yesmaximal, characteristic, complemented
14#20738C211yesmaximal, characteristic
15#20732C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2844—22no—
20#2628—22no—
21#2628—22no—
22#2628C2214yes—
23#2571—21no—
24#2571C2212yescomplemented
25#2571C2212yescomplemented
26#2551—21no—
27#2551C2212yescomplemented
28#2551C2212yescomplemented
29#2357—22no—
30#2357C2214yes—
31#2354—21no—
32#2354—21no—
33#2354C2212yes—
34#2354C2212yes—
35#2312—22no—
36#2309C2212yes—
37#2285C2211yescharacteristic
38#2285C2211yescharacteristic
39#2285C2211yescharacteristic
40#2285C2211yescharacteristic
41#2280—22no—
42#2274C2212yes—
43#2092C2214yes—
44#2092C2214yes—
45#2067C2214yes—
46#2067C2214yes—
47#2065C2214yes—
48#2065C2214yes—
49#2023C2214yes—
50#2023C2214yes—
51#1988C2214yescomplemented
52#1987C2214yescomplemented
53#1949C2214yescomplemented
54#1949C2214yescomplemented
55#1949C2214yescomplemented
56#1948C2212yescomplemented
57#1947C2212yescomplemented
58#1940C2214yescomplemented
59#1939C2212yescomplemented
60#1939C2212yescomplemented
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#1802C2212yes—
76#1802C2212yes—
77#1802C2212yes—
78#1802C2212yes—
79#1798C2214yes—
80#1786C2212yes—
81#1786C2212yes—
82#1762C2214yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8C2314yes—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—24no—
93C2×C4×D4Q818yescomplemented
94C2×C4×D4C2312yes—
95C2×C4×D4C2312yes—
96C2×#117—21no—
97C2×#117—21no—
98C2×#117—22no—
99C2×#117—22no—
100C2×#117C2312yes—
101C2×#117C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—21no—
104C22×C42C2311yescharacteristic
105#432—24no—
106#431—24no—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2314yes—
114C2×C4⋊C8C2314yes—
115C2×#102C2312yes—
116C2×#102C2312yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2312yes—
120C2×#102C2312yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×#97C2312yes—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#400—24no—
127#400—24no—
128#400—24no—
129#399—24no—
130#395—24no—
131#395—24no—
132#395—24no—
133#393—24no—
134#390—22no—
135#390—22no—
136#390C2314yes—
137#390C2314yes—
138#389—22no—
139#389—22no—
140#389—22no—
141#389—22no—
142#389C2314yes—
143#389C2314yes—
144#388—24no—
145#388—24no—
146#388—24no—
147#388—24no—
148#384D411yescharacteristic
149#384D411yescharacteristic
150#384C2311yescharacteristic
151#384C2311yescharacteristic
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#384C2311yescharacteristic
155#384C2311yescharacteristic
156#382—22no—
157#382—22no—
158#382—21no—
159#382—21no—
160#382D412yes—
161#382C2314yes—
162#382C2312yes—
163#381—24no—
164#380—21no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380—22no—
169#380—22no—
170#380D412yes—
171#380D412yes—
172#380C2312yes—
173#380C2312yes—
174#379—24no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2314yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C22⋊C4—24no—
188C4×C22⋊C4—24no—
189#340C2312yes—
190#340C2312yes—
191#340C2312yes—
192#340C2312yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C22×Q8—24no—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4C2×Q814yes—
208C22×D4C2412yes—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—28no—
212C23×C4—22no—
213C23×C4—82no—
214C23×C4—82no—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2411yescharacteristic
218C22×C8C2412yes—
219C22×C8C2412yes—
220C22×C8C2414yes—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223C4⋊Q8—84no—
224C4⋊Q8—84no—
225C4×D4—28no—
226C4×D4—28no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4C2×Q818yes—
230C4×D4C2×Q818yes—
231#117—44no—
232#117—44no—
233#117—82no—
234#117—82no—
235#117—84no—
236#117—84no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×Q814yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4C2×D414yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×Q814yes—
287C2×C42#4114yes—
288C2×C42C2414yes—
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—44no—
292#107—44no—
293#106—44no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8—24no—
297C4⋊C8—44no—
298C4⋊C8—44no—
299#102—42no—
300#102—44no—
301#102—42no—
302#102—42no—
303#102—44no—
304#102—42no—
305C22⋊C8—24no—
306C22⋊C8—24no—
307#97—24no—
308C4×C8—22no—
309C4×C8—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—22no—
325C24—24no—
326C24—42no—
327C24—42no—
328C24—81no—
329C24—81no—
330C24—81no—
331C24—81no—
332C24C22×D411yescharacteristic
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×D4—24no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—48no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4C22×Q814yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—816no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C22⋊Q814yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q812yes—
380C22×C4C2×#4112yes—
381C22×C4C2511yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4C22×Q814yes—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C42—24no—
434C42—24no—
435C42—44no—
436C42—44no—
437C42—44no—
438C42—84no—
439C42—84no—
440C42C22⋊Q814yes—
441C42C22⋊Q814yes—
442C42C22×Q812yes—
443C42C22×Q812yes—
444C42C2×#4112yes—
445C42C2×#4112yes—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—24no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—84no—
461C23—84no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23C2×C22⋊Q812yes—
467C23C23×D411yescharacteristic
468Q8—168no—
469D4—24no—
470D4—24no—
471D4—48no—
472D4—42no—
473D4—42no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
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—88no—
515C2×C4—84no—
516C2×C4C2×C22⋊Q812yes—
517C2×C4C2×C22⋊Q812yes—
518C2×C4C2×C22⋊Q812yes—
519C2×C4C2×C22⋊Q812yes—
520C2×C4C2×C22⋊Q812yes—
521C2×C4#57612yes—
522C2×C4#57712yes—
523C2×C4C23×Q811yescharacteristic
524C2×C4C22×#4111yescharacteristic
525C2×C4#58512yes—
526C2×C4#58512yes—
527C8—42no—
528C8—42no—
529C8—44no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—44no—
538C22—44no—
539C22—44no—
540C22—816no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C22×C22⋊Q811yescharacteristic
546C22#348511yescharacteristic
547C22#348711yescharacteristic
548C22C2×#57611yescharacteristic
549C22C2×#57711yescharacteristic
550C22#358511yescharacteristic
551C22#358611yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—44no—
557C4—44no—
558C4—44no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4#308114yes—
566C4#345812yes—
567C4#349312yes—
568C2—42no—
569C2—84no—
570C2#3515111yescharacteristic
571C2C2×#308111yescharacteristic
572C2#6049311yescharacteristic
573C2#6180211yescharacteristic
574C2#6180311yescharacteristic
575C2#6191211yescharacteristic
576C2#6191311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37642C2semidirect
3. #33159C2semidirect
5. #26103C2semidirect
6. #26091C2semidirect
7. #26087C2semidirect
8. #26036C2semidirect
9. #26024C2semidirect
10. #26020C2semidirect
11. #21988C2semidirect
12. #21984C2semidirect
13. #21954C2semidirect
24. #2571C22semidirect
25. #2571C22semidirect
27. #2551C22semidirect
28. #2551C22semidirect
51. #1988C22semidirect
52. #1987C22semidirect
53. #1949C22semidirect
54. #1949C22semidirect
55. #1949C22semidirect
56. #1948C22semidirect
57. #1947C22semidirect
58. #1940C22semidirect
59. #1939C22semidirect
60. #1939C22semidirect
61. #1921C22semidirect
62. #1920C22semidirect
63. #1919C22semidirect
64. #1918C22semidirect
67. #1847C22semidirect
68. #1845C22semidirect
69. #1844C22semidirect
70. #1836C22semidirect
71. #1834C22semidirect
72. #1834C22semidirect
93. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#376422
#341862
#331592
#299592
#261032
#260912
#260872
#260362
#260242
#260202
#219882
#219842
#219542
#207382
#207322