Finite Group Databasebeta

← #6500829

Subgroups — #6500829

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35563C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27620C212yesmaximal, complemented
4#27604C212yesmaximal, complemented
5#27594C212yesmaximal, complemented
6#26215C214yesmaximal, complemented
7#26199C214yesmaximal, complemented
8#26142C214yesmaximal, complemented
9#26126C214yesmaximal, complemented
10#24002C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic, complemented
13#20847C212yesmaximal, complemented
14C2×#535C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C2×#520C2211yescharacteristic
18C2×#480C2214yes—
19C2×#428C2212yescomplemented
20C22×#107C2212yescomplemented
21C2×#408C2212yes—
22#2534—22no—
23#2428—22no—
24#2426—22no—
25#2416C2214yes—
26#2414C2214yes—
27#2408—22no—
28#2376—22no—
29#2358—22no—
30#2352C2214yescomplemented
31D4×C4⋊C4C2214yescomplemented
32#2329—22no—
33#2325—22no—
34#2316C2211yescharacteristic
35#2314C2212yes—
36#2311C2212yes—
37#2309C2211yescharacteristic
38#2280C2211yescharacteristic
39C2×#380C2212yes—
40#2101C2214yescomplemented
41#2094C2212yescomplemented
42#2093C2212yescomplemented
43#2083C2214yescomplemented
44#2077C2214yescomplemented
45#2075C2214yescomplemented
46#2073C2212yescomplemented
47#2073C2212yescomplemented
48#2033C2214yes—
49#2033C2214yes—
50#1926C2214yescomplemented
51#1925C2214yescomplemented
52#1921C2214yescomplemented
53#1918C2214yescomplemented
54#1895C2214yescomplemented
55#1893C2214yescomplemented
56#1875C2214yescomplemented
57#1874C2214yes—
58#1874C2214yescomplemented
59#1870C2214yescomplemented
60#1869C2214yescomplemented
61#1854C2214yes—
62#1854C2214yescomplemented
63#1853C2212yescomplemented
64#1852C2214yes—
65#1852C2212yescomplemented
66#1852C2214yescomplemented
67#1825C2214yes—
68#1824C2212yescomplemented
69#1824C2212yescomplemented
70#1818C2212yes—
71#1786C2214yes—
72#1779C2212yes—
73#1736C2212yescomplemented
74#1735C2212yescomplemented
75C22×#41C2311yescharacteristic
76C23×D4—21no—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×M16C2312yes—
80C23×C8C2312yes—
81#535—28no—
82C2×#127—21no—
83C2×#124—21no—
84#525—28no—
85C2×#123D412yes—
86C2×C22⋊Q8D412yes—
87C2×#121—22no—
88C2×#121—24no—
89C2×#121—44no—
90C2×#121D412yes—
91C2×#121C2314yes—
92C2×#121C2314yes—
93C2×#120—22no—
94C2×#120D412yes—
95#520—44no—
96C2×C4×Q8—21no—
97C2×C4×D4—22no—
98C2×C4×D4—21no—
99C2×C4×D4—24no—
100C2×C4×D4C2314yes—
101C2×#117C2311yescharacteristic
102C2×#117C2312yes—
103C2×#117C2311yescharacteristic
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—22no—
113#480—48no—
114#428—44no—
115C2×#107—44no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124#408—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130#396—24no—
131#394—24no—
132#394—24no—
133#392—24no—
134#391—22no—
135#390—22no—
136#390—22no—
137#390—24no—
138#390C2312yes—
139#390C2312yes—
140#389—22no—
141#389—24no—
142#389C2314yes—
143#388—24no—
144#388—24no—
145#388—24no—
146#387—24no—
147#387—24no—
148#386—22no—
149#386—22no—
150#386—24no—
151#386—24no—
152#385—24no—
153#385—24no—
154#384C2312yes—
155#382C2312yes—
156#381—21no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380C2314yes—
161#380C2314yes—
162#380C2312yes—
163#380C2314yes—
164#380C2312yes—
165#379—21no—
166#379—22no—
167C4×C4⋊C4C2314yes—
168C4×C22⋊C4—22no—
169C4×C22⋊C4—24no—
170C2×#95—22no—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#336C2312yes—
180#336C2312yes—
181#336C2314yes—
182C25—21no—
183C25C2×D411yescharacteristic
184C2×#41—42no—
185C2×#41—41no—
186C2×#41—41no—
187C2×#41#4118yes—
188C22×Q8C2×D411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—21no—
192C22×D4—24no—
193C22×D4—42no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—44no—
197C22×D4—84no—
198C22×D4C2×D414yes—
199C22×D4C2×D412yes—
200C22×D4C2×D411yescharacteristic
201C22×D4C2412yes—
202C23×C4—21no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4—41no—
210C23×C4—41no—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C2×M16—24no—
217C2×M16—42no—
218C2×M16—42no—
219C22×C8—24no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8C2414yes—
223C22×C8C2414yes—
224#127—44no—
225#124—44no—
226#123—28no—
227C22⋊Q8—28no—
228#121—28no—
229#121—28no—
230#121—28no—
231#121—28no—
232#121—44no—
233#121—48no—
234#121—88no—
235#120—28no—
236#120—48no—
237C4×Q8—44no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—44no—
241C4×D4—48no—
242C4×D4—88no—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—42no—
247#117—42no—
248#117—42no—
249#117—42no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2411yescharacteristic
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—84no—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C42—21no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42C2412yes—
297#107—28no—
298#107—28no—
299#107—44no—
300#106—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304#102—44no—
305C22⋊C8—28no—
306C22⋊C8—28no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—44no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318C24—22no—
319C24—24no—
320C24—22no—
321C24—41no—
322C24—42no—
323C24—41no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—82no—
328C24—82no—
329C24C22×D411yescharacteristic
330C24C22×D412yes—
331C24C22×D412yes—
332#41—48no—
333#41—48no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8#12314yes—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—41no—
348C2×D4—48no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—42no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—816no—
360C2×D4#12318yes—
361C2×D4#12314yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4C22×D412yes—
400C22×C4C22×D414yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4C2511yescharacteristic
407M16—48no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—48no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—28no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4H(5,𝔽2)14yes—
433C22⋊C4—24no—
434C22⋊C4—28no—
435C22⋊C4—28no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—48no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—82no—
446C22⋊C4—84no—
447C22⋊C4—82no—
448C22⋊C4—84no—
449C42—24no—
450C42—44no—
451C42—42no—
452C42—42no—
453C42—82no—
454C42—84no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—22no—
467C23—24no—
468C23—22no—
469C23—44no—
470C23—48no—
471C23—44no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—82no—
477C23—84no—
478C23—84no—
479C23—82no—
480C23—84no—
481C23—84no—
482C23—164no—
483C23—164no—
484C23C2×#12312yes—
485C23C2×#12314yes—
486C23#57614yes—
487C23C23×D411yescharacteristic
488Q8—44no—
489D4—44no—
490D4—48no—
491D4—416no—
492D4—44no—
493D4—44no—
494D4—84no—
495D4—816no—
496D4—168no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—84no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—164no—
534C2×C4—164no—
535C2×C4—164no—
536C2×C4—164no—
537C2×C4C2×#12312yes—
538C2×C4C2×#12314yes—
539C2×C4C22×#4112yes—
540C2×C4C2×H(5,𝔽2)12yes—
541C2×C4#58511yescharacteristic
542C2×C4#58511yescharacteristic
543C8—44no—
544C8—44no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—28no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—41no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—88no—
563C22—82no—
564C22—84no—
565C22—84no—
566C22—82no—
567C22—82no—
568C22—168no—
569C22C22×#12311yescharacteristic
570C22#348411yescharacteristic
571C22#348511yescharacteristic
572C22C2×#57612yes—
573C22#358612yes—
574C4—22no—
575C4—24no—
576C4—42no—
577C4—44no—
578C4—44no—
579C4—84no—
580C4—82no—
581C4—81no—
582C4—84no—
583C4—81no—
584C4#350512yes—
585C4#350512yes—
586C2—44no—
587C2—44no—
588C2—42no—
589C2—162no—
590C2#3535111yescharacteristic
591C2#6052112yes—
592C2#6163011yescharacteristic
593C2#6165211yescharacteristic
594C2#6175912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35563C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27620C2semidirect
4. #27604C2semidirect
5. #27594C2semidirect
6. #26215C2semidirect
7. #26199C2semidirect
8. #26142C2semidirect
9. #26126C2semidirect
10. #24002C2semidirect
11. #21770C2semidirect
12. #21744C2semidirect
13. #20847C2semidirect
19. C2×#428C22semidirect
20. C22×#107C22semidirect
30. #2352C22semidirect
31. D4×C4⋊C4C22semidirect
40. #2101C22semidirect
41. #2094C22semidirect
42. #2093C22semidirect
43. #2083C22semidirect
44. #2077C22semidirect
45. #2075C22semidirect
46. #2073C22semidirect
47. #2073C22semidirect
50. #1926C22semidirect
51. #1925C22semidirect
52. #1921C22semidirect
53. #1918C22semidirect
54. #1895C22semidirect
55. #1893C22semidirect
56. #1875C22semidirect
58. #1874C22semidirect
59. #1870C22semidirect
60. #1869C22semidirect
62. #1854C22semidirect
63. #1853C22semidirect
65. #1852C22semidirect
66. #1852C22semidirect
68. #1824C22semidirect
69. #1824C22semidirect
73. #1736C22semidirect
74. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#355632
#343602
#276202
#276042
#275942
#262152
#261992
#261422
#261262
#240022
#217702
#217442
#208472