Finite Group Databasebeta

← #6500881

Subgroups — #6500881

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35533C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic, complemented
3#27612C212yesmaximal, complemented
4#27608C212yesmaximal, complemented
5#27582C212yesmaximal, complemented
6#26225C214yesmaximal, complemented
7#26213C214yesmaximal, complemented
8#26146C214yesmaximal, complemented
9#26126C214yesmaximal, complemented
10#24013C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#21745C211yesmaximal, characteristic, complemented
13#20847C212yesmaximal, complemented
14C2×#535C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#532C2211yescharacteristic
17C2×#520C2211yescharacteristic
18C2×#481C2212yes—
19C2×#480C2212yes—
20C2×#428C2212yescomplemented
21C22×#107C2212yescomplemented
22C2×#408C2212yes—
23#2548—21no—
24#2545—21no—
25#2544—21no—
26#2541—21no—
27#2424—24no—
28#2416C2214yes—
29#2414C2214yes—
30#2379—21no—
31#2379—21no—
32#2369—21no—
33#2366—21no—
34#2354C2214yescomplemented
35D4×C4⋊C4C2214yescomplemented
36#2324—24no—
37#2311C2212yes—
38#2311C2212yes—
39#2311C2211yescharacteristic
40#2296C2211yescharacteristic
41C2×#390C2211yescharacteristic
42#2285C2212yes—
43#2100C2212yescomplemented
44#2100C2212yescomplemented
45#2093C2214yes—
46#2084C2212yescomplemented
47#2083C2212yescomplemented
48#2076C2214yescomplemented
49#2075C2214yescomplemented
50#2071C2214yes—
51#2034C2214yes—
52#2033C2214yes—
53#1925C2214yes—
54#1925C2214yes—
55#1920C2214yescomplemented
56#1918C2214yescomplemented
57#1895C2214yes—
58#1893C2214yes—
59#1875C2214yes—
60#1875C2212yes—
61#1875C2212yes—
62#1874C2214yes—
63#1870C2214yescomplemented
64#1869C2214yescomplemented
65#1854C2214yes—
66#1854C2214yescomplemented
67#1852C2214yes—
68#1852C2214yescomplemented
69#1852C2214yescomplemented
70#1825C2214yes—
71#1825C2214yescomplemented
72#1817C2212yes—
73#1786C2214yes—
74#1780C2212yes—
75#1736C2212yescomplemented
76#1735C2212yescomplemented
77C22×#41C2311yescharacteristic
78C22×#41C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C22×M16C2312yes—
81C23×C8C2312yes—
82#535—28no—
83#535—42no—
84#535—42no—
85#532—28no—
86C2×C4⋊Q8D411yescharacteristic
87C2×#127—21no—
88C2×#127—21no—
89C2×#127D411yescharacteristic
90C2×#124—21no—
91C2×#124—21no—
92C2×#124D411yescharacteristic
93C2×#124D411yescharacteristic
94C2×#121—24no—
95C2×#121D414yes—
96C2×#121C2314yes—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×#120—22no—
100C2×#120—22no—
101#520—42no—
102#520—42no—
103C2×C4×Q8—21no—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—24no—
107C2×C4×D4C2314yes—
108C2×#117—22no—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—22no—
119C22×C42C2311yescharacteristic
120#481—44no—
121#480—44no—
122#428—44no—
123C2×#107—44no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132#408—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138#396—24no—
139#394—24no—
140#394—24no—
141#392—24no—
142#391—22no—
143#391—22no—
144#391—21no—
145#390—22no—
146#390—22no—
147#390—21no—
148#390—24no—
149#390—24no—
150#390—21no—
151#390—21no—
152#390C2314yes—
153#389—24no—
154#389C2312yes—
155#389C2312yes—
156#388—24no—
157#388—24no—
158#388—24no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#386—24no—
165#385—22no—
166#385—24no—
167#385—24no—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#380—22no—
172#380C2314yes—
173#380C2314yes—
174#380C2314yes—
175#380C2314yes—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2312yes—
191#336C2312yes—
192C25—21no—
193C2×#41—24no—
194C2×#41—41no—
195C2×#41—41no—
196C2×#41—42no—
197C2×#41—41no—
198C2×#41—41no—
199C2×#41—42no—
200C2×#41#4118yes—
201C22×Q8C2×D411yescharacteristic
202C22×Q8C2×D411yescharacteristic
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—24no—
208C22×D4—41no—
209C22×D4—41no—
210C22×D4—42no—
211C22×D4C2×D414yes—
212C22×D4C2×D412yes—
213C22×D4C2×D411yescharacteristic
214C22×D4C2×D412yes—
215C22×D4C2×D411yescharacteristic
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2×D412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C2×M16—24no—
226C2×M16—42no—
227C2×M16—42no—
228C22×C8—24no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—24no—
234#127—24no—
235#127—44no—
236#127—44no—
237#124—24no—
238#124—24no—
239#124—44no—
240#124—44no—
241#121—216no—
242#121—28no—
243#121—28no—
244#121—24no—
245#121—24no—
246#121—42no—
247#121—42no—
248#121—88no—
249#120—48no—
250#120—48no—
251C4×Q8—44no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—44no—
255C4×D4—48no—
256C4×D4—88no—
257#117—22no—
258#117—22no—
259#117—24no—
260#117—24no—
261#117—42no—
262#117—42no—
263#117—42no—
264#117—44no—
265#117—84no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—84no—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—21no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2412yes—
318#107—28no—
319#107—44no—
320#107—44no—
321#106—28no—
322#102—44no—
323#102—44no—
324#102—44no—
325#102—44no—
326C22⋊C8—28no—
327C22⋊C8—28no—
328C22⋊C8—44no—
329C22⋊C8—42no—
330C22⋊C8—42no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338C24—22no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24—82no—
343C24—82no—
344C24C22×D412yes—
345C24C22×D411yescharacteristic
346C24C22×D412yes—
347#41—48no—
348#41—48no—
349#41—84no—
350#41—84no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8#12314yes—
357C2×D4—28no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—48no—
379C2×D4—42no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4#12318yes—
387C2×D4#12314yes—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2511yescharacteristic
425M16—48no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—48no—
433C4⋊C4—28no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4H(5,𝔽2)12yes—
454C4⋊C4H(5,𝔽2)12yes—
455C22⋊C4—24no—
456C22⋊C4—28no—
457C22⋊C4—28no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—84no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—24no—
475C42—44no—
476C42—42no—
477C42—48no—
478C42—82no—
479C42—82no—
480C23—22no—
481C23—24no—
482C23—21no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—84no—
499C23—88no—
500C23—164no—
501C23—162no—
502C23—162no—
503C23C2×#12314yes—
504C23C2×#12312yes—
505C23C23×D411yescharacteristic
506Q8—44no—
507Q8—82no—
508D4—44no—
509D4—48no—
510D4—416no—
511D4—44no—
512D4—44no—
513D4—82no—
514D4—84no—
515D4—84no—
516D4—164no—
517D4—164no—
518D4—168no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—164no—
556C2×C4—162no—
557C2×C4—164no—
558C2×C4—164no—
559C2×C4C2×#12312yes—
560C2×C4C2×#12314yes—
561C2×C4#57614yes—
562C2×C4C22×#4112yes—
563C2×C4C2×H(5,𝔽2)11yescharacteristic
564C2×C4C2×H(5,𝔽2)11yescharacteristic
565C2×C4#58512yes—
566C8—44no—
567C8—44no—
568C22—22no—
569C22—24no—
570C22—24no—
571C22—28no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—41no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—82no—
584C22—84no—
585C22—84no—
586C22—82no—
587C22—82no—
588C22—168no—
589C22C22×#12311yescharacteristic
590C22#348411yescharacteristic
591C22#348511yescharacteristic
592C22C2×#57612yes—
593C22#358612yes—
594C4—22no—
595C4—24no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—82no—
602C4—84no—
603C4—82no—
604C4—84no—
605C4#350512yes—
606C4#350512yes—
607C2—44no—
608C2—42no—
609C2—82no—
610C2—162no—
611C2#3535111yescharacteristic
612C2#6052112yes—
613C2#6166111yescharacteristic
614C2#6167411yescharacteristic
615C2#6180712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35533C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34368C2semidirect
3. #27612C2semidirect
4. #27608C2semidirect
5. #27582C2semidirect
6. #26225C2semidirect
7. #26213C2semidirect
8. #26146C2semidirect
9. #26126C2semidirect
10. #24013C2semidirect
11. #21770C2semidirect
12. #21745C2semidirect
13. #20847C2semidirect
20. C2×#428C22semidirect
21. C22×#107C22semidirect
34. #2354C22semidirect
35. D4×C4⋊C4C22semidirect
43. #2100C22semidirect
44. #2100C22semidirect
46. #2084C22semidirect
47. #2083C22semidirect
48. #2076C22semidirect
49. #2075C22semidirect
55. #1920C22semidirect
56. #1918C22semidirect
63. #1870C22semidirect
64. #1869C22semidirect
66. #1854C22semidirect
68. #1852C22semidirect
69. #1852C22semidirect
71. #1825C22semidirect
75. #1736C22semidirect
76. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#355332
#343682
#276122
#276082
#275822
#262252
#262132
#261462
#261262
#240132
#217702
#217452
#208472