Finite Group Databasebeta

← #6519444

Subgroups — #6519444

Order 512.

1957 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35634C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#26249C214yesmaximal, complemented
4#26242C214yesmaximal, complemented
5#26240C214yesmaximal, complemented
6#26234C214yesmaximal, complemented
7#24016C212yesmaximal, complemented
8#24012C212yesmaximal, complemented
9#24007C212yesmaximal, complemented
10#24003C212yesmaximal, complemented
11#21814C212yesmaximal, complemented
12#21792C211yesmaximal, characteristic, complemented
13#21369C212yesmaximal, complemented
14C2×#537C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#525C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C2×#484C2212yes—
21C2×#481C2212yes—
22C2×#481C2212yescomplemented
23C2×#485C2214yescomplemented
24C2×#483C2214yes—
25C2×#483C2212yescomplemented
26C2×#433C2212yes—
27#2542—22no—
28#2542C2214yes—
29#2540—22no—
30#2540C2214yes—
31#2539—22no—
32#2501—22no—
33#2376—22no—
34#2358—22no—
35#2352C2214yes—
36D4×C4⋊C4C2214yes—
37#2329—22no—
38#2325—22no—
39#2314C2212yes—
40#2311C2212yes—
41C2×#380C2212yes—
42#2274C2211yescharacteristic
43#2036C2214yescomplemented
44#2035C2214yes—
45#2035C2214yes—
46#2035C2214yes—
47#2035C2214yescomplemented
48#2035C2214yescomplemented
49#2034C2214yescomplemented
50#2033C2214yes—
51#1983C2214yescomplemented
52#1981C2214yescomplemented
53#1960C2212yescomplemented
54#1959C2212yescomplemented
55#1958C2212yescomplemented
56#1956C2212yescomplemented
57#1955C2214yes—
58#1954C2214yes—
59#1949C2214yescomplemented
60#1948C2214yescomplemented
61#1875C2212yescomplemented
62#1874C2212yes—
63#1873C2212yescomplemented
64#1872C2212yes—
65#1872C2214yes—
66#1872C2214yescomplemented
67#1854C2214yes—
68#1853C2214yes—
69#1788C2214yes—
70#1787C2214yes—
71#1780C2212yes—
72#1779C2212yes—
73#1776C2214yescomplemented
74C4×#106C2214yescomplemented
75C22×#41C2311yescharacteristic
76C22×#41C2311yescharacteristic
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79#537—44no—
80#536—28no—
81#536—44no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8C2312yes—
84C2×#124—21no—
85C2×#124C2312yes—
86#525—28no—
87#525—44no—
88C2×#123—22no—
89C2×#123—21no—
90C2×#123—22no—
91C2×#123D412yes—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8D412yes—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102C2×#121—21no—
103C2×#121D412yes—
104C2×#121C2312yes—
105C2×#121C2312yes—
106C2×#120—21no—
107C2×#120D412yes—
108C2×C4×D4—22no—
109C2×C4×D4—24no—
110C2×C4×D4C2314yes—
111C2×#117C2311yescharacteristic
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—22no—
121#485—48no—
122#484—44no—
123#483—48no—
124#483—44no—
125#481—44no—
126#481—44no—
127#433—28no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#399—24no—
151#399—24no—
152#399—24no—
153#392—24no—
154#391—22no—
155#390—22no—
156#390—24no—
157#390—24no—
158#390—24no—
159#389—24no—
160#388—24no—
161#388—24no—
162#387—24no—
163#387—24no—
164#386—22no—
165#386—22no—
166#386—24no—
167#385—24no—
168#385—24no—
169#384—24no—
170#384C2312yes—
171#382C2312yes—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176#380C2314yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—24no—
181C4×C22⋊C4—22no—
182C2×#95—22no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187C25C2×D411yescharacteristic
188C2×#41—24no—
189C2×#41—24no—
190C2×#41—42no—
191C2×#41—42no—
192C2×#41—42no—
193C2×#41—42no—
194C22×Q8—21no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8C2×D411yescharacteristic
199C22×Q8C2412yes—
200C22×Q8C2411yescharacteristic
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4C2×D414yes—
205C22×D4C2×D412yes—
206C22×D4C2×D411yescharacteristic
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—24no—
210C23×C4—21no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—21no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—44no—
227#124—24no—
228#124—24no—
229#124—44no—
230#123—28no—
231#123—44no—
232#123—48no—
233#123—48no—
234C22⋊Q8—28no—
235C22⋊Q8—28no—
236C22⋊Q8—24no—
237C22⋊Q8—28no—
238C22⋊Q8—24no—
239C22⋊Q8—44no—
240C22⋊Q8—42no—
241C22⋊Q8—44no—
242C22⋊Q8—42no—
243C22⋊Q8—48no—
244C22⋊Q8—44no—
245C22⋊Q8—48no—
246#121—28no—
247#121—24no—
248#121—24no—
249#121—42no—
250#121—42no—
251#121—44no—
252#121—48no—
253#120—28no—
254#120—44no—
255C4×D4—28no—
256C4×D4—28no—
257C4×D4—48no—
258C4×D4—88no—
259#117—24no—
260#117—44no—
261#117#4118yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—84no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42C2412yes—
307#107—28no—
308#106—28no—
309#106—28no—
310#106—28no—
311C4⋊C8—28no—
312#103—44no—
313#103—44no—
314#103—44no—
315#103—44no—
316#103—44no—
317#103—44no—
318#102—44no—
319#102—44no—
320C22⋊C8—28no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323C22⋊C8—44no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—44no—
327#97—24no—
328C4×C8—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335C24—22no—
336C24—24no—
337C24—21no—
338C24—41no—
339C24—42no—
340C24—41no—
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343C24C22×D412yes—
344#41—88no—
345#41—88no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—88no—
360C2×D4—28no—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—44no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—82no—
379C2×D4—84no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—88no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—84no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D414yes—
423C22×C4C2×#4114yes—
424C22×C4C2511yescharacteristic
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C4⋊C4—24no—
432C4⋊C4—28no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—84no—
448C4⋊C4#12318yes—
449C4⋊C4H(5,𝔽2)14yes—
450C4⋊C4H(5,𝔽2)14yes—
451C22⋊C4—24no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—84no—
463C22⋊C4—82no—
464C22⋊C4—84no—
465C22⋊C4#12318yes—
466C42—24no—
467C42—84no—
468C42—82no—
469C42—84no—
470C42C2×#4114yes—
471C42C2×#4114yes—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—21no—
477C23—22no—
478C23—22no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—48no—
488C23—44no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—82no—
495C23—84no—
496C23—82no—
497C23C2×#12312yes—
498C23#57714yes—
499C23C23×D411yescharacteristic
500Q8—44no—
501Q8—44no—
502Q8—44no—
503Q8—82no—
504Q8—84no—
505D4—416no—
506D4—44no—
507D4—82no—
508D4—84no—
509D4—88no—
510D4—84no—
511D4—164no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—82no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—164no—
550C2×C4—164no—
551C2×C4—164no—
552C2×C4C2×#12318yes—
553C2×C4C2×#12312yes—
554C2×C4C22×#4112yes—
555C2×C4C2×H(5,𝔽2)12yes—
556C2×C4C2×H(5,𝔽2)12yes—
557C8—44no—
558C8—44no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—22no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—41no—
568C22—41no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—88no—
577C22—82no—
578C22—82no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22C22×#12311yescharacteristic
583C22#348411yescharacteristic
584C22#348411yescharacteristic
585C22C2×#57712yes—
586C22#358612yes—
587C4—28no—
588C4—22no—
589C4—44no—
590C4—44no—
591C4—42no—
592C4—42no—
593C4—84no—
594C4—84no—
595C4—81no—
596C4—82no—
597C4#350814yes—
598C2—44no—
599C2—42no—
600C2—82no—
601C2—161no—
602C2#3555411yescharacteristic
603C2#6052212yes—
604C2#6162012yes—
605C2#6163311yescharacteristic
606C2#6164911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35634C2
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. #26249C2semidirect
4. #26242C2semidirect
5. #26240C2semidirect
6. #26234C2semidirect
7. #24016C2semidirect
8. #24012C2semidirect
9. #24007C2semidirect
10. #24003C2semidirect
11. #21814C2semidirect
12. #21792C2semidirect
13. #21369C2semidirect
22. C2×#481C22semidirect
23. C2×#485C22semidirect
25. C2×#483C22semidirect
43. #2036C22semidirect
47. #2035C22semidirect
48. #2035C22semidirect
49. #2034C22semidirect
51. #1983C22semidirect
52. #1981C22semidirect
53. #1960C22semidirect
54. #1959C22semidirect
55. #1958C22semidirect
56. #1956C22semidirect
59. #1949C22semidirect
60. #1948C22semidirect
61. #1875C22semidirect
63. #1873C22semidirect
66. #1872C22semidirect
73. #1776C22semidirect
74. C4×#106C22semidirect

Maximal subgroups

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

subgroupindex
#356342
#343602
#262492
#262422
#262402
#262342
#240162
#240122
#240072
#240032
#218142
#217922
#213692