Finite Group Databasebeta

← #6660961

Subgroups — #6660961

Order 512.

1599 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35280C211yesmaximal, characteristic, complemented
2#35254C211yesmaximal, characteristic
3#32754C214yesmaximal, complemented
4#30365C212yesmaximal, complemented
5#30356C212yesmaximal, complemented
6#29971C212yesmaximal, complemented
7#29950C212yesmaximal, complemented
8#29906C212yesmaximal, complemented
9#29644C212yesmaximal, complemented
10#29306C212yesmaximal, complemented
11#29115C212yesmaximal, complemented
12#27608C212yesmaximal, complemented
13#24615C214yesmaximal, complemented
14#24015C212yesmaximal
15#20720C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#533C2211yescharacteristic, complemented
18C2×C4×#41C2211yescharacteristic
19C2×#482C2214yes—
20C2×#433C2212yescomplemented
21#2857C2212yes—
22#2857C2212yescomplemented
23#2854C2212yes—
24#2854C2212yes—
25#2629C2212yes—
26#2629C2212yescomplemented
27#2624C2212yes—
28#2624C2212yes—
29#2582—22no—
30#2538—22no—
31#2448—22no—
32#2445—22no—
33#2418—22no—
34#2418—22no—
35#2417—22no—
36#2417—22no—
37#2316C2211yescharacteristic
38#2316C2212yescomplemented
39#2311C2211yescharacteristic
40#2311C2212yes—
41#2309C2211yescharacteristic
42#2296C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#384C2211yescharacteristic, complemented
45#2099C2212yescomplemented
46#2099C2212yescomplemented
47#2093C2212yes—
48#2093C2212yes—
49#2088C2214yescomplemented
50#2087C2214yescomplemented
51#2083C2214yescomplemented
52#2076C2214yescomplemented
53#2075C2214yescomplemented
54#2071C2214yes—
55#2067C2214yes—
56#2067C2214yes—
57#2059C2214yescomplemented
58#2058C2212yescomplemented
59#2057C2212yescomplemented
60#2034C2214yes—
61#2033C2214yes—
62#2025C2214yes—
63#2023C2212yes—
64#2023C2212yes—
65#1943C2214yescomplemented
66#1942C2214yescomplemented
67#1936C2214yes—
68#1934C2214yes—
69#1874C2214yes—
70#1854C2214yes—
71#1852C2214yes—
72#1852C2214yescomplemented
73#1825C2214yes—
74#1825C2214yescomplemented
75#1817C2212yescomplemented
76#1798C2212yescomplemented
77#1797C2212yes—
78#1786C2214yes—
79#1780C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×D4C2311yescharacteristic
85#535—44no—
86#533—44no—
87C2×C4⋊Q8C2312yes—
88C2×#127—21no—
89C2×#125C2312yes—
90C2×#124—21no—
91C2×#123—22no—
92C2×#123—22no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×#121C2312yes—
97C2×#121C2312yes—
98C2×#120—21no—
99C2×#120—21no—
100C4×#41—44no—
101C2×C4×Q8—22no—
102C2×C4×Q8D412yes—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×C4×D4D412yes—
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C22×C4⋊C4—22no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—22no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#482—48no—
120#433—44no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2312yes—
135C2×C4×C8C2312yes—
136#399—22no—
137#399—22no—
138#398—24no—
139#398—24no—
140#397—22no—
141#397—22no—
142#396—24no—
143#396—24no—
144#394—22no—
145#394—22no—
146#394—22no—
147#394—22no—
148#391—22no—
149#391—21no—
150#391—22no—
151#391—22no—
152#390—21no—
153#390—22no—
154#390—22no—
155#390—22no—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—24no—
162#389—22no—
163#389—22no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#386—22no—
175#386—22no—
176#386—21no—
177#386—21no—
178#386—22no—
179#386—22no—
180#385—22no—
181#384—22no—
182#384C2312yes—
183#384C2312yes—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—24no—
188#380C2312yes—
189#380C2314yes—
190#380C2314yes—
191#380C2312yes—
192#379—22no—
193C4×C4⋊C4—22no—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C25—21no—
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—42no—
209C22×Q8—22no—
210C22×Q8C2×D411yescharacteristic
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4C2×D411yescharacteristic
218C22×D4C2×D411yescharacteristic
219C22×D4C2×D411yescharacteristic
220C22×D4C2412yes—
221C23×C4—21no—
222C23×C4—22no—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—44no—
232#127—44no—
233#125—42no—
234#125—42no—
235#124—44no—
236#123—84no—
237#123—84no—
238C22⋊Q8—48no—
239C22⋊Q8—84no—
240C22⋊Q8—84no—
241#121—28no—
242#121—44no—
243#120—44no—
244#120—44no—
245C4×Q8—44no—
246C4×Q8—48no—
247C4×D4—44no—
248C4×D4—44no—
249C4×D4—44no—
250#117—24no—
251#117—28no—
252#117—24no—
253#117—42no—
254#117—42no—
255#117—42no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—84no—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—84no—
299C2×C22⋊C4C2×D412yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C42—24no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315#107—44no—
316#107—44no—
317#107—44no—
318#106—44no—
319C4⋊C8—44no—
320#102—44no—
321#102—44no—
322#102—44no—
323#102—44no—
324C22⋊C8—28no—
325C22⋊C8—28no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328#97—42no—
329C4×C8—42no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—24no—
340C24—22no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—82no—
345C24—82no—
346C24C22×D411yescharacteristic
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349#41—416no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—42no—
373C2×D4—88no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C42−1+414yes—
412C22×C4C2511yescharacteristic
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—82no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—42no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C42—24no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—42no—
468C42—42no—
469C42—42no—
470C42—44no—
471C42—42no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—84no—
493C23—88no—
494C23—162no—
495C23—164no—
496C23—162no—
497C23#53412yes—
498C23#53612yes—
499C23C23×D411yescharacteristic
500Q8—44no—
501Q8—88no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—48no—
506D4—84no—
507D4—164no—
508D4—164no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—48no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—164no—
549C2×C4—162no—
550C2×C4#53412yes—
551C2×C4#53612yes—
552C2×C4#57614yes—
553C2×C4C2×H(5,𝔽2)12yes—
554C2×C4C2×2−1+412yes—
555C2×C4#58512yes—
556C8—44no—
557C8—44no—
558C22—22no—
559C22—22no—
560C22—24no—
561C22—24no—
562C22—41no—
563C22—41no—
564C22—42no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—41no—
569C22—41no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—168no—
581C22C2×#53411yescharacteristic
582C22C2×#53611yescharacteristic
583C22#344911yescharacteristic
584C22C2×#57612yes—
585C22#358612yes—
586C4—22no—
587C4—22no—
588C4—24no—
589C4—41no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4#353514yes—
599C2—42no—
600C2—42no—
601C2—81no—
602C2—162no—
603C2#3744111yescharacteristic
604C2#6076212yes—
605C2#6088812yes—
606C2#6101111yescharacteristic
607C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35280C2semidirect
3. #32754C2semidirect
4. #30365C2semidirect
5. #30356C2semidirect
6. #29971C2semidirect
7. #29950C2semidirect
8. #29906C2semidirect
9. #29644C2semidirect
10. #29306C2semidirect
11. #29115C2semidirect
12. #27608C2semidirect
13. #24615C2semidirect
15. #20720C2semidirect
17. C2×#533C22semidirect
20. C2×#433C22semidirect
22. #2857C22semidirect
26. #2629C22semidirect
38. #2316C22semidirect
44. C2×#384C22semidirect
45. #2099C22semidirect
46. #2099C22semidirect
49. #2088C22semidirect
50. #2087C22semidirect
51. #2083C22semidirect
52. #2076C22semidirect
53. #2075C22semidirect
57. #2059C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
65. #1943C22semidirect
66. #1942C22semidirect
72. #1852C22semidirect
74. #1825C22semidirect
75. #1817C22semidirect
76. #1798C22semidirect
81. #1749C22semidirect
82. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#352802
#352542
#327542
#303652
#303562
#299712
#299502
#299062
#296442
#293062
#291152
#276082
#246152
#240152
#207202