Finite Group Databasebeta

← #6477115

Subgroups — #6477115

Order 512.

1867 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic, complemented
2#34204C211yesmaximal, characteristic
3#33156C212yesmaximal, complemented
4#32966C212yesmaximal, complemented
5#30178C214yesmaximal, complemented
6#29306C212yesmaximal, complemented
7#29115C212yesmaximal, complemented
8#27644C212yesmaximal, complemented
9#27602C212yesmaximal, complemented
10#24713C214yesmaximal, complemented
11#24305C212yesmaximal, complemented
12#24002C212yesmaximal
13#22081C212yesmaximal, complemented
14#22075C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic, complemented
17C22×#121C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#480C2212yes—
21C2×#480C2212yes—
22C2×#478C2214yescomplemented
23C2×#433C2212yescomplemented
24#2629—22no—
25#2629—22no—
26#2629C2212yes—
27#2629C2212yes—
28#2629C2212yescomplemented
29#2629C2212yescomplemented
30C2×#398—22no—
31C2×#396—22no—
32#2363—22no—
33#2361—22no—
34#2342—22no—
35D4×C22⋊C4—22no—
36#2324C2212yes—
37#2324C2212yes—
38#2316C2211yescharacteristic
39#2309C2211yescharacteristic
40C4×#121C2212yes—
41C4×#121C2212yes—
42C2×#389C2212yescomplemented
43C2×#384C2211yescharacteristic, complemented
44#2280C2211yescharacteristic
45C2×#380C2212yes—
46#2274C2211yescharacteristic
47#2083C2214yescomplemented
48#2077C2214yescomplemented
49#2076C2214yescomplemented
50#2071C2214yescomplemented
51#2059C2214yescomplemented
52#2033C2214yes—
53#2033C2214yes—
54#2025C2214yescomplemented
55#1989C2214yescomplemented
56#1947C2214yescomplemented
57#1943C2214yescomplemented
58#1942C2214yescomplemented
59#1936C2214yescomplemented
60#1934C2214yescomplemented
61#1921C2214yescomplemented
62#1918C2214yescomplemented
63#1874C2212yes—
64#1874C2212yes—
65#1864C2214yescomplemented
66#1862C2214yescomplemented
67#1854C2214yes—
68#1853C2212yescomplemented
69#1852C2214yes—
70#1852C2212yescomplemented
71#1825C2214yes—
72#1823C2212yescomplemented
73#1823C2212yescomplemented
74#1818C2212yescomplemented
75#1802C2214yescomplemented
76#1798C2212yes—
77#1797C2212yescomplemented
78#1786C2214yes—
79#1779C2212yes—
80#1750C2212yes—
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C23×D4—21no—
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#125—22no—
87C2×#125—44no—
88C2×#125C2312yes—
89C2×#125C2312yes—
90C2×#121—21no—
91C2×#121—21no—
92C2×#121—28no—
93C2×#121C2312yes—
94C2×#121C2312yes—
95#518—44no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4C2312yes—
100C2×C4×D4C2312yes—
101C2×#117—44no—
102C2×#117D412yes—
103C2×#117D412yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—22no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—22no—
113C22×C4⋊C4D412yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—22no—
119C22×C22⋊C4—22no—
120C22×C22⋊C4D412yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#480—28no—
124#480—28no—
125#478—48no—
126#433—44no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#398—24no—
143#398—24no—
144#398—24no—
145#398—24no—
146#396—24no—
147#396—24no—
148#396—24no—
149#396—24no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#385—22no—
171#385—22no—
172#384—22no—
173#384C2312yes—
174#384C2312yes—
175#381—21no—
176#380—22no—
177#380—22no—
178#380—22no—
179#380C2312yes—
180#380C2314yes—
181#380C2312yes—
182#380C2314yes—
183#379—21no—
184#379—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C2×#95—22no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2312yes—
205#336C2312yes—
206C25—21no—
207C25C2×D411yescharacteristic
208C22×D4—22no—
209C22×D4—28no—
210C22×D4—22no—
211C22×D4—24no—
212C22×D4—21no—
213C22×D4—24no—
214C22×D4—42no—
215C22×D4—84no—
216C22×D4C2412yes—
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—42no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8C2414yes—
234C22×C8C2414yes—
235#125—28no—
236#125—42no—
237#125—42no—
238#125—48no—
239#121—28no—
240#121—28no—
241#121—24no—
242#121—24no—
243#121—24no—
244#121#4118yes—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—48no—
248C4×D4—84no—
249C4×D4—84no—
250#117—24no—
251#117—24no—
252#117—24no—
253#117—42no—
254#117—42no—
255#117—44no—
256#117—44no—
257#117—42no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—84no—
276C2×C4⋊C4—84no—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—84no—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C42—24no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—28no—
321#107—28no—
322#107—44no—
323#106—44no—
324C4⋊C8—28no—
325#102—28no—
326#102—28no—
327#102—44no—
328#102—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—28no—
332C22⋊C8—44no—
333#97—42no—
334C4×C8—42no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—41no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—82no—
355C24—82no—
356C24C22×D411yescharacteristic
357C24C22×D411yescharacteristic
358C24C22×D411yescharacteristic
359C2×D4—24no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—416no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—42no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—816no—
373C2×D4C2×#4114yes—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—416no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4114yes—
424C22×C42−1+414yes—
425C22×C4C2511yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—84no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4C2×#4114yes—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4—82no—
470C22⋊C4—82no—
471C22⋊C4—84no—
472C22⋊C4—84no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—82no—
481C42—82no—
482C42H(5,𝔽2)14yes—
483C23—22no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—84no—
507C23—82no—
508C23—84no—
509C23—84no—
510C23—164no—
511C23—164no—
512C23#52512yes—
513C23#53612yes—
514C23#57614yes—
515C23C23×D411yescharacteristic
516D4—28no—
517D4—48no—
518D4—416no—
519D4—44no—
520D4—88no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—88no—
564C2×C4—88no—
565C2×C4—164no—
566C2×C4—164no—
567C2×C4#52512yes—
568C2×C4#53414yes—
569C2×C4#53612yes—
570C2×C4C22×#4112yes—
571C2×C4C2×H(5,𝔽2)12yes—
572C2×C4C2×2−1+412yes—
573C8—44no—
574C8—44no—
575C22—22no—
576C22—24no—
577C22—22no—
578C22—24no—
579C22—22no—
580C22—24no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—41no—
589C22—41no—
590C22—44no—
591C22—42no—
592C22—88no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—168no—
598C22#312214yes—
599C22C2×#52511yescharacteristic
600C22C2×#53411yescharacteristic
601C22C2×#53611yescharacteristic
602C22C2×#57612yes—
603C22#358612yes—
604C4—22no—
605C4—22no—
606C4—24no—
607C4—41no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—44no—
612C4—84no—
613C4—82no—
614C4—84no—
615C4—82no—
616C4#352014yes—
617C2—24no—
618C2—42no—
619C2—81no—
620C2—84no—
621C2#3508411yescharacteristic
622C2C2×#312211yescharacteristic
623C2#6056911yescharacteristic
624C2#6076212yes—
625C2#6087112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2629C2semidirect
3. #33156C2semidirect
4. #32966C2semidirect
5. #30178C2semidirect
6. #29306C2semidirect
7. #29115C2semidirect
8. #27644C2semidirect
9. #27602C2semidirect
10. #24713C2semidirect
11. #24305C2semidirect
13. #22081C2semidirect
14. #22075C2semidirect
15. #20716C2semidirect
16. C22×#125C22semidirect
22. C2×#478C22semidirect
23. C2×#433C22semidirect
28. #2629C22semidirect
29. #2629C22semidirect
42. C2×#389C22semidirect
43. C2×#384C22semidirect
47. #2083C22semidirect
48. #2077C22semidirect
49. #2076C22semidirect
50. #2071C22semidirect
51. #2059C22semidirect
54. #2025C22semidirect
55. #1989C22semidirect
56. #1947C22semidirect
57. #1943C22semidirect
58. #1942C22semidirect
59. #1936C22semidirect
60. #1934C22semidirect
61. #1921C22semidirect
62. #1918C22semidirect
65. #1864C22semidirect
66. #1862C22semidirect
68. #1853C22semidirect
70. #1852C22semidirect
72. #1823C22semidirect
73. #1823C22semidirect
74. #1818C22semidirect
75. #1802C22semidirect
77. #1797C22semidirect
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
C2×#26292
#342042
#331562
#329662
#301782
#293062
#291152
#276442
#276022
#247132
#243052
#240022
#220812
#220752
#207162