Finite Group Databasebeta

← #6503203

Subgroups — #6503203

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35084C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27864C214yesmaximal, complemented
4#26211C214yesmaximal, complemented
5#26139C214yesmaximal, complemented
6#24958C214yesmaximal, complemented
7#24888C214yesmaximal, complemented
8#24561C214yesmaximal, complemented
9#21943C211yesmaximal, characteristic, complemented
10#21768C211yesmaximal, characteristic, complemented
11#20972C211yesmaximal, characteristic, complemented
12#20898C211yesmaximal, characteristic
13#20660C211yesmaximal, characteristic, complemented
14C23×M16C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#485C2214yescomplemented
18C2×#427C2212yescomplemented
19C2×#419C2212yescomplemented
20C2×#407C2211yescharacteristic
21C2×#407C2212yes—
22#2855C2212yes—
23#2848C2212yes—
24#2638—22no—
25#2583—21no—
26#2561—21no—
27#2486C2212yes—
28#2483—21no—
29#2415C2214yes—
30#2409C2212yes—
31#2405—21no—
32#2357—22no—
33#2357—22no—
34#2354C2214yes—
35#2354C2214yes—
36#2322—22no—
37#2312—21no—
38#2312—21no—
39#2309—21no—
40#2285C2212yes—
41#2280C2211yescharacteristic
42#2280C2211yescharacteristic
43#2280C2211yescharacteristic
44#2280C2212yes—
45#2274—21no—
46#2100C2214yescomplemented
47#2095C2214yescomplemented
48#2084C2214yescomplemented
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2072C2214yescomplemented
52#2036C2214yescomplemented
53#2035C2214yescomplemented
54#1927C2214yescomplemented
55#1927C2214yescomplemented
56#1918C2214yescomplemented
57#1918C2214yescomplemented
58#1894C2214yes—
59#1894C2214yes—
60#1892C2212yescomplemented
61#1892C2212yescomplemented
62#1873C2214yescomplemented
63#1873C2214yescomplemented
64#1870C2214yes—
65#1869C2212yes—
66#1869C2212yes—
67#1854C2214yescomplemented
68#1854C2214yescomplemented
69#1854C2214yescomplemented
70#1825C2214yescomplemented
71#1824C2212yescomplemented
72#1809C2214yes—
73#1809C2214yes—
74#1809C2214yescomplemented
75#1787C2212yescomplemented
76#1736C2212yescomplemented
77#1735C2212yescomplemented
78#1728C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4D411yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×M16—24no—
84C22×M16—44no—
85C22×M16C2312yes—
86C22×M16C2312yes—
87C22×M16C2312yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C2×#123—22no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2314yes—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4D412yes—
101C2×C4×D4D412yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C2×#117—22no—
105C2×#117C2314yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4D411yescharacteristic
114C22×C22⋊C4D411yescharacteristic
115C22×C42C2311yescharacteristic
116#485—48no—
117#427—44no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122#419—44no—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127#407—24no—
128#407—44no—
129C2×C22⋊C8C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2314yes—
133#398—22no—
134#397—22no—
135#397—24no—
136#395—22no—
137#394—24no—
138#393—22no—
139#391C2314yes—
140#390—24no—
141#390—24no—
142#389—24no—
143#389C2314yes—
144#388—24no—
145#388—24no—
146#388—24no—
147#388—24no—
148#388—24no—
149#387—24no—
150#386—22no—
151#385—24no—
152#385—22no—
153#385—22no—
154#385—22no—
155#384C2314yes—
156#382—22no—
157#381—22no—
158#381—22no—
159#381—21no—
160#381—21no—
161#381—21no—
162#381—22no—
163#380—22no—
164#380C2312yes—
165#380C2312yes—
166#380C2312yes—
167#380C2312yes—
168#380C2312yes—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172#379—22no—
173#379—22no—
174#379—21no—
175#379—21no—
176#379—21no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25C2×D411yescharacteristic
200C25C2×D411yescharacteristic
201C22×Q8—21no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8C2412yes—
207C22×D4—24no—
208C22×D4—44no—
209C22×D4C2×D412yes—
210C22×D4C2×D412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D412yes—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4#4114yes—
227C23×C4#4114yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C2×M16—24no—
233C2×M16—24no—
234C2×M16—24no—
235C2×M16—24no—
236C2×M16—24no—
237C2×M16—42no—
238C2×M16—42no—
239C2×M16—42no—
240C2×M16—42no—
241C2×M16—416no—
242C22×C8—22no—
243C22×C8—24no—
244C22×C8—22no—
245C22×C8—42no—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249#123—48no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C4×D4—28no—
259C4×D4—28no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—24no—
263C4×D4—24no—
264#117—48no—
265#117—42no—
266#117—44no—
267#117—42no—
268#117—42no—
269#117—42no—
270#117—42no—
271#117—42no—
272#117—44no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2414yes—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321#107—28no—
322#107—44no—
323#106—44no—
324#106—44no—
325#103—28no—
326#103—44no—
327#103—44no—
328#103—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331C22⋊C8—28no—
332C22⋊C8—44no—
333#95—22no—
334#95—22no—
335#95—24no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345C24—22no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—42no—
352C24#12314yes—
353C24C22×D412yes—
354C24C22×D411yescharacteristic
355C24C22×D411yescharacteristic
356C24C22×D411yescharacteristic
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—48no—
362C2×Q8—42no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—416no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4—82no—
409C22×C4#12314yes—
410C22×C4#12314yes—
411C22×C4#12314yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2511yescharacteristic
425M16—48no—
426M16—48no—
427M16—84no—
428C2×C8—28no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—44no—
438C2×C8H(5,𝔽2)14yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4—84no—
470C22⋊C4—82no—
471C22⋊C4—82no—
472C22⋊C4—84no—
473C22⋊C4—82no—
474C22⋊C4—84no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—48no—
479C42—82no—
480C42—82no—
481C42—82no—
482C42—84no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—82no—
502C23—82no—
503C23—88no—
504C23—88no—
505C23C2×#12312yes—
506C23C2×#12312yes—
507C23C2×#12312yes—
508C23C23×D411yescharacteristic
509Q8—42no—
510Q8—42no—
511Q8—44no—
512Q8—84no—
513D4—48no—
514D4—48no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4C2×#12312yes—
559C2×C4C2×#12312yes—
560C2×C4C2×#12312yes—
561C2×C4#57514yes—
562C2×C4C22×#4111yescharacteristic
563C2×C4C22×#4111yescharacteristic
564C2×C4C2×H(5,𝔽2)12yes—
565C2×C4#58512yes—
566C8—24no—
567C8—42no—
568C8—44no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—44no—
585C22—42no—
586C22—88no—
587C22—88no—
588C22C22×#12311yescharacteristic
589C22#349612yes—
590C22C2×#57512yes—
591C22#358612yes—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—24no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—82no—
600C4—84no—
601C4—84no—
602C4—84no—
603C4#349011yescharacteristic
604C4#349111yescharacteristic
605C4#350512yes—
606C2—42no—
607C2—42no—
608C2—42no—
609C2—82no—
610C2#3536011yescharacteristic
611C2#6051111yescharacteristic
612C2#6051911yescharacteristic
613C2#6230614yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27864C2semidirect
4. #26211C2semidirect
5. #26139C2semidirect
6. #24958C2semidirect
7. #24888C2semidirect
8. #24561C2semidirect
9. #21943C2semidirect
10. #21768C2semidirect
11. #20972C2semidirect
13. #20660C2semidirect
17. C2×#485C22semidirect
18. C2×#427C22semidirect
19. C2×#419C22semidirect
46. #2100C22semidirect
47. #2095C22semidirect
48. #2084C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2072C22semidirect
52. #2036C22semidirect
53. #2035C22semidirect
54. #1927C22semidirect
55. #1927C22semidirect
56. #1918C22semidirect
57. #1918C22semidirect
60. #1892C22semidirect
61. #1892C22semidirect
62. #1873C22semidirect
63. #1873C22semidirect
67. #1854C22semidirect
68. #1854C22semidirect
69. #1854C22semidirect
70. #1825C22semidirect
71. #1824C22semidirect
74. #1809C22semidirect
75. #1787C22semidirect
76. #1736C22semidirect
77. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#350842
#341862
#278642
#262112
#261392
#249582
#248882
#245612
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