Finite Group Databasebeta

← #6458977

Subgroups — #6458977

Order 512.

1887 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34204C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic, complemented
3#27684C212yesmaximal, complemented
4#27645C212yesmaximal, complemented
5#27637C214yesmaximal, complemented
6#27621C212yesmaximal, complemented
7#27604C212yesmaximal, complemented
8#27591C212yesmaximal, complemented
9#24005C212yesmaximal
10#23771C214yesmaximal, complemented
11#21580C212yesmaximal, complemented
12C4×#481C212yesmaximal, complemented
13#21369C212yesmaximal, complemented
14#20744C212yesmaximal, complemented
15#20716C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C2×#518C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#433C2212yes—
22#2363—22no—
23#2361—22no—
24#2342—22no—
25D4×C22⋊C4—22no—
26#2324C2212yes—
27#2324C2212yes—
28#2316C2211yescharacteristic
29#2314—22no—
30#2314—22no—
31#2314C2212yes—
32#2314C2212yes—
33#2314C2212yescomplemented
34#2314C2212yescomplemented
35#2309C2211yescharacteristic
36C4×#121C2212yes—
37C4×#121C2212yes—
38C2×#388—22no—
39C2×#387—22no—
40C2×#384C2211yescharacteristic, complemented
41C2×#382C2211yescharacteristic, complemented
42#2280C2211yescharacteristic
43C2×#380C2212yes—
44#2274C2211yescharacteristic
45C2×C4×C4⋊C4C2212yescomplemented
46#2101C2214yescomplemented
47#2092C2214yescomplemented
48#2084C2214yescomplemented
49#2077C2214yescomplemented
50#2075C2214yescomplemented
51#2071C2214yescomplemented
52#2034C2214yes—
53#2034C2214yes—
54#1949C2214yescomplemented
55#1948C2214yescomplemented
56#1926C2214yescomplemented
57#1925C2214yescomplemented
58#1920C2214yescomplemented
59#1919C2214yescomplemented
60#1874C2212yes—
61#1874C2212yes—
62#1854C2214yes—
63#1853C2214yes—
64#1853C2212yescomplemented
65#1852C2212yescomplemented
66#1825C2214yes—
67#1824C2212yescomplemented
68#1824C2212yescomplemented
69#1818C2212yes—
70#1804C2214yescomplemented
71#1802C2214yescomplemented
72#1798C2212yescomplemented
73#1797C2212yescomplemented
74#1786C2214yes—
75#1779C2212yes—
76#1776C2214yescomplemented
77C4×#106C2214yescomplemented
78#1762C2214yescomplemented
79C4×#102C2214yescomplemented
80#1750C2212yescomplemented
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C23×D4—21no—
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—28no—
89C2×#121C2312yes—
90C2×#121C2312yes—
91#518—28no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—22no—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C2×#117—28no—
98C2×#117D412yes—
99C2×#117D412yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C22×C4⋊C4—22no—
107C22×C4⋊C4D412yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—22no—
113C22×C22⋊C4—22no—
114C22×C22⋊C4D412yes—
115C22×C42—22no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#481—28no—
119#481—28no—
120#433—28no—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2312yes—
135C2×C4×C8C2312yes—
136#390—22no—
137#390—22no—
138#390—22no—
139#390C2312yes—
140#390C2312yes—
141#389—22no—
142#389C2312yes—
143#389C2312yes—
144#388—22no—
145#388—22no—
146#388—24no—
147#388—24no—
148#388—24no—
149#388—24no—
150#388—22no—
151#388—22no—
152#387—24no—
153#387—24no—
154#387—24no—
155#387—24no—
156#386—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#385—22no—
161#385—22no—
162#384—22no—
163#384C2312yes—
164#384C2312yes—
165#382—22no—
166#382C2312yes—
167#382C2312yes—
168#381—21no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380C2314yes—
173#380C2314yes—
174#380C2312yes—
175#380C2312yes—
176#379—21no—
177#379—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C2×#95—22no—
191C2×#95—22no—
192C2×#95—22no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2312yes—
203#336C2312yes—
204C25—21no—
205C25C2×D411yescharacteristic
206C22×D4—22no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—21no—
211C22×D4—24no—
212C22×D4—42no—
213C22×D4—84no—
214C22×D4C2412yes—
215C23×C4—21no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—42no—
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C22×C8C2414yes—
232C22×C8C2414yes—
233#121—28no—
234#121—28no—
235#121—24no—
236#121—24no—
237#121—24no—
238#121#4118yes—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—48no—
242C4×D4—84no—
243C4×D4—84no—
244#117—24no—
245#117—28no—
246#117—28no—
247#117—24no—
248#117—24no—
249#117—24no—
250#117—42no—
251#117#4118yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—84no—
265C2×C4⋊C4C2×D412yes—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—84no—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—84no—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311#107—44no—
312#106—28no—
313#106—28no—
314#106—44no—
315C4⋊C8—28no—
316#102—28no—
317#102—28no—
318#102—44no—
319#102—44no—
320C22⋊C8—28no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—44no—
324#97—24no—
325C4×C8—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—42no—
344C24—41no—
345C24—42no—
346C24—42no—
347C24—82no—
348C24—82no—
349C24C22×D411yescharacteristic
350C24C22×D411yescharacteristic
351C24C22×D411yescharacteristic
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—24no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—416no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—42no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—816no—
366C2×D4C2×#4114yes—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C2×#4114yes—
417C22×C4C2×#4114yes—
418C22×C4C2511yescharacteristic
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4#12318yes—
443C4⋊C4C2×#4114yes—
444C22⋊C4—28no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—28no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—82no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4—82no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4#12318yes—
462C42—24no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—42no—
467C42—42no—
468C42—84no—
469C42—84no—
470C42—82no—
471C42—82no—
472C42C2×#4114yes—
473C42C2×#4114yes—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—21no—
483C23—24no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—44no—
488C23—48no—
489C23—48no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—84no—
498C23—82no—
499C23—84no—
500C23—84no—
501C23—164no—
502C23—164no—
503C23C2×#12312yes—
504C23#52512yes—
505C23#57714yes—
506C23C23×D411yescharacteristic
507D4—28no—
508D4—48no—
509D4—416no—
510D4—44no—
511D4—88no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—44no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—84no—
553C2×C4—88no—
554C2×C4—88no—
555C2×C4—164no—
556C2×C4—164no—
557C2×C4C2×#12314yes—
558C2×C4C2×#12314yes—
559C2×C4C2×#12312yes—
560C2×C4#52512yes—
561C2×C4C22×#4112yes—
562C2×C4C22×#4112yes—
563C2×C4#58512yes—
564C8—44no—
565C8—44no—
566C22—24no—
567C22—24no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—22no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—41no—
579C22—41no—
580C22—44no—
581C22—41no—
582C22—44no—
583C22—88no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—168no—
589C22#312314yes—
590C22C22×#12311yescharacteristic
591C22C2×#52511yescharacteristic
592C22#344911yescharacteristic
593C22C2×#57712yes—
594C22#358512yes—
595C4—24no—
596C4—24no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—41no—
603C4—82no—
604C4—84no—
605C4—82no—
606C4—84no—
607C4#347514yes—
608C2—24no—
609C2—42no—
610C2—81no—
611C2—84no—
612C2#3480911yescharacteristic
613C2#6052512yes—
614C2C2×#312311yescharacteristic
615C2#6056811yescharacteristic
616C2#6098012yes—
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
2. C2×#2314C2semidirect
3. #27684C2semidirect
4. #27645C2semidirect
5. #27637C2semidirect
6. #27621C2semidirect
7. #27604C2semidirect
8. #27591C2semidirect
10. #23771C2semidirect
11. #21580C2semidirect
12. C4×#481C2semidirect
13. #21369C2semidirect
14. #20744C2semidirect
15. #20716C2semidirect
33. #2314C22semidirect
34. #2314C22semidirect
40. C2×#384C22semidirect
41. C2×#382C22semidirect
45. C2×C4×C4⋊C4C22semidirect
46. #2101C22semidirect
47. #2092C22semidirect
48. #2084C22semidirect
49. #2077C22semidirect
50. #2075C22semidirect
51. #2071C22semidirect
54. #1949C22semidirect
55. #1948C22semidirect
56. #1926C22semidirect
57. #1925C22semidirect
58. #1920C22semidirect
59. #1919C22semidirect
64. #1853C22semidirect
65. #1852C22semidirect
67. #1824C22semidirect
68. #1824C22semidirect
70. #1804C22semidirect
71. #1802C22semidirect
72. #1798C22semidirect
73. #1797C22semidirect
76. #1776C22semidirect
77. C4×#106C22semidirect
78. #1762C22semidirect
79. C4×#102C22semidirect
80. #1750C22semidirect
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
#342042
C2×#23142
#276842
#276452
#276372
#276212
#276042
#275912
#240052
#237712
#215802
C4×#4812
#213692
#207442
#207162