Finite Group Databasebeta

← #6508786

Subgroups — #6508786

Order 512.

1988 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic, complemented
5#28130C214yesmaximal, complemented
6#27657C214yesmaximal, complemented
7#25210C214yesmaximal, complemented
8#25204C214yesmaximal, complemented
9#24589C214yesmaximal, complemented
10#24541C214yesmaximal, complemented
11#21933C211yesmaximal, characteristic, complemented
12#21764C211yesmaximal, characteristic
13C2×#1825C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3187—28no—
18#3095—28no—
19C2×#484C2212yescomplemented
20C2×#481C2212yes—
21C2×#485C2214yes—
22C2×#483C2212yes—
23C2×#483C2212yescomplemented
24C2×#465C2214yes—
25C2×#448C2212yes—
26C2×#450C2212yes—
27C2×#428C2212yescomplemented
28C2×#419C2212yescomplemented
29C2×#407C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2854—22no—
32#2624—21no—
33#2624—21no—
34#2504C2212yes—
35#2501C2212yes—
36#2500C2214yes—
37#2482—22no—
38#2405—22no—
39#2378—22no—
40#2360C2214yescomplemented
41#2344C2212yescomplemented
42#2342—22no—
43Q8×C22⋊C4C2212yescomplemented
44#2329—22no—
45#2323C2214yes—
46#2317C2212yes—
47#2316—21no—
48#2313—21no—
49#2309C2212yes—
50#2280C2211yescharacteristic
51C2×#380C2211yescharacteristic
52#2085C2214yescomplemented
53#2084C2214yescomplemented
54#2074C2214yescomplemented
55#2072C2214yescomplemented
56#2067C2214yescomplemented
57#2065C2214yescomplemented
58#2055C2214yescomplemented
59#2055C2214yescomplemented
60#1949C2214yescomplemented
61#1948C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1895C2214yes—
65#1880C2214yescomplemented
66#1880C2214yescomplemented
67#1874C2214yes—
68#1851C2214yescomplemented
69#1850C2214yescomplemented
70#1825C2214yes—
71#1825C2214yescomplemented
72#1824C2212yes—
73#1805C2214yescomplemented
74#1803C2214yescomplemented
75#1798C2214yes—
76#1786C2212yes—
77C2×#340C2212yescomplemented
78C2×#340C2212yescomplemented
79C23×Q8C2311yescharacteristic
80C23×D4—21no—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C22×SD16C2314yes—
84C22×M16C2312yes—
85C23×C8C2312yes—
86C2×#123—24no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8D418yes—
91C2×C22⋊Q8C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×#121—21no—
94C2×#121—21no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#121C2312yes—
98C2×#121C2312yes—
99C2×C4×Q8C2314yes—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117—22no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42—21no—
119#485—216no—
120#484—28no—
121#483—28no—
122#483—28no—
123#481—28no—
124#465—216no—
125#450—28no—
126#448—28no—
127#428—28no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133#419—28no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140#407—24no—
141C2×C22⋊C8—24no—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146#399—22no—
147#399—22no—
148#397—22no—
149#397—22no—
150#394—24no—
151#394—24no—
152#391—24no—
153#391C2314yes—
154#390—22no—
155#390—22no—
156#390C2314yes—
157#390C2314yes—
158#389—24no—
159#389—22no—
160#389—22no—
161#389—24no—
162#389C2314yes—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#387—24no—
168#387—24no—
169#387—24no—
170#386—24no—
171#385—24no—
172#385—24no—
173#384C2314yes—
174#382C2314yes—
175#381—22no—
176#381D412yes—
177#380—21no—
178#380—21no—
179#380—22no—
180#380D411yescharacteristic
181#380D411yescharacteristic
182#380D412yes—
183#380C2312yes—
184#380C2312yes—
185#379—24no—
186#379D412yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—24no—
190C4×C22⋊C4C2312yes—
191C4×C22⋊C4C2312yes—
192C2×#95—21no—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2314yes—
200#336C2314yes—
201C25—21no—
202C25C2×D411yescharacteristic
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8C2×D414yes—
208C22×Q8C2412yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—42no—
215C22×D4—42no—
216C22×D4—44no—
217C22×D4—84no—
218C22×D4C2412yes—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D414yes—
231C23×C4#4114yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C2×SD16—28no—
236C2×SD16—44no—
237C2×SD16—44no—
238C2×M16—24no—
239C2×M16—42no—
240C2×M16—42no—
241C22×C8—24no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246#123—88no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—24no—
250C22⋊Q8—24no—
251C22⋊Q8C2×D418yes—
252C22⋊Q8C2×D418yes—
253#121—24no—
254#121—24no—
255#121—24no—
256#121—24no—
257#121—44no—
258#121—44no—
259C4×Q8—44no—
260C4×Q8—44no—
261#117—24no—
262#117—24no—
263#117—24no—
264#117—24no—
265#117—42no—
266#117—42no—
267#117—44no—
268#117—44no—
269#117—84no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—82no—
315C2×C22⋊C4C2×D414yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D414yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—44no—
324C2×C42C2414yes—
325#107—28no—
326#106—28no—
327#106—28no—
328C4⋊C8—44no—
329C4⋊C8—44no—
330#103—28no—
331#103—28no—
332#103—28no—
333#102—28no—
334#102—44no—
335#102—44no—
336C22⋊C8—24no—
337C22⋊C8—24no—
338C22⋊C8—24no—
339C22⋊C8—24no—
340#95—22no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D412yes—
349#95C2×D414yes—
350#95C2×D412yes—
351C24—24no—
352C24—22no—
353C24—41no—
354C24—42no—
355C24—41no—
356C24—44no—
357C24—42no—
358C24—82no—
359C24—81no—
360C24—81no—
361C24#12114yes—
362C24C22×D411yescharacteristic
363C24C22×D411yescharacteristic
364C24C22×D411yescharacteristic
365C2×Q8—24no—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—48no—
370C2×Q8—82no—
371C2×Q8—82no—
372C2×Q8—82no—
373C2×Q8C22×D414yes—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—48no—
382C2×D4—42no—
383C2×D4—84no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—84no—
387C2×D4—816no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4#12114yes—
422C22×C4#12114yes—
423C22×C4#12114yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2511yescharacteristic
436SD16—88no—
437M16—48no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—84no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4C22×D414yes—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—28no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—82no—
484C42—24no—
485C42—24no—
486C42—44no—
487C42—44no—
488C42—84no—
489C42—84no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23—84no—
513C23—84no—
514C23—164no—
515C23—164no—
516C23C2×#12112yes—
517C23C2×#12112yes—
518C23C2×#12112yes—
519C23#57714yes—
520C23C23×D411yescharacteristic
521Q8—28no—
522Q8—42no—
523Q8—42no—
524Q8—84no—
525D4—44no—
526D4—42no—
527D4—42no—
528D4—84no—
529D4—88no—
530D4—88no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—48no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—164no—
574C2×C4C2×#12112yes—
575C2×C4C2×#12112yes—
576C2×C4C2×#12112yes—
577C2×C4C23×D411yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4#58512yes—
580C2×C4#58512yes—
581C8—44no—
582C8—44no—
583C22—22no—
584C22—24no—
585C22—24no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—88no—
598C22—82no—
599C22—82no—
600C22—84no—
601C22—84no—
602C22—82no—
603C22—168no—
604C22#304314yes—
605C22C22×#12111yescharacteristic
606C22#348512yes—
607C22C2×#57712yes—
608C22#358511yescharacteristic
609C22#358611yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4#348512yes—
622C4#349111yescharacteristic
623C4#349111yescharacteristic
624C2—24no—
625C2—42no—
626C2—42no—
627C2—162no—
628C2#3544711yescharacteristic
629C2C2×#304311yescharacteristic
630C2#6042511yescharacteristic
631C2#6175312yes—
632C2#6175812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
4. #34380C2semidirect
5. #28130C2semidirect
6. #27657C2semidirect
7. #25210C2semidirect
8. #25204C2semidirect
9. #24589C2semidirect
10. #24541C2semidirect
11. #21933C2semidirect
13. C2×#1825C2semidirect
19. C2×#484C22semidirect
23. C2×#483C22semidirect
27. C2×#428C22semidirect
28. C2×#419C22semidirect
40. #2360C22semidirect
41. #2344C22semidirect
43. Q8×C22⋊C4C22semidirect
52. #2085C22semidirect
53. #2084C22semidirect
54. #2074C22semidirect
55. #2072C22semidirect
56. #2067C22semidirect
57. #2065C22semidirect
58. #2055C22semidirect
59. #2055C22semidirect
60. #1949C22semidirect
61. #1948C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
65. #1880C22semidirect
66. #1880C22semidirect
68. #1851C22semidirect
69. #1850C22semidirect
71. #1825C22semidirect
73. #1805C22semidirect
74. #1803C22semidirect
77. C2×#340C22semidirect
78. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C2×#30952
#352622
#343802
#281302
#276572
#252102
#252042
#245892
#245412
#219332
#217642
C2×#18252