Finite Group Databasebeta

← #6508845

Subgroups — #6508845

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic
3#35084C211yesmaximal, characteristic
4#34809C211yesmaximal, characteristic
5#28112C214yesmaximal, complemented
6#27670C214yesmaximal, complemented
7#25262C214yesmaximal, complemented
8#25256C214yesmaximal, complemented
9#24604C214yesmaximal, complemented
10#24561C214yesmaximal, complemented
11#21943C211yesmaximal, characteristic, complemented
12#21768C211yesmaximal, characteristic, complemented
13#20701C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17#3095—28no—
18C2×#484C2212yescomplemented
19C2×#485C2214yes—
20C2×#483C2212yescomplemented
21#2975—28no—
22C2×#436C2214yes—
23C2×#436C2214yes—
24C2×#431C2212yes—
25C2×#428C2212yescomplemented
26C2×#419C2212yescomplemented
27C2×#407C2211yescharacteristic
28C2×#407C2212yes—
29C2×#407C2211yescharacteristic
30#2855C2212yes—
31#2848C2212yes—
32#2638—22no—
33#2583—21no—
34#2561—21no—
35#2487C2212yes—
36#2486C2212yes—
37#2486C2212yes—
38#2485C2214yes—
39#2484—21no—
40#2483—21no—
41#2483—21no—
42#2482—21no—
43#2415C2214yes—
44#2409C2212yes—
45#2405—21no—
46#2405—21no—
47#2395—22no—
48#2322—22no—
49C4×#123—22no—
50#2296C2212yes—
51#2280C2211yescharacteristic
52#2280C2211yescharacteristic
53#2280C2212yes—
54#2088C2214yescomplemented
55#2086C2214yescomplemented
56#2077C2214yescomplemented
57#2076C2214yescomplemented
58#2068C2214yescomplemented
59#2066C2214yescomplemented
60#2036C2214yescomplemented
61#2035C2214yescomplemented
62#1942C2214yescomplemented
63#1941C2214yescomplemented
64#1933C2214yescomplemented
65#1933C2214yescomplemented
66#1894C2214yescomplemented
67#1894C2214yescomplemented
68#1894C2214yescomplemented
69#1873C2214yes—
70#1854C2214yes—
71#1854C2214yescomplemented
72#1825C2214yes—
73#1825C2214yescomplemented
74#1824C2212yescomplemented
75#1809C2214yescomplemented
76#1809C2214yescomplemented
77#1809C2214yescomplemented
78#1787C2212yescomplemented
79#1736C2212yescomplemented
80#1736C2212yescomplemented
81C23×Q8C2311yescharacteristic
82C23×D4D411yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×M16C2314yes—
86C22×M16C2312yes—
87C22×M16C2312yes—
88C2×#123—22no—
89C2×#123—22no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8C2312yes—
95C2×C22⋊Q8C2312yes—
96C2×C22⋊Q8C2314yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122#485—216no—
123#484—28no—
124#483—28no—
125#436—216no—
126#436—216no—
127#431—28no—
128#428—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2314yes—
134#419—28no—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139#407—24no—
140#407—28no—
141#407D418yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C4×C8C2314yes—
148#400—22no—
149#399—22no—
150#398—22no—
151#398—22no—
152#397—22no—
153#397—24no—
154#397—22no—
155#397—24no—
156#395—22no—
157#394—24no—
158#394—24no—
159#393—22no—
160#391C2314yes—
161#389—24no—
162#389C2314yes—
163#389C2314yes—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#387—24no—
168#387—24no—
169#386—22no—
170#386—22no—
171#385—24no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#381—22no—
179#381—21no—
180#381—21no—
181#381—22no—
182#381—22no—
183#381—22no—
184#380C2314yes—
185#380C2312yes—
186#380C2312yes—
187#380C2314yes—
188#380C2314yes—
189#380C2312yes—
190#380C2312yes—
191#379—22no—
192#379—22no—
193#379—21no—
194#379—21no—
195C4×C4⋊C4—24no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—24no—
200C4×C22⋊C4—21no—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209C25C2×D411yescharacteristic
210C25C2×D411yescharacteristic
211C22×Q8—21no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8C2412yes—
217C22×D4—24no—
218C22×D4—44no—
219C22×D4C2×D412yes—
220C22×D4C2×D412yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D414yes—
239C23×C4#4114yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×M16—24no—
245C2×M16—28no—
246C2×M16—42no—
247C2×M16—42no—
248C2×M16—44no—
249C2×M16—44no—
250C2×M16—42no—
251C2×M16—42no—
252C2×M16C2×D418yes—
253C22×C8C2414yes—
254C22×C8C2414yes—
255#123—48no—
256#123—48no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271#117—24no—
272#117—24no—
273#117—24no—
274#117—24no—
275#117—42no—
276#117—42no—
277#117—44no—
278#117—44no—
279#117—42no—
280#117—42no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2414yes—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336#107—28no—
337#107—44no—
338#106—28no—
339#106—44no—
340C4⋊C8—28no—
341#103—28no—
342#103—28no—
343#103—44no—
344#103—44no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8—28no—
348C22⋊C8C2×D418yes—
349C22⋊C8C2×D418yes—
350C4×C8—28no—
351#95—24no—
352#95—22no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24#12114yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C24C22×D411yescharacteristic
374C24C22×D411yescharacteristic
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—48no—
380C2×Q8—42no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×Q8—82no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—416no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—48no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4#12114yes—
433C22×C4#12114yes—
434C22×C4#12114yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2511yescharacteristic
448M16—48no—
449M16—88no—
450M16—84no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8C22×D418yes—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—84no—
494C22⋊C4—82no—
495C22⋊C4—84no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—82no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—82no—
522C23—82no—
523C23—88no—
524C23—88no—
525C23C2×#12112yes—
526C23C2×#12112yes—
527C23C2×#12112yes—
528C23C23×D411yescharacteristic
529Q8—44no—
530Q8—42no—
531Q8—42no—
532Q8—84no—
533D4—48no—
534D4—48no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—84no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4C2×#12112yes—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4#57514yes—
585C2×C4C23×D411yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4#58512yes—
588C2×C4#58512yes—
589C8—28no—
590C8—44no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—42no—
604C22—44no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—88no—
609C22—88no—
610C22C22×#12111yescharacteristic
611C22#348512yes—
612C22C2×#57512yes—
613C22#358612yes—
614C4—24no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4#348512yes—
627C4#349111yescharacteristic
628C4#349111yescharacteristic
629C2—42no—
630C2—42no—
631C2—42no—
632C2—82no—
633C2#3544711yescharacteristic
634C2#6044411yescharacteristic
635C2#6045511yescharacteristic
636C2#6172414yes—
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
1. C2×#3095C2semidirect
5. #28112C2semidirect
6. #27670C2semidirect
7. #25262C2semidirect
8. #25256C2semidirect
9. #24604C2semidirect
10. #24561C2semidirect
11. #21943C2semidirect
12. #21768C2semidirect
13. #20701C2semidirect
18. C2×#484C22semidirect
20. C2×#483C22semidirect
25. C2×#428C22semidirect
26. C2×#419C22semidirect
54. #2088C22semidirect
55. #2086C22semidirect
56. #2077C22semidirect
57. #2076C22semidirect
58. #2068C22semidirect
59. #2066C22semidirect
60. #2036C22semidirect
61. #2035C22semidirect
62. #1942C22semidirect
63. #1941C22semidirect
64. #1933C22semidirect
65. #1933C22semidirect
66. #1894C22semidirect
67. #1894C22semidirect
68. #1894C22semidirect
71. #1854C22semidirect
73. #1825C22semidirect
74. #1824C22semidirect
75. #1809C22semidirect
76. #1809C22semidirect
77. #1809C22semidirect
78. #1787C22semidirect
79. #1736C22semidirect
80. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C2×#29752
#350842
#348092
#281122
#276702
#252622
#252562
#246042
#245612
#219432
#217682
#207012