Finite Group Databasebeta

← #6508737

Subgroups — #6508737

Order 512.

1943 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3196C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic, complemented
3#35227C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic, complemented
5#28133C214yesmaximal, complemented
6#27657C214yesmaximal, complemented
7#25231C214yesmaximal, complemented
8#25219C214yesmaximal, complemented
9#24579C212yesmaximal, complemented
10#24578C212yesmaximal, complemented
11#24529C212yesmaximal, complemented
12#24528C212yesmaximal, complemented
13#21933C211yesmaximal, characteristic, complemented
14#21760C211yesmaximal, characteristic
15#20700C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3196—28no—
20C2×#484C2211yescharacteristic
21C2×#481C2211yescharacteristic
22C2×#485C2212yes—
23C2×#483—28no—
24C2×#483C2214yes—
25C2×#483C2214yes—
26C2×#470C2212yes—
27C2×#468C2212yes—
28C2×#452C2212yes—
29C2×#451C2211yescharacteristic
30C2×#448C2211yescharacteristic
31C22×#106C2212yes—
32C22×#103C2212yes—
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2849—21no—
36#2849—21no—
37#2623—22no—
38#2504C2212yes—
39#2503C2212yes—
40#2502C2212yes—
41#2501C2212yes—
42#2487—22no—
43#2486—22no—
44#2378—22no—
45#2362C2214yes—
46#2345—22no—
47#2345C2212yes—
48Q8×C22⋊C4C2212yes—
49#2332—22no—
50#2323C2214yes—
51#2317C2212yes—
52#2312—21no—
53#2312—21no—
54#2311C2212yes—
55#2280C2211yescharacteristic
56#2280C2211yescharacteristic
57#2085C2214yescomplemented
58#2084C2214yescomplemented
59#2074C2214yescomplemented
60#2072C2214yescomplemented
61#2066C2214yescomplemented
62#2065C2214yescomplemented
63#2056C2214yescomplemented
64#2055C2214yescomplemented
65#1949C2214yescomplemented
66#1948C2214yescomplemented
67#1927C2214yescomplemented
68#1926C2214yescomplemented
69#1893C2212yes—
70#1892C2212yes—
71#1880C2214yescomplemented
72#1878C2214yescomplemented
73#1875C2212yes—
74#1873C2212yes—
75#1851C2214yes—
76#1850C2212yes—
77#1850C2212yes—
78#1825C2212yes—
79#1824C2214yes—
80#1823C2212yes—
81#1823C2212yes—
82#1805C2214yescomplemented
83#1803C2214yescomplemented
84#1798C2214yes—
85#1788C2212yes—
86#1736C2214yes—
87C22×#41C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×Q16C2312yes—
92C22×SD16C2312yes—
93C22×M16C2312yes—
94C23×C8C2312yes—
95#525—24no—
96#525—24no—
97C2×#123—21no—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8D418yes—
104C2×C22⋊Q8C2311yescharacteristic
105C2×C22⋊Q8C2311yescharacteristic
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121C2311yescharacteristic
109C2×#121C2311yescharacteristic
110C2×#120—21no—
111C2×C4×Q8C2314yes—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×#117C2311yescharacteristic
115C2×#117C2312yes—
116C2×#117C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128#485—28no—
129#484—24no—
130#483—216no—
131#483—216no—
132#481—24no—
133#470—28no—
134#468—28no—
135#452—28no—
136#451—24no—
137#448—24no—
138C2×#107C2312yes—
139C2×#106—28no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—28no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154#407—24no—
155C2×C22⋊C8—24no—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161#400—24no—
162#399—24no—
163#398—24no—
164#397—24no—
165#391—24no—
166#391C2314yes—
167#391C2314yes—
168#390C2314yes—
169#389—24no—
170#389—24no—
171#389C2314yes—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#387—24no—
177#387—24no—
178#387—24no—
179#386—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384C2314yes—
188#382C2314yes—
189#381—21no—
190#381—22no—
191#381—22no—
192#381D412yes—
193#380—24no—
194#380—22no—
195#380—22no—
196#380D412yes—
197#380D412yes—
198#380C2312yes—
199#380C2312yes—
200#379—21no—
201#379D411yescharacteristic
202#379D411yescharacteristic
203C4×C4⋊C4—24no—
204C4×C4⋊C4C2314yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4C2312yes—
210C4×C22⋊C4C2312yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C2×#41—22no—
221C2×#41—22no—
222C2×#41—44no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—44no—
226C22×Q8C2×D414yes—
227C22×Q8C2411yescharacteristic
228C22×Q8C2412yes—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—84no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D414yes—
246C23×C4#4114yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×Q16—24no—
251C2×Q16—44no—
252C2×SD16—24no—
253C2×SD16—44no—
254C2×M16—24no—
255C2×M16—44no—
256C22×C8—24no—
257C22×C8—44no—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261#123—44no—
262#123—84no—
263#123—84no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8C2×D418yes—
273C22⋊Q8C2×D418yes—
274#121—22no—
275#121—22no—
276#121—22no—
277#121—22no—
278#120—44no—
279C4×Q8—44no—
280C4×Q8—44no—
281C4×D4—44no—
282C4×D4—44no—
283#117—22no—
284#117—22no—
285#117—44no—
286#117—44no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42C2414yes—
341#107—24no—
342#106—28no—
343#106—28no—
344#106—24no—
345C4⋊C8—44no—
346C4⋊C8—44no—
347#103—28no—
348#103—28no—
349#103—24no—
350#103—24no—
351#103—24no—
352#103—44no—
353#102—24no—
354#102—44no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8—22no—
359C22⋊C8—24no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D414yes—
371#95C2×D414yes—
372C24—24no—
373C24—21no—
374C24—42no—
375C24—42no—
376C24#12114yes—
377C24C22×D411yescharacteristic
378C24C22×D411yescharacteristic
379C24C22×D411yescharacteristic
380#41—84no—
381#41—84no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—48no—
390C2×Q8—82no—
391C2×Q8—84no—
392C2×Q8C22×D414yes—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—24no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—44no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—82no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—82no—
430C22×C4—816no—
431C22×C4#12114yes—
432C22×C4#12114yes—
433C22×C4#12114yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4112yes—
445C22×C4C2511yescharacteristic
446Q16—84no—
447SD16—84no—
448M16—84no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—48no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4C22×D414yes—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—24no—
499C42—24no—
500C42—42no—
501C42—84no—
502C42—82no—
503C42—84no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—28no—
511C23—22no—
512C23—22no—
513C23—44no—
514C23—44no—
515C23—416no—
516C23—42no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23C2×#12112yes—
521C23C2×#12112yes—
522C23C2×#12112yes—
523C23#57714yes—
524C23C23×D411yescharacteristic
525Q8—24no—
526Q8—24no—
527Q8—44no—
528Q8—42no—
529Q8—42no—
530Q8—84no—
531D4—42no—
532D4—42no—
533D4—84no—
534D4—84no—
535D4—84no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
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—24no—
552C2×C4—24no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—88no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4C2×#12112yes—
577C2×C4C2×#12112yes—
578C2×C4C2×#12112yes—
579C2×C4C23×D411yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4#58514yes—
582C8—44no—
583C8—42no—
584C8—42no—
585C22—22no—
586C22—28no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—416no—
595C22—44no—
596C22—44no—
597C22—81no—
598C22—81no—
599C22—84no—
600C22—81no—
601C22#304314yes—
602C22C22×#12111yescharacteristic
603C22#348512yes—
604C22C2×#57712yes—
605C22#358612yes—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—84no—
614C4—81no—
615C4—81no—
616C4—84no—
617C4—81no—
618C4#348512yes—
619C4#349112yes—
620C2—24no—
621C2—42no—
622C2—42no—
623C2—161no—
624C2#3544711yescharacteristic
625C2C2×#304311yescharacteristic
626C2#6042711yescharacteristic
627C2#6174514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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. C22×#483C2semidirect
4. #34370C2semidirect
5. #28133C2semidirect
6. #27657C2semidirect
7. #25231C2semidirect
8. #25219C2semidirect
9. #24579C2semidirect
10. #24578C2semidirect
11. #24529C2semidirect
12. #24528C2semidirect
13. #21933C2semidirect
15. #20700C2semidirect
57. #2085C22semidirect
58. #2084C22semidirect
59. #2074C22semidirect
60. #2072C22semidirect
61. #2066C22semidirect
62. #2065C22semidirect
63. #2056C22semidirect
64. #2055C22semidirect
65. #1949C22semidirect
66. #1948C22semidirect
67. #1927C22semidirect
68. #1926C22semidirect
71. #1880C22semidirect
72. #1878C22semidirect
82. #1805C22semidirect
83. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31962
C22×#4832
#352272
#343702
#281332
#276572
#252312
#252192
#245792
#245782
#245292
#245282
#219332
#217602
#207002