Finite Group Databasebeta

← #6508752

Subgroups — #6508752

Order 512.

1985 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3190C211yesmaximal, characteristic
2C22×#484C211yesmaximal, characteristic, complemented
3#35227C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic, complemented
5#28131C214yesmaximal, complemented
6#27656C214yesmaximal, complemented
7#25210C214yesmaximal, complemented
8#25206C214yesmaximal, complemented
9#24577C212yesmaximal, complemented
10#24576C212yesmaximal, complemented
11#24529C212yesmaximal, complemented
12#24528C212yesmaximal, complemented
13#21931C211yesmaximal, characteristic, complemented
14#21763C211yesmaximal, characteristic
15C2×#1823C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#3190—28no—
20C2×#484—28no—
21C2×#484C2214yes—
22C2×#484C2214yes—
23C2×#485C2212yes—
24C2×#483C2211yescharacteristic
25C2×#480C2211yescharacteristic
26C2×#467C2212yes—
27C2×#465C2212yes—
28C2×#452C2212yes—
29C2×#451C2211yescharacteristic
30C2×#448C2211yescharacteristic
31C22×#107C2212yes—
32C22×#103C2212yes—
33C22×C22⋊C8C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2849—22no—
36#2623—22no—
37#2504C2212yes—
38#2503C2212yes—
39#2502C2212yes—
40#2501C2212yes—
41#2487—22no—
42#2486—21no—
43#2486—21no—
44#2378—22no—
45#2360C2214yes—
46#2344C2212yes—
47#2342—22no—
48Q8×C22⋊C4C2212yes—
49#2329—22no—
50#2323C2214yes—
51#2317C2212yes—
52#2316—21no—
53#2313—21no—
54#2311C2212yes—
55#2280C2211yescharacteristic
56C2×#380C2211yescharacteristic
57#2085C2214yescomplemented
58#2083C2214yescomplemented
59#2074C2214yescomplemented
60#2072C2214yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2056C2214yescomplemented
64#2055C2214yescomplemented
65#1949C2214yescomplemented
66#1947C2214yescomplemented
67#1928C2214yescomplemented
68#1925C2214yescomplemented
69#1893C2212yes—
70#1892C2212yes—
71#1880C2214yescomplemented
72#1878C2214yescomplemented
73#1874C2212yes—
74#1872C2212yes—
75#1850C2212yes—
76#1850C2212yes—
77#1850C2214yes—
78#1825C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1823C2214yes—
82#1805C2214yescomplemented
83#1803C2214yescomplemented
84#1798C2214yes—
85#1788C2212yes—
86C2×#340C2214yes—
87C22×#41C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×Q16C2312yes—
92C22×SD16C2312yes—
93C23×C8C2312yes—
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×#117—22no—
115C2×#117C2311yescharacteristic
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×C4⋊C4C2312yes—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130#485—28no—
131#484—216no—
132#484—216no—
133#483—24no—
134#480—24no—
135#467—28no—
136#465—28no—
137#452—28no—
138#451—24no—
139#448—24no—
140C2×#107—28no—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2314yes—
146C2×C4⋊C8C2314yes—
147C2×#103—28no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×C22⋊C8—24no—
157C2×C22⋊C8—24no—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2311yescharacteristic
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2312yes—
163#400—24no—
164#399—24no—
165#398—22no—
166#398—22no—
167#397—22no—
168#397—22no—
169#391—24no—
170#391C2314yes—
171#391C2314yes—
172#390C2314yes—
173#389—24no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389C2314yes—
178#388—24no—
179#388—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#386—24no—
184#386—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#384C2314yes—
189#382C2314yes—
190#381—22no—
191#381—22no—
192#381—22no—
193#381D412yes—
194#380—21no—
195#380—21no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380D411yescharacteristic
200#380D411yescharacteristic
201#380D412yes—
202#380C2312yes—
203#380C2312yes—
204#379D412yes—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—24no—
208C4×C22⋊C4C2312yes—
209C4×C22⋊C4C2312yes—
210C2×#95—21no—
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—22no—
239C23×C4—21no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D414yes—
248C23×C4#4114yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—24no—
253C2×Q16—44no—
254C2×SD16—24no—
255C2×SD16—44no—
256C22×C8—24no—
257C22×C8—24no—
258C22×C8—44no—
259C22×C8—44no—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2414yes—
263#123—44no—
264#123—84no—
265#123—84no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—22no—
273C22⋊Q8—22no—
274C22⋊Q8C2×D418yes—
275C22⋊Q8C2×D418yes—
276#121—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#120—44no—
281C4×Q8—44no—
282C4×Q8—44no—
283C4×D4—44no—
284C4×D4—44no—
285#117—22no—
286#117—22no—
287#117—44no—
288#117—84no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D414yes—
338C2×C22⋊C4C2×D414yes—
339C2×C42—21no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42C2414yes—
343#107—28no—
344#107—28no—
345#107—24no—
346#106—24no—
347C4⋊C8—44no—
348C4⋊C8—44no—
349#103—28no—
350#103—28no—
351#103—24no—
352#103—24no—
353#103—24no—
354#103—44no—
355#102—24no—
356#102—44no—
357C22⋊C8—22no—
358C22⋊C8—22no—
359C22⋊C8—24no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D412yes—
373#95C2×D414yes—
374#95C2×D412yes—
375C24—22no—
376C24—22no—
377C24—24no—
378C24—21no—
379C24—42no—
380C24#12114yes—
381C24C22×D411yescharacteristic
382C24C22×D411yescharacteristic
383C24C22×D411yescharacteristic
384#41—84no—
385#41—84no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—48no—
394C2×Q8—82no—
395C2×Q8—84no—
396C2×Q8C22×D414yes—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—24no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—44no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4—82no—
441C22×C4—816no—
442C22×C4#12114yes—
443C22×C4#12114yes—
444C22×C4#12114yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457Q16—84no—
458SD16—84no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—48no—
467C2×C8—84no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—21no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4C22×D414yes—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C22⋊C4—82no—
508C42—24no—
509C42—24no—
510C42—42no—
511C42—84no—
512C42—84no—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—28no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—44no—
525C23—48no—
526C23—48no—
527C23—41no—
528C23—42no—
529C23—41no—
530C23—41no—
531C23—42no—
532C23C2×#12112yes—
533C23C2×#12112yes—
534C23C2×#12112yes—
535C23#57612yes—
536C23#57712yes—
537C23#57814yes—
538C23C23×D411yescharacteristic
539Q8—24no—
540Q8—24no—
541Q8—44no—
542Q8—42no—
543Q8—42no—
544Q8—84no—
545D4—42no—
546D4—42no—
547D4—84no—
548D4—84no—
549D4—84no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4—88no—
591C2×C4—88no—
592C2×C4—164no—
593C2×C4C2×#12112yes—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4C23×D411yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4#58514yes—
599C8—44no—
600C8—42no—
601C8—42no—
602C22—22no—
603C22—22no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—416no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—82no—
619C22—82no—
620C22—81no—
621C22#304414yes—
622C22#304514yes—
623C22C22×#12111yescharacteristic
624C22#348512yes—
625C22C2×#57611yescharacteristic
626C22C2×#57711yescharacteristic
627C22C2×#57812yes—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—81no—
636C4—84no—
637C4—81no—
638C4—84no—
639C4—81no—
640C4#348512yes—
641C4#349112yes—
642C2—24no—
643C2—24no—
644C2—42no—
645C2—161no—
646C2#3544711yescharacteristic
647C2C2×#304411yescharacteristic
648C2C2×#304511yescharacteristic
649C2#6176412yes—
650C2#6176712yes—
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×#484C2semidirect
4. #34380C2semidirect
5. #28131C2semidirect
6. #27656C2semidirect
7. #25210C2semidirect
8. #25206C2semidirect
9. #24577C2semidirect
10. #24576C2semidirect
11. #24529C2semidirect
12. #24528C2semidirect
13. #21931C2semidirect
15. C2×#1823C2semidirect
57. #2085C22semidirect
58. #2083C22semidirect
59. #2074C22semidirect
60. #2072C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2056C22semidirect
64. #2055C22semidirect
65. #1949C22semidirect
66. #1947C22semidirect
67. #1928C22semidirect
68. #1925C22semidirect
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×#31902
C22×#4842
#352272
#343802
#281312
#276562
#252102
#252062
#245772
#245762
#245292
#245282
#219312
#217632
C2×#18232