Finite Group Databasebeta

← #6508847

Subgroups — #6508847

Order 512.

1950 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic, complemented
2C2×#2976C211yesmaximal, characteristic
3#37438C211yesmaximal, characteristic
4#35227C211yesmaximal, characteristic
5#28105C212yesmaximal, complemented
6#28098C212yesmaximal, complemented
7#27663C212yesmaximal, complemented
8#27658C212yesmaximal, complemented
9#25262C214yesmaximal, complemented
10#25244C214yesmaximal, complemented
11#24616C214yesmaximal, complemented
12#24566C214yesmaximal, complemented
13#21939C211yesmaximal, characteristic, complemented
14#21768C211yesmaximal, characteristic, complemented
15#20700C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C2×#483—28no—
20C2×#483C2214yes—
21C2×#483C2214yes—
22#2976—28no—
23C2×#436C2212yes—
24C2×#435C2212yes—
25C2×#435C2212yes—
26C2×C8×D4C2212yes—
27C2×#433C2212yes—
28C22×#106C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C2×#408C2212yes—
33C2×#407C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2849—22no—
36#2817—21no—
37#2816—21no—
38#2663—22no—
39#2625C2212yes—
40#2623—22no—
41#2622—22no—
42#2616C2212yes—
43#2591—21no—
44#2583—21no—
45#2504—21no—
46#2503—21no—
47#2502—21no—
48#2501—21no—
49#2487C2212yes—
50#2487C2214yes—
51#2486C2212yes—
52#2486C2212yes—
53#2415C2214yes—
54#2409C2212yes—
55#2317C2212yes—
56#2311C2212yes—
57#2280C2211yescharacteristic
58#2280C2211yescharacteristic
59#2088C2214yescomplemented
60#2088C2214yescomplemented
61#2077C2214yescomplemented
62#2077C2214yescomplemented
63#2068C2214yescomplemented
64#2066C2214yescomplemented
65#2036C2214yescomplemented
66#2035C2214yescomplemented
67#1943C2214yescomplemented
68#1941C2214yescomplemented
69#1935C2214yescomplemented
70#1933C2214yescomplemented
71#1894C2212yes—
72#1894C2212yes—
73#1894C2214yescomplemented
74#1892C2214yescomplemented
75#1873C2214yes—
76#1854C2214yes—
77#1854C2214yes—
78#1824C2212yes—
79#1824C2214yes—
80#1823C2212yes—
81#1823C2212yes—
82#1809C2214yescomplemented
83#1808C2212yes—
84#1807C2212yes—
85#1807C2214yescomplemented
86#1787C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89C22×#41C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93C22×M16C2312yes—
94C22×M16C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C23×C8C2312yes—
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99#525—24no—
100#525—24no—
101C2×#123—22no—
102C2×#123—22no—
103C2×#123D412yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8D411yescharacteristic
109C2×C22⋊Q8D411yescharacteristic
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2314yes—
113C2×#121D411yescharacteristic
114C2×#121D411yescharacteristic
115C2×#120D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133#483—216no—
134#483—216no—
135#436—28no—
136#435—28no—
137#435—28no—
138C8×D4—28no—
139#433—28no—
140C2×#107C2314yes—
141C2×#106—24no—
142C2×#106—24no—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×#103—24no—
149C2×#103—24no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154#408—28no—
155#407—24no—
156C2×C22⋊C8D418yes—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2312yes—
165#400—22no—
166#400—22no—
167#400—24no—
168#399—22no—
169#399—24no—
170#398—22no—
171#398—22no—
172#398—22no—
173#397—24no—
174#397—24no—
175#397—22no—
176#397—22no—
177#395—22no—
178#395—22no—
179#394—22no—
180#394—24no—
181#394—24no—
182#393—22no—
183#391—22no—
184#391C2314yes—
185#390—22no—
186#389—24no—
187#389C2314yes—
188#389C2314yes—
189#389C2314yes—
190#388—24no—
191#386—24no—
192#386—22no—
193#386—22no—
194#386—24no—
195#385—24no—
196#385—22no—
197#385—22no—
198#385—24no—
199#384—22no—
200#382—22no—
201#381—21no—
202#381—21no—
203#381—22no—
204#381—22no—
205#380C2314yes—
206#380C2314yes—
207#380C2312yes—
208#380C2312yes—
209#380C2314yes—
210#380C2312yes—
211#380C2312yes—
212#379—21no—
213#379—21no—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—21no—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226C25C2×D411yescharacteristic
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—42no—
230C2×#41—42no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×D4C2×D412yes—
239C22×D4C2×D412yes—
240C22×D4C2×D412yes—
241C22×D4C2×D411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—42no—
249C23×C4—42no—
250C23×C4—44no—
251C23×C4—84no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D414yes—
256C23×C4#4114yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—24no—
264C2×M16—44no—
265C2×M16—42no—
266C2×M16—42no—
267C22×C8—24no—
268C22×C8—42no—
269C22×C8—42no—
270C22×C8—44no—
271C22×C8C2×D414yes—
272C22×C8C2×D414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277#123—28no—
278#123—84no—
279#123—84no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—44no—
287C22⋊Q8—44no—
288C22⋊Q8—44no—
289C22⋊Q8—44no—
290#121—24no—
291#121—24no—
292#120—28no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297#117—22no—
298#117—22no—
299#117—42no—
300#117—42no—
301#117—42no—
302#117—42no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2414yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42C2412yes—
358#107—44no—
359#106—28no—
360#106—28no—
361#106—44no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364#103—28no—
365#103—28no—
366#103—44no—
367#103—44no—
368C22⋊C8—24no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8C2×D418yes—
373C22⋊C8C2×D418yes—
374#97—24no—
375C4×C8—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388C24—24no—
389C24—42no—
390C24—42no—
391C24#12114yes—
392C24C22×D412yes—
393C24C22×D411yescharacteristic
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396#41—82no—
397#41—82no—
398#41—84no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—82no—
409C2×Q8—84no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—44no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—82no—
453C22×C4—816no—
454C22×C4#12114yes—
455C22×C4#12114yes—
456C22×C4#12114yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2511yescharacteristic
470M16—84no—
471M16—82no—
472M16—82no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—44no—
480C2×C8—44no—
481C2×C8—84no—
482C2×C8C22×D414yes—
483C2×C8C22×D414yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—24no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—22no—
527C42—22no—
528C42—44no—
529C42—82no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—28no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—22no—
543C23—22no—
544C23—44no—
545C23—44no—
546C23—416no—
547C23C2×#12112yes—
548C23C2×#12112yes—
549C23C2×#12112yes—
550C23#57714yes—
551C23C23×D411yescharacteristic
552Q8—42no—
553Q8—42no—
554Q8—44no—
555Q8—82no—
556Q8—84no—
557D4—48no—
558D4—82no—
559D4—84no—
560D4—84no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—84no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4C2×#12112yes—
608C2×C4C2×#12112yes—
609C2×C4C2×#12112yes—
610C2×C4C23×D411yescharacteristic
611C2×C4C22×#4111yescharacteristic
612C2×C4#58512yes—
613C2×C4#58512yes—
614C8—24no—
615C8—24no—
616C8—42no—
617C8—42no—
618C22—22no—
619C22—28no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—416no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22#305714yes—
636C22C22×#12111yescharacteristic
637C22#348512yes—
638C22C2×#57712yes—
639C22#358612yes—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—84no—
651C4—82no—
652C4#348511yescharacteristic
653C4#348511yescharacteristic
654C4#349112yes—
655C2—24no—
656C2—42no—
657C2—42no—
658C2—82no—
659C2#3544711yescharacteristic
660C2C2×#305711yescharacteristic
661C2#6045511yescharacteristic
662C2#6174514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#483C2semidirect
5. #28105C2semidirect
6. #28098C2semidirect
7. #27663C2semidirect
8. #27658C2semidirect
9. #25262C2semidirect
10. #25244C2semidirect
11. #24616C2semidirect
12. #24566C2semidirect
13. #21939C2semidirect
14. #21768C2semidirect
15. #20700C2semidirect
59. #2088C22semidirect
60. #2088C22semidirect
61. #2077C22semidirect
62. #2077C22semidirect
63. #2068C22semidirect
64. #2066C22semidirect
65. #2036C22semidirect
66. #2035C22semidirect
67. #1943C22semidirect
68. #1941C22semidirect
69. #1935C22semidirect
70. #1933C22semidirect
73. #1894C22semidirect
74. #1892C22semidirect
82. #1809C22semidirect
85. #1807C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4832
C2×#29762
#374382
#352272
#281052
#280982
#276632
#276582
#252622
#252442
#246162
#245662
#219392
#217682
#207002