Finite Group Databasebeta

← #6508972

Subgroups — #6508972

Order 512.

1700 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27614C211yesmaximal, characteristic, complemented
4#27613C211yesmaximal, characteristic, complemented
5#27611C211yesmaximal, characteristic, complemented
6#27610C211yesmaximal, characteristic, complemented
7#27604C212yesmaximal, complemented
8#27602C212yesmaximal, complemented
9#24948C214yesmaximal, complemented
10#24946C214yesmaximal, complemented
11#24944C214yesmaximal, complemented
12#24942C214yesmaximal, complemented
13#21941C212yesmaximal, complemented
14#21738C211yesmaximal, characteristic
15#20699C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×C22⋊C8C2212yes—
19#2485—22no—
20#2485—22no—
21#2485C2214yes—
22#2485C2214yes—
23C2×#397—21no—
24C2×#394—21no—
25#2357—22no—
26#2357—22no—
27#2354C2212yes—
28#2354C2212yes—
29#2354C2212yes—
30#2354C2212yes—
31#2312—21no—
32#2312—21no—
33#2309—21no—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388—21no—
37C2×#387—21no—
38#2285C2211yescharacteristic
39#2285C2211yescharacteristic
40#2280C2212yes—
41#2280C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45#2274—21no—
46#2100C2212yescomplemented
47#2100C2212yescomplemented
48#2099C2212yescomplemented
49#2099C2212yescomplemented
50#2084C2212yescomplemented
51#2084C2212yescomplemented
52#2083C2212yescomplemented
53#2083C2212yescomplemented
54#2077C2214yescomplemented
55#2077C2214yescomplemented
56#2076C2214yescomplemented
57#2075C2214yescomplemented
58#1921C2214yescomplemented
59#1920C2214yescomplemented
60#1919C2214yescomplemented
61#1918C2214yescomplemented
62#1869C2214yes—
63#1869C2214yes—
64#1854C2214yescomplemented
65#1854C2214yescomplemented
66#1854C2214yescomplemented
67#1854C2214yescomplemented
68#1853C2212yes—
69#1853C2212yes—
70#1852C2212yes—
71#1852C2212yes—
72#1825C2214yescomplemented
73#1825C2214yescomplemented
74#1825C2214yescomplemented
75#1825C2214yescomplemented
76#1824C2212yes—
77#1824C2212yes—
78#1823C2212yes—
79#1823C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1808C2214yescomplemented
83#1807C2214yescomplemented
84#1779C2212yes—
85#1735C2214yescomplemented
86#1735C2214yescomplemented
87C23×D4D411yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C23×C8C2314yes—
91C2×C4×D4—28no—
92C2×C4×D4D412yes—
93C2×C4×D4D412yes—
94C2×C4×D4C2312yes—
95C2×C4×D4C2312yes—
96C2×#117—21no—
97C2×#117—21no—
98C2×#117—22no—
99C2×#117—28no—
100C2×#117C2314yes—
101C2×#117C2312yes—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C22⋊C8—28no—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2314yes—
130#397—24no—
131#397—24no—
132#394—24no—
133#394—24no—
134#390—22no—
135#390—22no—
136#390—22no—
137#390—22no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#389C2312yes—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2311yescharacteristic
161#384C2311yescharacteristic
162#382—21no—
163#382—21no—
164#381—22no—
165#381—22no—
166#380—21no—
167#380—21no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380C2312yes—
172#380C2314yes—
173#380C2312yes—
174#380C2312yes—
175#380C2314yes—
176#380C2314yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#379—22no—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2312yes—
212#336C2312yes—
213#336C2312yes—
214#336C2312yes—
215C25C2×D411yescharacteristic
216C25C2×D411yescharacteristic
217C22×D4—24no—
218C22×D4—24no—
219C22×D4—42no—
220C22×D4C2×D412yes—
221C22×D4C2×D412yes—
222C23×C4—24no—
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×D412yes—
237C23×C4#4118yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C22×C8—28no—
242C22×C8—44no—
243C22×C8—44no—
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4×D4—28no—
247C4×D4—28no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252#117—28no—
253#117—28no—
254#117—44no—
255#117—48no—
256#117—44no—
257#117—44no—
258#117—44no—
259#117—44no—
260#117—44no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
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—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—82no—
313C2×C22⋊C4—82no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—44no—
332#107—44no—
333#106—44no—
334#106—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337C22⋊C8—28no—
338C22⋊C8—44no—
339C22⋊C8—44no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—41no—
361C24—41no—
362C24—41no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24#12314yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C22×D411yescharacteristic
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—24no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—416no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—416no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—88no—
424C22×C4#12314yes—
425C22×C4#12318yes—
426C22×C4#13514yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2511yescharacteristic
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—88no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
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—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—44no—
496C42—48no—
497C42—44no—
498C42—82no—
499C42—82no—
500C42—82no—
501C42—82no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—28no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23C2×#12314yes—
534C23C2×#12312yes—
535C23C2×#13512yes—
536C23#57512yes—
537C23C23×D411yescharacteristic
538D4—48no—
539D4—48no—
540C2×C4—28no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—84no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4C2×#12312yes—
597C2×C4C2×#12314yes—
598C2×C4C2×#13512yes—
599C2×C4C2×#13512yes—
600C2×C4C22×#4112yes—
601C2×C4#58512yes—
602C2×C4#58512yes—
603C8—44no—
604C8—44no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—88no—
626C22—88no—
627C22#309114yes—
628C22C22×#12311yescharacteristic
629C22#348511yescharacteristic
630C22#348511yescharacteristic
631C22C22×#13511yescharacteristic
632C22C2×#57511yescharacteristic
633C22#358511yescharacteristic
634C22#358611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4#349114yes—
651C2—24no—
652C2—44no—
653C2—82no—
654C2#3544811yescharacteristic
655C2C2×#309111yescharacteristic
656C2#6052011yescharacteristic
657C2#6170011yescharacteristic
658C2#6170111yescharacteristic
659C2#6171111yescharacteristic
660C2#6171311yescharacteristic
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
3. #27614C2semidirect
4. #27613C2semidirect
5. #27611C2semidirect
6. #27610C2semidirect
7. #27604C2semidirect
8. #27602C2semidirect
9. #24948C2semidirect
10. #24946C2semidirect
11. #24944C2semidirect
12. #24942C2semidirect
13. #21941C2semidirect
15. #20699C2semidirect
46. #2100C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2099C22semidirect
50. #2084C22semidirect
51. #2084C22semidirect
52. #2083C22semidirect
53. #2083C22semidirect
54. #2077C22semidirect
55. #2077C22semidirect
56. #2076C22semidirect
57. #2075C22semidirect
58. #1921C22semidirect
59. #1920C22semidirect
60. #1919C22semidirect
61. #1918C22semidirect
64. #1854C22semidirect
65. #1854C22semidirect
66. #1854C22semidirect
67. #1854C22semidirect
72. #1825C22semidirect
73. #1825C22semidirect
74. #1825C22semidirect
75. #1825C22semidirect
82. #1808C22semidirect
83. #1807C22semidirect
85. #1735C22semidirect
86. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
#341862
#276142
#276132
#276112
#276102
#276042
#276022
#249482
#249462
#249442
#249422
#219412
#217382
#206992