Finite Group Databasebeta

← #6479540

Subgroups — #6479540

Order 512.

1765 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#33100C212yesmaximal, complemented
4#33096C212yesmaximal, complemented
5#29913C212yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#27866C214yesmaximal, complemented
8#27865C214yesmaximal, complemented
9#27618C214yesmaximal, complemented
10#27590C214yesmaximal, complemented
11#21944C211yesmaximal, characteristic, complemented
12#21944C211yesmaximal, characteristic, complemented
13#21769C211yesmaximal, characteristic
14#21769C211yesmaximal, characteristic
15#20682C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#407C2211yescharacteristic
20#2855C2211yescharacteristic
21#2855C2211yescharacteristic
22#2855C2212yes—
23#2854C2211yescharacteristic
24#2854C2211yescharacteristic
25#2854C2212yes—
26#2853—22no—
27#2849—21no—
28#2849—21no—
29C4×#123—21no—
30C4×#123—21no—
31C4×#123—22no—
32#2297C2212yes—
33#2296C2212yes—
34#2283—22no—
35#2283—22no—
36#2283C2214yes—
37#2283C2214yes—
38C2×#383—21no—
39C2×#382C2212yes—
40#2280C2211yescharacteristic
41C2×#381—21no—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4—21no—
46C2×#376—21no—
47#2101C2214yescomplemented
48#2101C2214yescomplemented
49#2097C2214yescomplemented
50#2092C2214yescomplemented
51#2085C2214yescomplemented
52#2085C2214yescomplemented
53#2074C2214yescomplemented
54#2073C2214yescomplemented
55#1928C2214yescomplemented
56#1927C2214yescomplemented
57#1926C2214yescomplemented
58#1925C2214yescomplemented
59#1921C2214yescomplemented
60#1920C2214yescomplemented
61#1919C2214yescomplemented
62#1918C2214yescomplemented
63#1895C2212yes—
64#1895C2212yes—
65#1894C2212yes—
66#1894C2212yes—
67#1874C2212yes—
68#1874C2212yes—
69#1872C2212yes—
70#1872C2212yes—
71#1854C2214yes—
72#1854C2214yes—
73#1853C2212yes—
74#1853C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1825C2212yes—
78#1825C2212yes—
79#1817C2212yes—
80#1817C2214yes—
81#1817C2212yes—
82#1809C2214yes—
83#1809C2214yes—
84#1788C2212yes—
85#1788C2212yes—
86#1728C2214yes—
87C22×#41C2311yescharacteristic
88C24×C4D411yescharacteristic
89C24×C4D411yescharacteristic
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C22×M16C2312yes—
93#525—24no—
94#525—24no—
95C2×#123—21no—
96C2×#123—21no—
97C2×#123—22no—
98C2×#123—21no—
99C2×C22⋊Q8C2311yescharacteristic
100C2×C22⋊Q8C2311yescharacteristic
101C2×#121C2311yescharacteristic
102C2×#121C2311yescharacteristic
103C2×#120—21no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C42—24no—
113C22×C42—24no—
114C22×C42—22no—
115C22×C42D412yes—
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D412yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131#407—24no—
132#407—24no—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137#391C2312yes—
138#391C2312yes—
139#390C2312yes—
140#390C2312yes—
141#389C2312yes—
142#389C2312yes—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—22no—
147#388—22no—
148#388—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#386—22no—
154#386—22no—
155#386—21no—
156#386—21no—
157#386—22no—
158#385—22no—
159#385—22no—
160#385—21no—
161#385—21no—
162#385—22no—
163#383—24no—
164#383—24no—
165#382—22no—
166#382—22no—
167#382C2314yes—
168#382C2314yes—
169#381—21no—
170#381—21no—
171#381—22no—
172#381—21no—
173#381—24no—
174#381—24no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182#380C2312yes—
183#380C2312yes—
184#380C2311yescharacteristic
185#380C2311yescharacteristic
186#379—21no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C22⋊C4—24no—
194C4×C22⋊C4—24no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199#376—24no—
200#376—24no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—22no—
204C2×#95—22no—
205C2×#95—22no—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—44no—
222C22×Q8C2411yescharacteristic
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2411yescharacteristic
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—21no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4#4114yes—
248C23×C4#4114yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×M16—24no—
253C2×M16—24no—
254C2×M16—44no—
255C2×M16—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258#123—44no—
259#123—44no—
260#123—48no—
261#123—44no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266#121—22no—
267#121—22no—
268#121—22no—
269#121—22no—
270#120—44no—
271C4×D4—44no—
272C4×D4—44no—
273#117—22no—
274#117—22no—
275#117—44no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2414yes—
288C2×C4⋊C4C2411yescharacteristic
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—48no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336#107—44no—
337#107—44no—
338#106—44no—
339#106—44no—
340#103—44no—
341#103—44no—
342#102—44no—
343#102—44no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C22⋊C8—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—42no—
373C24—42no—
374C24#12314yes—
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378#41—84no—
379#41—84no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—24no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—24no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—416no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4#12314yes—
441C22×C4#12314yes—
442C22×C4#12314yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457M16—84no—
458M16—84no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—44no—
500C42—24no—
501C42—24no—
502C42—24no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—48no—
511C42—44no—
512C42—42no—
513C42—48no—
514C23—22no—
515C23—22no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—22no—
525C23—44no—
526C23—44no—
527C23—48no—
528C23—48no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23C2×#12312yes—
535C23C2×#12312yes—
536C23C2×#12312yes—
537C23#57512yes—
538C23#57512yes—
539C23C23×D411yescharacteristic
540Q8—42no—
541Q8—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545D4—84no—
546D4—84no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
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—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—84no—
598C2×C4—88no—
599C2×C4—88no—
600C2×C4C2×#12312yes—
601C2×C4C2×#12312yes—
602C2×C4C2×#12312yes—
603C2×C4#57514yes—
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4#58514yes—
607C8—44no—
608C8—44no—
609C22—22no—
610C22—24no—
611C22—24no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—416no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—81no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22#309314yes—
630C22C22×#12311yescharacteristic
631C22#344912yes—
632C22C2×#57511yescharacteristic
633C22C2×#57512yes—
634C22C2×#57511yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—84no—
647C4—81no—
648C4—81no—
649C4#349112yes—
650C4#349112yes—
651C2—24no—
652C2—42no—
653C2—42no—
654C2—161no—
655C2#3509711yescharacteristic
656C2C2×#309311yescharacteristic
657C2#6051111yescharacteristic
658C2#6095412yes—
659C2#6097012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #33100C2semidirect
4. #33096C2semidirect
5. #29913C2semidirect
6. #29912C2semidirect
7. #27866C2semidirect
8. #27865C2semidirect
9. #27618C2semidirect
10. #27590C2semidirect
11. #21944C2semidirect
12. #21944C2semidirect
15. #20682C2semidirect
47. #2101C22semidirect
48. #2101C22semidirect
49. #2097C22semidirect
50. #2092C22semidirect
51. #2085C22semidirect
52. #2085C22semidirect
53. #2074C22semidirect
54. #2073C22semidirect
55. #1928C22semidirect
56. #1927C22semidirect
57. #1926C22semidirect
58. #1925C22semidirect
59. #1921C22semidirect
60. #1920C22semidirect
61. #1919C22semidirect
62. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#22832
#331002
#330962
#299132
#299122
#278662
#278652
#276182
#275902
#219442
#217692
#206822