Finite Group Databasebeta

← #6482122

Subgroups — #6482122

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3232C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic
3C2×#2370C211yesmaximal, characteristic, complemented
4#33110C212yesmaximal, complemented
5#33108C211yesmaximal, characteristic, complemented
6#33106C211yesmaximal, characteristic
7#29760C212yesmaximal, complemented
8#29757C211yesmaximal, characteristic, complemented
9#29755C211yesmaximal, characteristic
10#25460C214yesmaximal, complemented
11#25458C214yesmaximal, complemented
12#25283C214yesmaximal, complemented
13#25281C214yesmaximal, complemented
14#21951C211yesmaximal, characteristic, complemented
15#21388C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×#125C2211yescharacteristic
18#3232—216no—
19C2×#499C2214yes—
20C2×#488C2214yes—
21C2×#484C2212yes—
22C2×#481C2212yes—
23C2×#483C2212yes—
24C2×#480C2212yes—
25C2×#479C2214yes—
26C2×#478C2214yes—
27C2×#431C2212yes—
28#2857C2211yescharacteristic
29#2856C2211yescharacteristic
30#2850C2212yes—
31#2618—21no—
32#2618—21no—
33#2601—22no—
34#2601—22no—
35#2566C2212yes—
36#2565C2212yes—
37#2545C2212yes—
38#2544C2212yes—
39#2370—22no—
40#2370—22no—
41#2370C2214yes—
42#2370C2214yes—
43#2325—21no—
44#2325—21no—
45C2×#389—21no—
46C2×#389C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#387—21no—
49C2×#387—21no—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2085C2214yes—
56#2084C2212yes—
57#2083C2212yes—
58#2068C2214yes—
59#2068C2214yes—
60#2065C2214yes—
61#2065C2214yes—
62#2043C2212yes—
63#2042C2214yes—
64#2041C2212yes—
65#1989C2214yes—
66#1983C2214yes—
67#1983C2214yes—
68#1982C2212yes—
69#1981C2212yes—
70#1949C2214yes—
71#1948C2212yes—
72#1947C2212yes—
73#1939C2214yes—
74#1939C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1925C2214yes—
78#1925C2214yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1797C2211yescharacteristic
84#1797C2211yescharacteristic
85#1797C2212yes—
86C4×#103C2214yes—
87C4×#102C2214yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127C2311yescharacteristic
90#529—28no—
91#529—28no—
92C2×#126—22no—
93C2×#125—22no—
94C2×#125—24no—
95C2×#125D418yescomplemented
96C2×#125C2312yes—
97C2×#125C2314yes—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#499—216no—
126#488—216no—
127#484—28no—
128#483—28no—
129#481—28no—
130#480—28no—
131#479—216no—
132#478—216no—
133#431—28no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×#106C2314yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C4×C8C2314yes—
153#398—22no—
154#398—22no—
155#396—22no—
156#396—22no—
157#396—22no—
158#396—22no—
159#394—22no—
160#394—22no—
161#391C2312yes—
162#391C2312yes—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#390C2312yes—
167#389—24no—
168#389—24no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#385—22no—
192#385—22no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382C2312yes—
196#380—21no—
197#380—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—24no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215C2×#95—21no—
216C2×#95—21no—
217C2×#95D411yescharacteristic
218C2×#95D411yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2312yes—
224#336C2312yes—
225#336C2314yes—
226C22×Q8C2412yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—42no—
246#127—42no—
247#126—48no—
248#125—216no—
249#125—24no—
250#125—28no—
251#125—24no—
252#125C2×D4116yes—
253#124—24no—
254#124—24no—
255#124—44no—
256#123—48no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259#121—24no—
260#121—24no—
261#121—44no—
262#121—44no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—48no—
268#117—48no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D414yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42—82no—
330C2×C42—82no—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D414yes—
336C2×C42#4114yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341#107—28no—
342#107—28no—
343#106—28no—
344#106—28no—
345C4⋊C8—28no—
346C4⋊C8—44no—
347C4⋊C8—42no—
348C4⋊C8—42no—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—44no—
353#102—28no—
354#102—28no—
355#102—28no—
356#102—44no—
357C22⋊C8—24no—
358C22⋊C8—24no—
359C4×C8—28no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—22no—
378C24C22×D411yescharacteristic
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—81no—
395C2×D4—81no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—82no—
434C22×C4—82no—
435C22×C4#12114yes—
436C22×C4#12114yes—
437C22×C4#13514yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2511yescharacteristic
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C4⋊C4—24no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4C22×D414yes—
488C4⋊C4C22×D414yes—
489C4⋊C42−1+414yes—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—41no—
506C42—42no—
507C42—44no—
508C42—44no—
509C42—44no—
510C42—84no—
511C42—84no—
512C42#12118yes—
513C42C22×D414yes—
514C42C2×#4114yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C2×#12112yes—
527C23C2×#13512yes—
528C23#57512yes—
529C23C23×D411yescharacteristic
530Q8—44no—
531Q8—44no—
532D4—44no—
533D4—44no—
534D4—88no—
535D4—162no—
536D4—162no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—48no—
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—88no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—84no—
593C2×C4C2×#12114yes—
594C2×C4C2×#12112yes—
595C2×C4C2×#12114yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C23×D411yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×2−1+412yes—
601C2×C4#58512yes—
602C8—44no—
603C8—44no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#303514yes—
618C22C22×#12111yescharacteristic
619C22#348511yescharacteristic
620C22#348611yescharacteristic
621C22C22×#13511yescharacteristic
622C22C2×#57511yescharacteristic
623C22#358511yescharacteristic
624C22#358611yescharacteristic
625C4—22no—
626C4—22no—
627C4—24no—
628C4—22no—
629C4—24no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#346112yes—
642C4#349712yes—
643C2—24no—
644C2—162no—
645C2#3515311yescharacteristic
646C2C2×#303511yescharacteristic
647C2#6042511yescharacteristic
648C2#6169611yescharacteristic
649C2#6171811yescharacteristic
650C2#6181911yescharacteristic
651C2#6183211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3232C2semidirect
3. C2×#2370C2semidirect
4. #33110C2semidirect
5. #33108C2semidirect
7. #29760C2semidirect
8. #29757C2semidirect
10. #25460C2semidirect
11. #25458C2semidirect
12. #25283C2semidirect
13. #25281C2semidirect
14. #21951C2semidirect
15. #21388C2semidirect
95. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32322
#351032
C2×#23702
#331102
#331082
#331062
#297602
#297572
#297552
#254602
#254582
#252832
#252812
#219512
#213882