Finite Group Databasebeta

← #6506768

Subgroups — #6506768

Order 512.

1618 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2441C211yesmaximal, characteristic
3#29303C212yesmaximal, complemented
4#29302C211yesmaximal, characteristic, complemented
5#29301C211yesmaximal, characteristic, complemented
6#28780C214yesmaximal, complemented
7#28778C214yesmaximal, complemented
8#27953C212yesmaximal, complemented
9#26070C214yesmaximal, complemented
10#26069C214yesmaximal, complemented
11#25819C211yesmaximal, characteristic, complemented
12#25819C211yesmaximal, characteristic, complemented
13#21941C212yesmaximal
14#20843C212yesmaximal
15#20526C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#501C2212yescomplemented
19C2×#501C2212yescomplemented
20C2×#427C2214yes—
21C22×C22⋊C8C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2626—21no—
26#2626—21no—
27#2608—22no—
28#2607—22no—
29#2441—22no—
30#2441—22no—
31#2441C2212yes—
32#2441C2214yes—
33#2441C2212yes—
34#2441C2214yes—
35#2439C2212yes—
36#2439C2212yes—
37C2×#399—21no—
38C2×#396—21no—
39#2329—21no—
40#2329—21no—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#384C2211yescharacteristic
46#2280C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2085C2214yescomplemented
50#2084C2214yescomplemented
51#2084C2212yescomplemented
52#2083C2214yescomplemented
53#2083C2212yescomplemented
54#2059C2214yescomplemented
55#2059C2214yescomplemented
56#2059C2214yescomplemented
57#2058C2212yescomplemented
58#2057C2212yescomplemented
59#1992C2214yescomplemented
60#1992C2212yescomplemented
61#1992C2212yescomplemented
62#1944C2214yescomplemented
63#1944C2214yescomplemented
64#1943C2214yescomplemented
65#1943C2214yescomplemented
66#1854C2214yes—
67#1854C2214yes—
68#1854C2214yes—
69#1854C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1847C2214yescomplemented
79#1808C2214yes—
80#1807C2214yes—
81#1752C2214yes—
82#1752C2214yes—
83#1749C2212yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86#1735C2214yes—
87C24×C4C2311yescharacteristic
88C23×C8C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125—21no—
92C2×#125—21no—
93C2×#125—44no—
94C2×#125C2312yes—
95C2×#125C2312yes—
96C2×#125C2312yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×#117—22no—
100C2×#117—44no—
101C2×#117C2314yes—
102C2×#117C2312yes—
103C2×#117C2312yes—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#501—44no—
122#501—44no—
123#427—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C22⋊C8Q8116yescomplemented
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2314yes—
137C2×#97C2312yes—
138C2×#97C2312yes—
139#399—22no—
140#399—22no—
141#399—24no—
142#399—24no—
143#396—22no—
144#396—22no—
145#396—22no—
146#396—22no—
147#396—24no—
148#396—24no—
149#393—22no—
150#393—22no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391—22no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—24no—
163#389—24no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384C2311yescharacteristic
183#384C2314yes—
184#384C2311yescharacteristic
185#382C2312yes—
186#381—22no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380C2312yes—
191#380C2314yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#379—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4C2312yes—
201C4×C22⋊C4C2312yes—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95D411yescharacteristic
206C2×#95D411yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C22×Q8—21no—
220C22×Q8—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D412yes—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×Q818yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C22×C8—44no—
237C22×C8—44no—
238C22×C8C2×Q8116yes—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—42no—
242C4⋊Q8—42no—
243#125—24no—
244#125—24no—
245#125—42no—
246#125—42no—
247#125—44no—
248#125—44no—
249#125—44no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252#117—28no—
253#117—44no—
254#117—48no—
255#117—44no—
256#117—44no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—82no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—82no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2412yes—
328#107—28no—
329#107—44no—
330#107—44no—
331#107—44no—
332#106—28no—
333#106—44no—
334#106—44no—
335#106—44no—
336C22⋊C8C2×Q818yes—
337C22⋊C8C2×Q818yes—
338C22⋊C8C2×Q8116yes—
339#97—44no—
340#97—42no—
341#97—42no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—24no—
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—
355#95—22no—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24—22no—
361C24—22no—
362C24—41no—
363C24—41no—
364C24—41no—
365C24—41no—
366C24C4⋊Q814yes—
367C24C22×D412yes—
368C24C22×D411yescharacteristic
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×Q8—81no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—416no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C4⋊Q818yes—
417C22×C4C4⋊Q814yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q814yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q814yes—
429C22×C4C2511yescharacteristic
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—416no—
435C2×C8C22×Q818yes—
436C2×C8C22×Q818yes—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—81no—
462C4⋊C4—82no—
463C4⋊C4—81no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4H(5,𝔽2)14yes—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—84no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—42no—
489C42—41no—
490C42—41no—
491C42—48no—
492C42—82no—
493C42—82no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—44no—
506C23—44no—
507C23—82no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23C2×C4⋊Q814yes—
512C23C2×C4⋊Q812yes—
513C23#57512yes—
514C23#57512yes—
515C23C23×D411yescharacteristic
516Q8—162no—
517Q8—162no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4C2×C4⋊Q814yes—
578C2×C4C2×C4⋊Q812yes—
579C2×C4C23×Q812yes—
580C2×C4C2×H(5,𝔽2)12yes—
581C2×C4#58512yes—
582C8—28no—
583C8—28no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—88no—
598C22#316214yes—
599C22C22×C4⋊Q811yescharacteristic
600C22#348411yescharacteristic
601C22#348511yescharacteristic
602C22C2×#57511yescharacteristic
603C22C2×#57511yescharacteristic
604C22#358511yescharacteristic
605C22#358611yescharacteristic
606C4—24no—
607C4—22no—
608C4—42no—
609C4—42no—
610C4—41no—
611C4—41no—
612C4—44no—
613C4—41no—
614C4—41no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#350114yes—
623C2—24no—
624C2—44no—
625C2#3539111yescharacteristic
626C2C2×#316211yescharacteristic
627C2#6066411yescharacteristic
628C2#6157511yescharacteristic
629C2#6157711yescharacteristic
630C2#6171111yescharacteristic
631C2#6171311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29303C2semidirect
4. #29302C2semidirect
5. #29301C2semidirect
6. #28780C2semidirect
7. #28778C2semidirect
8. #27953C2semidirect
9. #26070C2semidirect
10. #26069C2semidirect
11. #25819C2semidirect
12. #25819C2semidirect
18. C2×#501C22semidirect
19. C2×#501C22semidirect
49. #2085C22semidirect
50. #2084C22semidirect
51. #2084C22semidirect
52. #2083C22semidirect
53. #2083C22semidirect
54. #2059C22semidirect
55. #2059C22semidirect
56. #2059C22semidirect
57. #2058C22semidirect
58. #2057C22semidirect
59. #1992C22semidirect
60. #1992C22semidirect
61. #1992C22semidirect
62. #1944C22semidirect
63. #1944C22semidirect
64. #1943C22semidirect
65. #1943C22semidirect
78. #1847C22semidirect
132. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24412
#293032
#293022
#293012
#287802
#287782
#279532
#260702
#260692
#258192
#219412
#208432
#205262