Finite Group Databasebeta

← #6506775

Subgroups — #6506775

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic
3#29303C212yesmaximal, complemented
4#29297C211yesmaximal, characteristic, complemented
5#29296C211yesmaximal, characteristic, complemented
6#28784C214yesmaximal, complemented
7#28782C214yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27948C211yesmaximal, characteristic, complemented
10#26078C214yesmaximal, complemented
11#26077C214yesmaximal, complemented
12#25816C212yesmaximal, complemented
13#21944C212yesmaximal
14#20844C212yesmaximal
15#20524C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C23×C4⋊C4C2211yescharacteristic
19C2×#501C2214yescomplemented
20C2×#428C2214yes—
21C2×#407C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2626—21no—
26#2626—21no—
27#2608—22no—
28#2607—22no—
29#2441C2212yes—
30#2441C2212yes—
31#2440—22no—
32#2440—22no—
33#2440C2214yes—
34#2440C2214yes—
35#2439C2212yes—
36#2439C2212yes—
37C2×#398—21no—
38C2×#395—21no—
39#2329—21no—
40#2329—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#385—21no—
44C2×#384C2211yescharacteristic
45#2280C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#2085C2214yescomplemented
49#2084C2214yescomplemented
50#2084C2212yescomplemented
51#2083C2214yescomplemented
52#2083C2212yescomplemented
53#2059C2214yescomplemented
54#2058C2214yescomplemented
55#2058C2212yescomplemented
56#2057C2212yescomplemented
57#2057C2214yescomplemented
58#1993C2214yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1944C2214yescomplemented
62#1943C2214yescomplemented
63#1942C2214yescomplemented
64#1941C2214yescomplemented
65#1854C2214yes—
66#1854C2214yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2214yes—
75#1852C2212yes—
76#1852C2212yes—
77#1845C2212yescomplemented
78#1844C2212yescomplemented
79#1809C2214yes—
80#1809C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86#1736C2214yes—
87C23×Q8—21no—
88C24×C4—21no—
89C24×C4C2311yescharacteristic
90C22×M16C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125C2312yes—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
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—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#501—48no—
123#428—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132#407Q8116yescomplemented
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138C2×C4×C8C2312yes—
139#399—22no—
140#399—22no—
141#398—24no—
142#398—24no—
143#396—22no—
144#396—22no—
145#396—22no—
146#396—22no—
147#395—24no—
148#395—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#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2314yes—
167#389C2312yes—
168#389C2312yes—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#385—22no—
176#385—22no—
177#385—24no—
178#385—24no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2312yes—
185#382C2312yes—
186#381—22no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380C2312yes—
191#380C2312yes—
192#380C2314yes—
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#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218C25C2×D411yescharacteristic
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—44no—
223C22×Q8—44no—
224C22×Q8—82no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—82no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×Q818yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C2×M16—44no—
245C2×M16—44no—
246C2×M16C2×Q8116yescomplemented
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—28no—
250C4⋊Q8—44no—
251C4⋊Q8—42no—
252C4⋊Q8—42no—
253C4⋊Q8—48no—
254#125—44no—
255C22⋊Q8—416no—
256C22⋊Q8—416no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259#117—44no—
260#117—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
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—44no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—28no—
332#107—44no—
333#107—44no—
334#107—44no—
335#106—28no—
336#106—44no—
337#106—44no—
338#106—44no—
339C22⋊C8C2×Q818yes—
340C22⋊C8C2×Q818yes—
341C22⋊C8C2×Q8116yescomplemented
342#97—44no—
343C4×C8—42no—
344C4×C8—42no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24—22no—
364C24—22no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24C4⋊Q814yes—
370C24C22×D412yes—
371C24C22×D411yescharacteristic
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—84no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×Q8—816no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—416no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—48no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—84no—
424C22×C4—816no—
425C22×C4C4⋊Q818yes—
426C22×C4C4⋊Q814yes—
427C22×C4AGL(1,ℤ/8)14yes—
428C22×C4#13714yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×Q814yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q814yes—
440C22×C4C2511yescharacteristic
441M16—416no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8C22×Q818yes—
447C2×C8C22×Q818yes—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4H(5,𝔽2)14yes—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C22⋊C4—82no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—42no—
499C42—82no—
500C42—82no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×C4⋊Q814yes—
519C23C2×C4⋊Q812yes—
520C23C2×AGL(1,ℤ/8)12yes—
521C23C2×#13712yes—
522C23C23×D411yescharacteristic
523Q8—816no—
524Q8—162no—
525Q8—162no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—48no—
554C2×C4—44no—
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—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—816no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4C2×C4⋊Q814yes—
583C2×C4C2×C4⋊Q812yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×AGL(1,ℤ/8)12yes—
586C2×C4C2×#13712yes—
587C2×C4C2×#13712yes—
588C2×C4C23×Q812yes—
589C2×C4C2×H(5,𝔽2)12yes—
590C2×C4#58512yes—
591C8—28no—
592C8—28no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—416no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22#316514yes—
608C22C22×C4⋊Q811yescharacteristic
609C22#348411yescharacteristic
610C22#348511yescharacteristic
611C22C22×AGL(1,ℤ/8)11yescharacteristic
612C22C22×#13711yescharacteristic
613C22#358511yescharacteristic
614C22#358611yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—24no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4#350114yes—
632C2—24no—
633C2—44no—
634C2#3539111yescharacteristic
635C2C2×#316511yescharacteristic
636C2#6066411yescharacteristic
637C2#6158211yescharacteristic
638C2#6158411yescharacteristic
639C2#6170611yescharacteristic
640C2#6170911yescharacteristic
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. #29297C2semidirect
5. #29296C2semidirect
6. #28784C2semidirect
7. #28782C2semidirect
8. #27950C2semidirect
9. #27948C2semidirect
10. #26078C2semidirect
11. #26077C2semidirect
12. #25816C2semidirect
19. C2×#501C22semidirect
48. #2085C22semidirect
49. #2084C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
52. #2083C22semidirect
53. #2059C22semidirect
54. #2058C22semidirect
55. #2058C22semidirect
56. #2057C22semidirect
57. #2057C22semidirect
58. #1993C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1944C22semidirect
62. #1943C22semidirect
63. #1942C22semidirect
64. #1941C22semidirect
77. #1845C22semidirect
78. #1844C22semidirect
132. #407Q8semidirect
246. C2×M16C2×Q8semidirect
341. C22⋊C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24402
#293032
#292972
#292962
#287842
#287822
#279502
#279482
#260782
#260772
#258162
#219442
#208442
#205242