Finite Group Databasebeta

← #6506761

Subgroups — #6506761

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic
3#29298C212yesmaximal, complemented
4#29297C211yesmaximal, characteristic, complemented
5#29296C211yesmaximal, characteristic, complemented
6#28774C214yesmaximal, complemented
7#28773C214yesmaximal, complemented
8#27952C212yesmaximal, complemented
9#26064C214yesmaximal, complemented
10#26061C214yesmaximal, complemented
11#25813C211yesmaximal, characteristic, complemented
12#25811C211yesmaximal, characteristic, complemented
13#21941C212yesmaximal
14#20843C212yesmaximal
15#20523C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#500C2212yescomplemented
19C2×#498C2212yescomplemented
20C2×#427C2214yes—
21C22×C22⋊C8C2212yes—
22#2857C2211yescharacteristic
23#2857C2211yescharacteristic
24#2851C2212yes—
25#2626—21no—
26#2626—21no—
27#2608—22no—
28#2607—22no—
29#2441C2212yes—
30#2441C2212yes—
31#2439—22no—
32#2439—22no—
33#2439C2214yes—
34#2439C2214yes—
35#2439C2212yes—
36#2439C2212yes—
37C2×#396—21no—
38C2×#393—21no—
39#2329—21no—
40#2329—21no—
41C2×#391—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#382C2211yescharacteristic
46#2280C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2085C2214yescomplemented
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2084C2212yescomplemented
53#2083C2212yescomplemented
54#2059C2214yescomplemented
55#2058C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2214yescomplemented
58#2057C2212yescomplemented
59#1991C2214yescomplemented
60#1990C2212yescomplemented
61#1990C2212yescomplemented
62#1942C2214yescomplemented
63#1942C2214yescomplemented
64#1941C2214yescomplemented
65#1941C2214yescomplemented
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#1846C2214yescomplemented
79#1808C2214yes—
80#1807C2214yes—
81C8×C22⋊C4C2214yes—
82C8×C22⋊C4C2214yes—
83C4×C22⋊C8C2212yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86#1735C2214yes—
87C23×Q8—21no—
88C24×C4C2311yescharacteristic
89C23×C8C2314yes—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8—21no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125C2312yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×#117—22no—
101C2×#117—44no—
102C2×#117C2314yes—
103C2×#117C2312yes—
104C2×#117C2312yes—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#500—44no—
120#498—44no—
121#427—48no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C22⋊C8Q8116yescomplemented
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C4×C8C2314yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2312yes—
137#399—22no—
138#399—22no—
139#396—22no—
140#396—22no—
141#396—22no—
142#396—22no—
143#396—24no—
144#396—24no—
145#393—22no—
146#393—22no—
147#393—24no—
148#393—24no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391—24no—
154#391—24no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—21no—
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#387—24no—
176#387—24no—
177#385—22no—
178#385—22no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#382—22no—
182#382C2312yes—
183#382C2314yes—
184#381—22no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380C2312yes—
189#380C2314yes—
190#380C2312yes—
191#380C2312yes—
192#380C2312yes—
193#379—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4C2312yes—
199C4×C22⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
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—22no—
220C22×Q8—22no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—82no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×Q818yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C22×C8—44no—
242C22×C8—44no—
243C22×C8C2×Q8116yescomplemented
244C22×C8C2414yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254#125—44no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#117—28no—
258#117—44no—
259#117—48no—
260#117—44no—
261#117—44no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—82no—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
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—44no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326#107—28no—
327#107—44no—
328#107—44no—
329#107—44no—
330#106—28no—
331#106—44no—
332#106—44no—
333#106—44no—
334C22⋊C8C2×Q818yes—
335C22⋊C8C2×Q818yes—
336C22⋊C8C2×Q8116yescomplemented
337C4×C8—44no—
338C4×C8—42no—
339C4×C8—42no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
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—
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—44no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—82no—
376C2×Q8—82no—
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—42no—
402C22×C4—42no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
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×C4C4⋊Q818yes—
422C22×C4C4⋊Q814yes—
423C22×C4#13514yes—
424C22×C4#13514yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C2511yescharacteristic
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—416no—
442C2×C8C22×Q818yes—
443C2×C8C22×Q818yes—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4H(5,𝔽2)14yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—84no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—42no—
492C42—41no—
493C42—41no—
494C42—48no—
495C42—82no—
496C42—82no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—44no—
509C23—44no—
510C23—82no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23C2×C4⋊Q814yes—
515C23C2×C4⋊Q812yes—
516C23C2×#13512yes—
517C23C2×#13512yes—
518C23C23×D411yescharacteristic
519Q8—88no—
520Q8—88no—
521Q8—162no—
522Q8—162no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
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—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
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—42no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4C2×C4⋊Q814yes—
579C2×C4C2×C4⋊Q812yes—
580C2×C4C2×#13512yes—
581C2×C4C2×#13512yes—
582C2×C4C2×#13512yes—
583C2×C4C2×#13512yes—
584C2×C4C23×Q812yes—
585C2×C4C2×H(5,𝔽2)12yes—
586C2×C4#58512yes—
587C8—28no—
588C8—28no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—88no—
603C22#316014yes—
604C22C22×C4⋊Q811yescharacteristic
605C22#348411yescharacteristic
606C22#348511yescharacteristic
607C22C22×#13511yescharacteristic
608C22C22×#13511yescharacteristic
609C22#358511yescharacteristic
610C22#358611yescharacteristic
611C4—24no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#350114yes—
628C2—24no—
629C2—44no—
630C2#3539111yescharacteristic
631C2C2×#316011yescharacteristic
632C2#6066411yescharacteristic
633C2#6156111yescharacteristic
634C2#6156411yescharacteristic
635C2#6170011yescharacteristic
636C2#6170111yescharacteristic
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. #29298C2semidirect
4. #29297C2semidirect
5. #29296C2semidirect
6. #28774C2semidirect
7. #28773C2semidirect
8. #27952C2semidirect
9. #26064C2semidirect
10. #26061C2semidirect
11. #25813C2semidirect
12. #25811C2semidirect
18. C2×#500C22semidirect
19. C2×#498C22semidirect
49. #2085C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
54. #2059C22semidirect
55. #2058C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #2057C22semidirect
59. #1991C22semidirect
60. #1990C22semidirect
61. #1990C22semidirect
62. #1942C22semidirect
63. #1942C22semidirect
64. #1941C22semidirect
65. #1941C22semidirect
78. #1846C22semidirect
130. C2×C22⋊C8Q8semidirect
243. C22×C8C2×Q8semidirect
336. 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×#24392
#292982
#292972
#292962
#287742
#287732
#279522
#260642
#260612
#258132
#258112
#219412
#208432
#205232