Finite Group Databasebeta

← #6470957

Subgroups — #6470957

Order 512.

1752 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#33146C211yesmaximal, characteristic, complemented
5#32960C212yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#29953C211yesmaximal, characteristic, complemented
8#29951C211yesmaximal, characteristic, complemented
9#26415C214yesmaximal, complemented
10#26413C214yesmaximal, complemented
11#26351C214yesmaximal, complemented
12#26349C214yesmaximal, complemented
13#21930C211yesmaximal, characteristic
14#21762C212yesmaximal
15#20843C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#479C2214yescomplemented
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#2439C2212yes—
32#2439C2212yes—
33#2362—22no—
34#2362—22no—
35#2362C2214yes—
36#2362C2214yes—
37#2329—21no—
38#2329—21no—
39C2×#391C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#388—21no—
42C2×#388—21no—
43C2×#387—21no—
44#2280C2212yes—
45C2×C4×C4⋊C4—21no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#2101C2214yescomplemented
49#2096C2212yescomplemented
50#2095C2212yescomplemented
51#2088C2214yescomplemented
52#2087C2214yescomplemented
53#2068C2214yescomplemented
54#2066C2214yescomplemented
55#2059C2214yescomplemented
56#2024C2212yescomplemented
57#2024C2212yescomplemented
58#1988C2212yescomplemented
59#1987C2212yescomplemented
60#1948C2212yescomplemented
61#1947C2212yescomplemented
62#1920C2214yescomplemented
63#1918C2214yescomplemented
64#1892C2214yes—
65#1892C2214yes—
66#1873C2214yes—
67#1873C2214yes—
68#1865C2214yescomplemented
69#1863C2214yescomplemented
70#1854C2214yes—
71#1854C2214yes—
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1850C2214yes—
77#1850C2214yes—
78#1823C2212yes—
79#1823C2212yes—
80#1805C2214yescomplemented
81#1797C2211yescharacteristic
82#1797C2212yes—
83#1797C2211yescharacteristic
84#1787C2212yes—
85#1787C2212yes—
86#1735C2214yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C23×C8C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125C2312yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8Q818yescomplemented
98C2×C22⋊Q8C2314yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×#117—22no—
103C2×#117—44no—
104C2×#117C2314yes—
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42—21no—
118C22×C42—21no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#479—48no—
123#427—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×C22⋊C8—28no—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139#399—22no—
140#399—22no—
141#396—22no—
142#396—22no—
143#396—22no—
144#396—22no—
145#393—22no—
146#393—22no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2312yes—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—24no—
166#388—24no—
167#388—24no—
168#388—24no—
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#382C2312yes—
182#381—22no—
183#380—21no—
184#380—21no—
185#380C2312yes—
186#380C2312yes—
187#380C2312yes—
188#380C2312yes—
189#379—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—24no—
193C4×C4⋊C4—24no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
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×#95—21no—
206C2×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C25C2×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×Q814yes—
227C22×Q8C2412yes—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×Q814yes—
241C23×C4#4114yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C22×C8—28no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8C2414yes—
248C22×C8C2414yes—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251#125—44no—
252C22⋊Q8—216no—
253C22⋊Q8—28no—
254C22⋊Q8—44no—
255C22⋊Q8—44no—
256C22⋊Q8C2×Q8116yes—
257C4×Q8—28no—
258C4×Q8—44no—
259C4×Q8—48no—
260#117—28no—
261#117—44no—
262#117—48no—
263#117—44no—
264#117—44no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
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—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×Q814yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—28no—
331#107—44no—
332#106—28no—
333#106—44no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336C4⋊C8—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340#103—44no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—28no—
344#95—22no—
345#95—22no—
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#95—22no—
360#95—22no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365C24—22no—
366C24—22no—
367C24—41no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24C22⋊Q814yes—
372C24C22×D412yes—
373C24C22×D411yescharacteristic
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—48no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8C22×Q814yes—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
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×C4C22⋊Q814yes—
431C22×C4C22⋊Q818yes—
432C22×C4#13514yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×Q812yes—
442C22×C4C22×Q814yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2511yescharacteristic
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—416no—
453C4⋊C4—24no—
454C4⋊C4—216no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4C22×Q814yes—
479C4⋊C4H(5,𝔽2)14yes—
480C22⋊C4—216no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42—42no—
498C42—48no—
499C42—48no—
500C42—44no—
501C42—44no—
502C42—82no—
503C42—82no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—28no—
515C23—44no—
516C23—44no—
517C23—82no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23C2×C22⋊Q814yes—
522C23C2×C22⋊Q812yes—
523C23C2×#13512yes—
524C23#57512yes—
525C23C23×D411yescharacteristic
526Q8—28no—
527Q8—48no—
528Q8—44no—
529Q8—162no—
530Q8—162no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—48no—
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—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4C2×C22⋊Q814yes—
590C2×C4C2×C22⋊Q812yes—
591C2×C4C2×#13512yes—
592C2×C4C2×#13512yes—
593C2×C4C23×Q811yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4C2×H(5,𝔽2)12yes—
596C2×C4#58512yes—
597C8—44no—
598C8—44no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—88no—
613C22#307614yes—
614C22C22×C22⋊Q811yescharacteristic
615C22#348411yescharacteristic
616C22#348511yescharacteristic
617C22C22×#13511yescharacteristic
618C22C2×#57511yescharacteristic
619C22#358611yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—24no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—44no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4#345912yes—
638C4#350112yes—
639C2—24no—
640C2—44no—
641C2#3498511yescharacteristic
642C2C2×#307611yescharacteristic
643C2#6049111yescharacteristic
644C2#6156511yescharacteristic
645C2#6157811yescharacteristic
646C2#6169611yescharacteristic
647C2#6171211yescharacteristic
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. #33148C2semidirect
4. #33146C2semidirect
5. #32960C2semidirect
6. #29972C2semidirect
7. #29953C2semidirect
8. #29951C2semidirect
9. #26415C2semidirect
10. #26413C2semidirect
11. #26351C2semidirect
12. #26349C2semidirect
19. C2×#479C22semidirect
48. #2101C22semidirect
49. #2096C22semidirect
50. #2095C22semidirect
51. #2088C22semidirect
52. #2087C22semidirect
53. #2068C22semidirect
54. #2066C22semidirect
55. #2059C22semidirect
56. #2024C22semidirect
57. #2024C22semidirect
58. #1988C22semidirect
59. #1987C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
62. #1920C22semidirect
63. #1918C22semidirect
68. #1865C22semidirect
69. #1863C22semidirect
80. #1805C22semidirect
97. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#23622
#331482
#331462
#329602
#299722
#299532
#299512
#264152
#264132
#263512
#263492
#219302
#217622
#208432