Finite Group Databasebeta

← #6643206

Subgroups — #6643206

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2497C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#31712C214yesmaximal, complemented
5#31710C214yesmaximal, complemented
6#31440C214yesmaximal, complemented
7#31438C214yesmaximal, complemented
8#29973C212yesmaximal, complemented
9#29303C212yesmaximal, complemented
10#27965C212yesmaximal, complemented
11#27603C211yesmaximal, characteristic
12#27603C211yesmaximal, characteristic
13#27601C211yesmaximal, characteristic
14#27601C211yesmaximal, characteristic
15#21906C211yesmaximal, characteristic
16#2851C2212yes—
17#2851C2212yes—
18#2646—21no—
19#2646—21no—
20#2569—22no—
21#2569—22no—
22#2497—22no—
23#2497—22no—
24#2497C2214yes—
25#2497C2214yes—
26#2442C2212yes—
27#2442C2212yes—
28#2442C2212yes—
29#2442C2212yes—
30C2×#400—21no—
31C2×#396—21no—
32#2315—21no—
33#2315—21no—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388—21no—
37C2×#387—21no—
38C2×#384C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#2101C2214yescomplemented
46#2099C2214yescomplemented
47#2099C2214yescomplemented
48#2088C2214yescomplemented
49#2086C2214yescomplemented
50#2085C2214yescomplemented
51#2077C2214yes—
52#2077C2214yes—
53#2076C2214yes—
54#2076C2214yes—
55#2059C2214yescomplemented
56#2059C2214yescomplemented
57#1992C2214yescomplemented
58#1989C2214yescomplemented
59#1949C2214yescomplemented
60#1944C2214yescomplemented
61#1943C2214yescomplemented
62#1924C2214yes—
63#1924C2214yes—
64#1920C2214yescomplemented
65#1918C2214yescomplemented
66#1853C2212yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2212yes—
70#1852C2212yes—
71#1852C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1836C2214yescomplemented
75#1824C2212yes—
76#1824C2212yes—
77#1824C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1797C2212yes—
84#1779C2211yescharacteristic
85#1779C2211yescharacteristic
86#1749C2212yes—
87C24×C4C2311yescharacteristic
88C2×#125C2312yes—
89C2×#125C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C2×#107C2314yes—
112C2×#107C2314yes—
113C2×#106C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2314yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2312yes—
122C2×#97C2314yes—
123#400—22no—
124#400—22no—
125#400—22no—
126#400—22no—
127#400—24no—
128#400—24no—
129#398—22no—
130#398—22no—
131#398—22no—
132#398—22no—
133#396—24no—
134#396—24no—
135#389—22no—
136#389—22no—
137#389C2314yes—
138#389C2314yes—
139#389C2312yes—
140#389C2312yes—
141#389C2312yes—
142#389C2312yes—
143#388—24no—
144#388—24no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—24no—
150#387—24no—
151#385—22no—
152#385—22no—
153#385—22no—
154#385—22no—
155#385—22no—
156#385—22no—
157#385—22no—
158#385—22no—
159#384—22no—
160#384C2314yes—
161#382C2312yes—
162#382C2312yes—
163#380—21no—
164#380—21no—
165#380—21no—
166#380—21no—
167#380—21no—
168#380—21no—
169#380—21no—
170#380—21no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380C2312yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182#380C2312yes—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188C4×C22⋊C4C2312yes—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95D411yescharacteristic
197C2×#95D411yescharacteristic
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2314yes—
211#336C2314yes—
212C25C2×D411yescharacteristic
213C23×C4—24no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C22×C8C2414yes—
235C22×C8C2414yes—
236#125—44no—
237#125—44no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—82no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C22⋊C4C2×D412yes—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—82no—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2412yes—
303C2×C42C2412yes—
304#107—44no—
305#107—44no—
306#106—44no—
307#106—44no—
308C4⋊C8—44no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—42no—
314C22⋊C8—42no—
315#97—44no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338#95C2×D412yes—
339C24—22no—
340C24—22no—
341C24—41no—
342C24—41no—
343C24—41no—
344C24—41no—
345C24#13714yes—
346C24C22×D411yescharacteristic
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—88no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C42−1+414yes—
419C22×C4C2511yescharacteristic
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C42—24no—
467C42—24no—
468C42—42no—
469C42—42no—
470C42—84no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—28no—
488C23—44no—
489C23—44no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23#53612yes—
495C23C2×#13712yes—
496C23C2×#13712yes—
497C23C2×#13712yes—
498C23#57512yes—
499C23C23×D411yescharacteristic
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—28no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—162no—
559C2×C4—162no—
560C2×C4—162no—
561C2×C4—162no—
562C2×C4#53612yes—
563C2×C4C2×2−1+412yes—
564C2×C4#58512yes—
565C2×C4#58512yes—
566C8—44no—
567C8—44no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—88no—
589C22#323614yes—
590C22C2×#53611yescharacteristic
591C22#348511yescharacteristic
592C22#348511yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57511yescharacteristic
595C22#358511yescharacteristic
596C22#358511yescharacteristic
597C4—22no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4#356414yes—
613C2—24no—
614C2—42no—
615C2—42no—
616C2#3697011yescharacteristic
617C2C2×#323611yescharacteristic
618C2#6086811yescharacteristic
619C2#6170911yescharacteristic
620C2#6170911yescharacteristic
621C2#6171011yescharacteristic
622C2#6171811yescharacteristic
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. #33150C2semidirect
4. #31712C2semidirect
5. #31710C2semidirect
6. #31440C2semidirect
7. #31438C2semidirect
8. #29973C2semidirect
9. #29303C2semidirect
10. #27965C2semidirect
45. #2101C22semidirect
46. #2099C22semidirect
47. #2099C22semidirect
48. #2088C22semidirect
49. #2086C22semidirect
50. #2085C22semidirect
55. #2059C22semidirect
56. #2059C22semidirect
57. #1992C22semidirect
58. #1989C22semidirect
59. #1949C22semidirect
60. #1944C22semidirect
61. #1943C22semidirect
64. #1920C22semidirect
65. #1918C22semidirect
74. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#24972
#331502
#317122
#317102
#314402
#314382
#299732
#293032
#279652
#276032
#276012
#219062