Finite Group Databasebeta

← #6528571

Subgroups — #6528571

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35103C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29329C211yesmaximal, characteristic, complemented
5#29328C211yesmaximal, characteristic
6#27978C212yesmaximal, complemented
7#27973C211yesmaximal, characteristic
8#27972C211yesmaximal, characteristic, complemented
9#27765C214yesmaximal, complemented
10#27763C214yesmaximal, complemented
11#27741C214yesmaximal, complemented
12#27739C214yesmaximal, complemented
13#21867C212yesmaximal, complemented
14#21391C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#529C2212yes—
17C22×#125—21no—
18C2×#432C2212yes—
19#2857C2211yescharacteristic
20#2856C2211yescharacteristic
21#2850C2212yes—
22#2618—21no—
23#2618—21no—
24#2601—22no—
25#2601—22no—
26#2566C2212yes—
27#2565C2212yes—
28#2552—22no—
29#2552—22no—
30#2552C2214yes—
31#2552C2214yes—
32#2545C2212yes—
33#2544C2212yes—
34C2×#398—21no—
35C2×#398—21no—
36#2325—21no—
37#2325—21no—
38C2×#389—21no—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2100C2212yes—
48#2099C2212yes—
49#2072C2214yes—
50#2072C2214yes—
51#2071C2214yes—
52#2071C2214yes—
53#2043C2212yes—
54#2042C2214yes—
55#2041C2212yes—
56#1992C2214yes—
57#1989C2214yes—
58#1983C2214yes—
59#1959C2214yes—
60#1959C2214yes—
61#1957C2212yes—
62#1956C2212yes—
63#1936C2214yes—
64#1935C2214yes—
65#1934C2214yes—
66#1933C2214yes—
67#1874C2212yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1872C2212yes—
72#1872C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1836C2214yes—
76#1834C2212yes—
77#1834C2212yes—
78#1803C2214yes—
79#1802C2214yes—
80#1771C2214yes—
81#1771C2214yes—
82#1763C2214yes—
83#1762C2214yes—
84#1749C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C2×C4⋊Q8C2311yescharacteristic
88C2×#127C2311yescharacteristic
89#529—28no—
90#529—28no—
91C2×#126—22no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—48no—
95C2×#125C2312yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#123—22no—
99C2×C22⋊Q8C2312yes—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#432—28no—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2314yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#398—22no—
142#398—22no—
143#398—24no—
144#398—24no—
145#398—24no—
146#398—24no—
147#396—22no—
148#396—22no—
149#396—22no—
150#396—22no—
151#394—22no—
152#394—22no—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#389—24no—
160#389—24no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2314yes—
188#384C2312yes—
189#384C2314yes—
190#384C2312yes—
191#382C2312yes—
192#380—21no—
193#380—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2314yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95D411yescharacteristic
204C2×#95D411yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2314yes—
215#336C2312yes—
216C22×Q8C2412yes—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—21no—
220C22×D4C2412yes—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—22no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2411yescharacteristic
233C22×C8C2414yes—
234C22×C8C2414yes—
235C4⋊Q8—42no—
236#127—42no—
237#126—48no—
238#125—24no—
239#125—24no—
240#125—416no—
241#125—416no—
242#124—24no—
243#124—24no—
244#124—44no—
245#123—48no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248#121—24no—
249#121—24no—
250#121—44no—
251#121—44no—
252C4×Q8—24no—
253C4×Q8—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—48no—
257#117—48no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
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—22no—
317C2×C22⋊C4—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42#4118yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332C4⋊C8—28no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#102—44no—
338#102—44no—
339#102—44no—
340#102—44no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343#97—28no—
344#97—44no—
345C4×C8—42no—
346C4×C8—42no—
347#95—24no—
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#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—22no—
362C24C22×D411yescharacteristic
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—81no—
379C2×D4—81no—
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—22no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
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—44no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4#12414yes—
419C22×C4#12414yes—
420C22×C4AGL(1,ℤ/8)14yes—
421C22×C4#13714yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4114yes—
432C22×C4C2511yescharacteristic
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—84no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—48no—
495C42—44no—
496C42#12418yes—
497C42C2×#4114yes—
498C42C2×#4114yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23C2×#12412yes—
511C23C2×AGL(1,ℤ/8)12yes—
512C23C2×#13712yes—
513C23C23×D411yescharacteristic
514Q8—44no—
515Q8—44no—
516D4—44no—
517D4—44no—
518D4—88no—
519D4—162no—
520D4—162no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—28no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
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—88no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4C2×#12414yes—
575C2×C4C2×#12412yes—
576C2×C4C2×#12414yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4C22×#4112yes—
582C2×C4#58512yes—
583C2×C4#58512yes—
584C8—44no—
585C8—44no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22#314114yes—
600C22C22×#12411yescharacteristic
601C22#348511yescharacteristic
602C22#348711yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22C22×#13711yescharacteristic
605C22#358511yescharacteristic
606C22#358611yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—24no—
613C4—24no—
614C4—42no—
615C4—42no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#351214yes—
624C2—24no—
625C2—162no—
626C2#3564911yescharacteristic
627C2C2×#314111yescharacteristic
628C2#6062111yescharacteristic
629C2#6170611yescharacteristic
630C2#6170911yescharacteristic
631C2#6189011yescharacteristic
632C2#6189311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2552C2semidirect
3. #29336C2semidirect
4. #29329C2semidirect
6. #27978C2semidirect
8. #27972C2semidirect
9. #27765C2semidirect
10. #27763C2semidirect
11. #27741C2semidirect
12. #27739C2semidirect
13. #21867C2semidirect
14. #21391C2semidirect
15. #20534C2semidirect

Maximal subgroups

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

subgroupindex
#351032
C2×#25522
#293362
#293292
#293282
#279782
#279732
#279722
#277652
#277632
#277412
#277392
#218672
#213912
#205342