Finite Group Databasebeta

← #6802842

Subgroups — #6802842

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#32383C214yesmaximal, complemented
5#31579C214yesmaximal, complemented
6#29761C212yesmaximal
7#28828C212yesmaximal, complemented
8#27912C212yesmaximal
9#27910C211yesmaximal, characteristic, complemented
10#27748C212yesmaximal, complemented
11#27653C212yesmaximal, complemented
12#27645C211yesmaximal, characteristic
13#26527C212yesmaximal, complemented
14#26087C212yesmaximal, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22079C212yesmaximal, complemented
17C4×#480C211yesmaximal, characteristic
18C2×#484C2212yes—
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#483C2212yes—
22C2×#480C2211yescharacteristic
23#2857C2211yescharacteristic
24#2850C2212yes—
25#2596—22no—
26#2571—22no—
27#2571—21no—
28#2571—21no—
29#2571C2214yes—
30#2571C2212yes—
31#2571C2212yes—
32#2571C2211yescharacteristic
33#2547C2211yescharacteristic
34#2514C2212yes—
35#2510—22no—
36#2494—21no—
37#2487—21no—
38#2448C2212yes—
39C2×#400—22no—
40#2365C2211yescharacteristic
41#2354C2211yescharacteristic
42#2352—21no—
43#2347—21no—
44C4×C22⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2212yes—
48C2×#384C2211yescharacteristic
49C2×#382—21no—
50C2×#382C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2097C2214yes—
54#2095C2214yes—
55#2092C2214yes—
56#2085C2214yes—
57#2085C2214yes—
58#2084C2212yes—
59#2074C2214yes—
60#2071C2212yes—
61#2067C2214yes—
62#2066C2214yes—
63#2042C2214yes—
64#1981C2212yes—
65#1958C2214yes—
66#1956C2212yes—
67#1949C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1942C2214yes—
71#1939C2212yes—
72#1933C2214yes—
73#1927C2214yes—
74#1925C2212yes—
75#1921C2214yes—
76#1918C2212yes—
77#1916C2214yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1863C2214yes—
81#1852C2211yescharacteristic
82#1846C2214yes—
83#1845C2212yes—
84#1824C2212yes—
85#1823C2211yescharacteristic
86#1805C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94C4×C22⋊C8C2212yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#124C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×#121C2311yescharacteristic
99C2×C4×Q8C2312yes—
100C2×C4×D4—21no—
101C2×C4×D4C2311yescharacteristic
102C2×#117—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—22no—
106C22×C4⋊C4D411yescharacteristic
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C42—21no—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114#484—28no—
115#484—44no—
116#483—44no—
117#481—44no—
118#480—24no—
119C2×#107C2312yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2312yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2311yescharacteristic
136C2×C22⋊C8C2311yescharacteristic
137C2×C4×C8C2312yes—
138C2×C4×C8C2314yes—
139#401—22no—
140#400—22no—
141#400—24no—
142#400—24no—
143#400—24no—
144#400—24no—
145#400—22no—
146#399—22no—
147#397—22no—
148#397—22no—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#390C2312yes—
153#390C2311yescharacteristic
154#389—22no—
155#389—22no—
156#389—24no—
157#389—21no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389C2314yes—
164#389C2314yes—
165#389C2312yes—
166#389C2311yescharacteristic
167#388—22no—
168#388—22no—
169#388—22no—
170#388—21no—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—22no—
177#386—22no—
178#386—21no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2312yes—
183#384C2312yes—
184#384C2311yescharacteristic
185#383—22no—
186#382—22no—
187#382—22no—
188#382—24no—
189#382—21no—
190#382—22no—
191#382—22no—
192#382C2314yes—
193#382C2314yes—
194#382C2312yes—
195#380—21no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380—21no—
200#380C2311yescharacteristic
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—21no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8C2412yes—
234C22×D4—21no—
235C22×D4C2411yescharacteristic
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2411yescharacteristic
245C22×C8C2414yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—24no—
249#124—44no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252#121—22no—
253#121—22no—
254C4×Q8—44no—
255C4×D4—24no—
256C4×D4—44no—
257#117—44no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—82no—
284C2×C4⋊C4—84no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2412yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—24no—
325#107—28no—
326#107—42no—
327#107—44no—
328#106—44no—
329#106—44no—
330C4⋊C8—24no—
331C4⋊C8—44no—
332#103—28no—
333#103—44no—
334#103—44no—
335#103—44no—
336#102—24no—
337#102—42no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8—22no—
341C22⋊C8#4114yes—
342C4×C8—24no—
343C4×C8—44no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—21no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C24C22×D411yescharacteristic
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—42no—
407C22×C4—44no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4#13514yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4C2511yescharacteristic
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8C2×#4114yes—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—24no—
486C42—24no—
487C42—42no—
488C42—42no—
489C42—44no—
490C42—44no—
491C42—44no—
492C42—42no—
493C42—82no—
494C422−1+414yes—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23#55012yes—
507C23C2×#13512yes—
508C23#57812yes—
509C23C23×D411yescharacteristic
510Q8—44no—
511Q8—44no—
512D4—42no—
513D4—42no—
514D4—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—88no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4#53612yes—
571C2×C4#53612yes—
572C2×C4#55012yes—
573C2×C4C2×#13512yes—
574C2×C4C2×#13512yes—
575C2×C4#57512yes—
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×H(5,𝔽2)11yescharacteristic
578C2×C4C2×2−1+412yes—
579C2×C4#58512yes—
580C8—24no—
581C8—44no—
582C8—42no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22#339314yes—
597C22C2×#53611yescharacteristic
598C22C2×#55011yescharacteristic
599C22#348511yescharacteristic
600C22C22×#13511yescharacteristic
601C22C2×#57511yescharacteristic
602C22C2×#57811yescharacteristic
603C22#358511yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—81no—
617C4—82no—
618C4—84no—
619C4—82no—
620C4#323214yes—
621C4#351512yes—
622C4#355912yes—
623C2—24no—
624C2—161no—
625C2#3913411yescharacteristic
626C2C2×#323211yescharacteristic
627C2#6087411yescharacteristic
628C2C2×#339311yescharacteristic
629C2#6140511yescharacteristic
630C2#6170011yescharacteristic
631C2#6173011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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×#2571C2semidirect
3. #33108C2semidirect
4. #32383C2semidirect
5. #31579C2semidirect
7. #28828C2semidirect
9. #27910C2semidirect
10. #27748C2semidirect
11. #27653C2semidirect
13. #26527C2semidirect
14. #26087C2semidirect
15. #22625C2semidirect
16. #22079C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25712
#331082
#323832
#315792
#297612
#288282
#279122
#279102
#277482
#276532
#276452
#265272
#260872
#226252
#220792
C4×#4802