Finite Group Databasebeta

← #6471250

Subgroups — #6471250

Order 512.

1694 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#35015C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33098C212yesmaximal
4#32954C211yesmaximal, characteristic
5#32906C211yesmaximal, characteristic
6#29951C211yesmaximal, characteristic
7#29715C211yesmaximal, characteristic
8#29627C212yesmaximal
9#27085C214yesmaximal, complemented
10#26911C214yesmaximal, complemented
11#26775C214yesmaximal, complemented
12#26607C214yesmaximal, complemented
13#22050C212yesmaximal
14#21916C211yesmaximal, characteristic
15#21455C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#475C2212yes—
20C2×#462C2212yes—
21C2×#451C2212yes—
22C2×#451C2212yes—
23C2×#441C2214yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2511C2212yes—
32#2507C2212yes—
33#2451C2212yes—
34#2446C2212yes—
35#2365—22no—
36#2365—22no—
37#2365C2214yes—
38#2365C2214yes—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389—21no—
45C2×#387—21no—
46C2×#387—21no—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2096C2212yes—
51#2072C2212yes—
52#2066C2214yes—
53#2056C2214yes—
54#2043C2214yes—
55#2043C2214yes—
56#2024C2212yes—
57#2016C2212yes—
58#2016C2214yes—
59#2016C2214yes—
60#1971C2214yes—
61#1935C2214yes—
62#1927C2214yes—
63#1927C2214yes—
64#1927C2214yes—
65#1920C2214yes—
66#1920C2214yes—
67#1917C2214yes—
68#1917C2214yes—
69#1878C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1865C2214yes—
75#1865C2214yes—
76#1853C2212yes—
77#1853C2212yes—
78#1817C2212yes—
79#1803C2212yes—
80#1803C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1787C2212yes—
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C22×Q16C2314yes—
90C22×Q16C2314yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8C2312yes—
99C2×C4×Q8—22no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—44no—
103C2×C4×Q8—44no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2312yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#475—44no—
123#462—44no—
124#451—28no—
125#451—28no—
126#441—48no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×#97C2314yes—
142#400—22no—
143#400—22no—
144#400—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#395—22no—
150#391—21no—
151#391—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2314yes—
161#391C2312yes—
162#389—24no—
163#389—24no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#385—22no—
186#385—22no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#381—22no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380C2312yes—
194C4×C4⋊C4—24no—
195C4×C4⋊C4—24no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2314yes—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8#4114yes—
231C22×Q8#4114yes—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×Q16—28no—
247C2×Q16—28no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×Q16—44no—
251C2×Q16—44no—
252C22×C8C2414yes—
253C22×C8C2414yes—
254C4⋊Q8—216no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—44no—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—48no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
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×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332#106—28no—
333#106—44no—
334C4⋊C8—42no—
335C4⋊C8—42no—
336#103—28no—
337#103—28no—
338#103—28no—
339#103—44no—
340#103—44no—
341#103—44no—
342#103—44no—
343#103—44no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346#97—44no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
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#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24C22×D411yescharacteristic
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—22no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—48no—
394C2×Q8—48no—
395C2×Q8—42no—
396C2×Q8—81no—
397C2×Q8—81no—
398C2×Q8C2×#4114yes—
399C2×Q8C2×#4114yes—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4#13714yes—
439C22×C4#13714yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2511yescharacteristic
452Q16—416no—
453Q16—88no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C4⋊C4—24no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4H(5,𝔽2)14yes—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—24no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—42no—
502C42—44no—
503C42—44no—
504C42—48no—
505C42—48no—
506C42—82no—
507C42—82no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#52512yes—
513C23C2×#13712yes—
514C23C2×#13712yes—
515C23C23×D411yescharacteristic
516Q8—28no—
517Q8—24no—
518Q8—24no—
519Q8—48no—
520Q8—48no—
521Q8—42no—
522Q8—42no—
523Q8—44no—
524Q8—44no—
525Q8—44no—
526Q8—162no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—48no—
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—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—82no—
584C2×C4—82no—
585C2×C4—84no—
586C2×C4#52512yes—
587C2×C4C2×#13714yes—
588C2×C4C2×#13712yes—
589C2×C4C2×#13712yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4#58512yes—
594C8—44no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22#311314yes—
603C22C2×#52511yescharacteristic
604C22#348411yescharacteristic
605C22#348711yescharacteristic
606C22C22×#13711yescharacteristic
607C22C22×#13711yescharacteristic
608C22#358611yescharacteristic
609C22#358611yescharacteristic
610C4—24no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—44no—
619C4—42no—
620C4—44no—
621C4—44no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#345912yes—
628C4#350512yes—
629C2—24no—
630C2#3498711yescharacteristic
631C2C2×#311311yescharacteristic
632C2#6056611yescharacteristic
633C2#6157411yescharacteristic
634C2#6158611yescharacteristic
635C2#6188911yescharacteristic
636C2#6189311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
9. #27085C2semidirect
10. #26911C2semidirect
11. #26775C2semidirect
12. #26607C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#23652
#330982
#329542
#329062
#299512
#297152
#296272
#270852
#269112
#267752
#266072
#220502
#219162
#214552