Finite Group Databasebeta

← #6471603

Subgroups — #6471603

Order 512.

1670 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic
3#33008C212yesmaximal
4#32957C211yesmaximal, characteristic
5#32906C211yesmaximal, characteristic
6#29968C211yesmaximal, characteristic
7#29913C212yesmaximal
8#29756C211yesmaximal, characteristic
9#27580C214yesmaximal, complemented
10#27324C214yesmaximal, complemented
11#26393C214yesmaximal, complemented
12#26183C214yesmaximal, complemented
13#22084C212yesmaximal
14#21916C211yesmaximal, characteristic
15#21455C212yesmaximal
16C22×C4×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#483C2212yes—
20C2×#477C2212yes—
21C2×#462C2212yes—
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#2351—22no—
36#2351—22no—
37#2351C2214yes—
38#2351C2214yes—
39#2329—21no—
40#2313—21no—
41C4×C22⋊Q8C2212yes—
42C2×#389—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#382—21no—
47C2×#382C2211yescharacteristic
48C2×C4×C4⋊C4—21no—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2100C2212yes—
52#2097C2214yes—
53#2088C2214yes—
54#2085C2214yes—
55#2084C2212yes—
56#2074C2214yes—
57#2066C2214yes—
58#2058C2212yes—
59#2043C2212yes—
60#1944C2214yes—
61#1935C2214yes—
62#1920C2214yes—
63#1917C2214yes—
64#1917C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1878C2214yes—
68#1878C2214yes—
69#1872C2212yes—
70#1872C2212yes—
71#1865C2214yes—
72#1865C2214yes—
73#1865C2214yes—
74#1863C2214yes—
75#1863C2214yes—
76#1853C2212yes—
77#1852C2212yes—
78#1823C2212yes—
79#1817C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1797C2211yescharacteristic
83#1797C2211yescharacteristic
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C22×Q16C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×Q8—21no—
98C2×C4×Q8—21no—
99C2×C4×Q8—44no—
100C2×C4×Q8—44no—
101C2×C4×Q8C2314yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C42—21no—
117C22×C42—21no—
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#483—28no—
123#477—44no—
124#462—44no—
125#451—28no—
126#441—48no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141#400—22no—
142#400—22no—
143#400—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#395—22no—
149#391—21no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#389—24no—
156#389—24no—
157#389—24no—
158#389—24no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
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—22no—
182#385—22no—
183#385—22no—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#382—22no—
187#382—24no—
188#382—24no—
189#382C2314yes—
190#381—22no—
191#380—21no—
192#380—21no—
193#380—21no—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—24no—
197C4×C4⋊C4—24no—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219#336C2314yes—
220#336C2314yes—
221C22×Q8—24no—
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—22no—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—28no—
241C2×Q16—28no—
242C2×Q16—28no—
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—44no—
250C4×Q8—216no—
251C4×Q8—28no—
252C4×Q8—28no—
253C4×Q8—28no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—44no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—48no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
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—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4#4114yes—
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×C42—28no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2412yes—
330#107—44no—
331#106—28no—
332#106—44no—
333C4⋊C8—42no—
334C4⋊C8—42no—
335#103—28no—
336#103—28no—
337#103—28no—
338#103—28no—
339#103—44no—
340#103—44no—
341C22⋊C8—24no—
342C22⋊C8—24no—
343#97—44no—
344#95—24no—
345#95—24no—
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#95—22no—
362#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24C22×D411yescharacteristic
368C2×Q8—24no—
369C2×Q8—216no—
370C2×Q8—24no—
371C2×Q8—216no—
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—42no—
381C2×Q8—81no—
382C2×Q8—81no—
383C2×Q8C2×#4114yes—
384C2×Q8H(5,𝔽2)14yes—
385C2×Q82−1+414yes—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
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—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4#13714yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2511yescharacteristic
438Q16—416no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—24no—
443C2×C8—24no—
444C2×C8—24no—
445C4⋊C4—216no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4C2×#4114yes—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C42—216no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42—44no—
494C42—44no—
495C42—48no—
496C42—82no—
497C42—82no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23#52512yes—
503C23C2×#13712yes—
504C23#57512yes—
505C23C23×D411yescharacteristic
506Q8—216no—
507Q8—24no—
508Q8—24no—
509Q8—24no—
510Q8—44no—
511Q8—44no—
512Q8—162no—
513Q8#53918yes—
514C2×C4—216no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—21no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
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—42no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4#52512yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4C22×#4111yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×H(5,𝔽2)12yes—
580C2×C4C2×2−1+412yes—
581C8—44no—
582C8—44no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22#311614yes—
590C22C2×#52511yescharacteristic
591C22#348411yescharacteristic
592C22#348611yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57511yescharacteristic
595C22#358611yescharacteristic
596C22#358611yescharacteristic
597C4—22no—
598C4—22no—
599C4—28no—
600C4—22no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4C2×#53912yes—
615C4#349212yes—
616C2—24no—
617C2#3498911yescharacteristic
618C2C2×#311611yescharacteristic
619C2#6056611yescharacteristic
620C2#6157311yescharacteristic
621C2#6158311yescharacteristic
622C2#6182911yescharacteristic
623C2#6183211yescharacteristic
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. #27580C2semidirect
10. #27324C2semidirect
11. #26393C2semidirect
12. #26183C2semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#23512
#330082
#329572
#329062
#299682
#299132
#297562
#275802
#273242
#263932
#261832
#220842
#219162
#214552