Finite Group Databasebeta

← #6593965

Subgroups — #6593965

Order 512.

1711 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36193C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#33008C212yesmaximal
4#32957C211yesmaximal, characteristic
5#27960C211yesmaximal, characteristic
6#27647C211yesmaximal, characteristic
7#27610C212yesmaximal
8#27569C214yesmaximal, complemented
9#27313C214yesmaximal, complemented
10#26108C212yesmaximal, complemented
11#24817C212yesmaximal, complemented
12#24609C212yesmaximal
13#22971C212yesmaximal, complemented
14#22763C212yesmaximal, complemented
15#22603C211yesmaximal, characteristic
16C4×#451C211yesmaximal, characteristic
17#20740C212yesmaximal
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#483C2212yes—
21C2×#477C2212yes—
22C2×#451C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2567C2211yescharacteristic
28#2552C2211yescharacteristic
29#2520—22no—
30#2511C2212yes—
31#2508—21no—
32#2507C2212yes—
33#2442—21no—
34#2438—22no—
35#2370—21no—
36#2369C2211yescharacteristic
37#2355C2211yescharacteristic
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—22no—
41Q8×C4⋊C4C2214yes—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44#2344—21no—
45C4×C22⋊Q8C2211yescharacteristic
46C2×#389—22no—
47C2×#389C2212yes—
48#2285C2212yes—
49C2×#384—21no—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2099C2214yes—
54#2099C2214yes—
55#2088C2214yes—
56#2084C2212yes—
57#2083C2214yes—
58#2072C2212yes—
59#2068C2214yes—
60#1990C2212yes—
61#1948C2212yes—
62#1943C2214yes—
63#1939C2212yes—
64#1935C2214yes—
65#1918C2214yes—
66#1918C2214yes—
67#1917C2214yes—
68#1878C2212yes—
69#1878C2214yes—
70#1878C2214yes—
71#1872C2211yescharacteristic
72#1865C2214yes—
73#1865C2214yes—
74#1863C2212yes—
75#1863C2214yes—
76#1852C2212yes—
77#1852C2211yescharacteristic
78#1834C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1823C2212yes—
82#1817C2212yes—
83#1803C2214yes—
84#1803C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1797C2211yescharacteristic
88#1787C2211yescharacteristic
89#1787C2212yes—
90#1770C2212yes—
91#1763C2214yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C2×C4⋊Q8C2311yescharacteristic
98C2×#125—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8C2311yescharacteristic
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—28no—
107C2×C4×Q8—44no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2314yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2312yes—
113C2×#117—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C42—21no—
124C22×C42—22no—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127#483—44no—
128#477—28no—
129#451—24no—
130C4×Q16—28no—
131#432—44no—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#400—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#395—22no—
155#391—21no—
156#391—22no—
157#391C2312yes—
158#391C2311yescharacteristic
159#391C2312yes—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389—24no—
168#389—24no—
169#389—24no—
170#389—24no—
171#389—22no—
172#389C2314yes—
173#389C2311yescharacteristic
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—21no—
187#384—22no—
188#384—24no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#382—21no—
198#382—22no—
199#381—22no—
200#380—21no—
201#380—22no—
202#380C2312yes—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—24no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—21no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8#4114yes—
242C22×Q8C2411yescharacteristic
243C22×Q8C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—24no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—24no—
259#125—44no—
260C22⋊Q8—22no—
261C22⋊Q8—22no—
262C22⋊Q8—82no—
263C4×Q8—28no—
264C4×Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—48no—
275C4×Q8#4118yes—
276#117—84no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—48no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—84no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—28no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—48no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2412yes—
348C2×C42C2411yescharacteristic
349#107—24no—
350#107—28no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#103—24no—
355#103—24no—
356#103—28no—
357#103—28no—
358#103—24no—
359#103—24no—
360#103—44no—
361C22⋊C8—22no—
362C22⋊C8—22no—
363C22⋊C8—24no—
364#97—44no—
365C4×C8—24no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—216no—
390C2×Q8—24no—
391C2×Q8—216no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×Q8C2×#4114yes—
404C2×Q8C2×#4114yes—
405C2×Q82−1+414yes—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4#13714yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4C2511yescharacteristic
450Q16—48no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—22no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4#13714yes—
500C4⋊C4C2×#4114yes—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—44no—
504C42—28no—
505C42—28no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—48no—
516C42—48no—
517C42—42no—
518C42—42no—
519C42—82no—
520C42—82no—
521C42C2×#4114yes—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#54612yes—
527C23C2×#13712yes—
528C23#57612yes—
529C23C23×D411yescharacteristic
530Q8—216no—
531Q8—24no—
532Q8—24no—
533Q8—24no—
534Q8—42no—
535Q8—42no—
536Q8—162no—
537Q8#55218yes—
538C2×C4—216no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—48no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4#52512yes—
595C2×C4#52512yes—
596C2×C4#54612yes—
597C2×C4C2×#13712yes—
598C2×C4C2×#13714yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C22×#4112yes—
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+412yes—
605C8—42no—
606C8—42no—
607C8—44no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22#333314yes—
615C22C2×#52511yescharacteristic
616C22C2×#54611yescharacteristic
617C22#348611yescharacteristic
618C22C22×#13711yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57611yescharacteristic
621C22#358611yescharacteristic
622C4—28no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—41no—
628C4—44no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—82no—
635C4—81no—
636C4—82no—
637C4—84no—
638C4—81no—
639C4#311314yes—
640C4C2×#55212yes—
641C4#349212yes—
642C2—24no—
643C2#3616311yescharacteristic
644C2C2×#311311yescharacteristic
645C2#6058411yescharacteristic
646C2C2×#333311yescharacteristic
647C2#6115711yescharacteristic
648C2#6182911yescharacteristic
649C2#6184411yescharacteristic
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
8. #27569C2semidirect
9. #27313C2semidirect
10. #26108C2semidirect
11. #24817C2semidirect
13. #22971C2semidirect
14. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#361932
C2×Q8×C4⋊C42
#330082
#329572
#279602
#276472
#276102
#275692
#273132
#261082
#248172
#246092
#229712
#227632
#226032
C4×#4512
#207402