Finite Group Databasebeta

← #6795014

Subgroups — #6795014

Order 512.

1543 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32439C214yesmaximal, complemented
5#32355C214yesmaximal, complemented
6#29913C212yesmaximal
7#28532C212yesmaximal, complemented
8#27730C212yesmaximal, complemented
9#27589C212yesmaximal
10#25813C211yesmaximal, characteristic
11#25456C212yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#24502C212yesmaximal
14#24301C211yesmaximal, characteristic
15#22599C211yesmaximal, characteristic
16#21983C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#500C2212yes—
21C2×#475C2212yes—
22C2×#475C2212yes—
23C2×#451C2211yescharacteristic
24C2×C4×Q16C2212yes—
25C2×#431C2212yes—
26#2857C2211yescharacteristic
27#2855C2212yes—
28#2571C2211yescharacteristic
29#2568—21no—
30#2568—21no—
31#2568—22no—
32#2568C2211yescharacteristic
33#2568C2212yes—
34#2568C2214yes—
35#2568C2212yes—
36#2533—22no—
37#2514—21no—
38#2514C2212yes—
39#2509C2212yes—
40#2503—21no—
41#2493—22no—
42C2×#400—22no—
43#2365C2211yescharacteristic
44#2360—21no—
45#2356—21no—
46#2355C2211yescharacteristic
47C4×C22⋊Q8C2211yescharacteristic
48C2×#391—21no—
49C2×#391—21no—
50C2×#391C2212yes—
51#2283C2212yes—
52C2×#382C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2097C2214yes—
57#2085C2214yes—
58#2074C2214yes—
59#2074C2214yes—
60#2068C2214yes—
61#2058C2212yes—
62#2057C2214yes—
63#2056C2214yes—
64#2024C2212yes—
65#1990C2212yes—
66#1988C2212yes—
67#1969C2214yes—
68#1949C2214yes—
69#1949C2214yes—
70#1939C2212yes—
71#1933C2214yes—
72#1928C2214yes—
73#1927C2214yes—
74#1916C2214yes—
75#1878C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1865C2214yes—
79#1863C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1852C2211yescharacteristic
85#1846C2214yes—
86#1835C2214yes—
87#1817C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91#1787C2211yescharacteristic
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#500—44no—
123#475—44no—
124#475—44no—
125#451—24no—
126C4×Q16—44no—
127#431—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2311yescharacteristic
144C2×C22⋊C8C2311yescharacteristic
145C2×C4×C8C2314yes—
146C2×C4×C8C2312yes—
147#401—22no—
148#400—22no—
149#400—22no—
150#400—24no—
151#400—24no—
152#400—24no—
153#400—24no—
154#400—22no—
155#400—22no—
156#400—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—24no—
162#391—24no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—22no—
168#391—21no—
169#391C2312yes—
170#391C2314yes—
171#391C2314yes—
172#391C2312yes—
173#391C2311yescharacteristic
174#389—21no—
175#389—22no—
176#389—21no—
177#389C2312yes—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#385—22no—
189#385—21no—
190#384C2311yescharacteristic
191#383—22no—
192#383—22no—
193#382—21no—
194#382—21no—
195#382—22no—
196#382—22no—
197#382C2314yes—
198#382C2314yes—
199#382C2312yes—
200#382C2312yes—
201#381—22no—
202#381—22no—
203#380—21no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219#376—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2411yescharacteristic
248C22×Q8C2412yes—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—21no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—42no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—44no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—42no—
276C4×Q8—42no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—44no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335#107—42no—
336#107—44no—
337#107—44no—
338#106—44no—
339#106—44no—
340C4⋊C8—28no—
341C4⋊C8—24no—
342#103—28no—
343#103—24no—
344#103—24no—
345#103—42no—
346#103—42no—
347#103—44no—
348#103—44no—
349C22⋊C8—24no—
350C22⋊C8—22no—
351C22⋊C8#4114yes—
352C4×C8—28no—
353C4×C8—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
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—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—816no—
410C2×Q82−1+414yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4#13514yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455Q16—48no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8C2×#4114yes—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—24no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—44no—
509C42—44no—
510C42—44no—
511C42—42no—
512C42—42no—
513C42—82no—
514C42H(5,𝔽2)14yes—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#54712yes—
520C23C2×#13512yes—
521C23#57812yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—48no—
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—88no—
531Q8—84no—
532Q8—88no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—44no—
559C2×C4—48no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—88no—
590C2×C4#53512yes—
591C2×C4#53512yes—
592C2×C4#54712yes—
593C2×C4C2×#13512yes—
594C2×C4C2×#13512yes—
595C2×C4#57512yes—
596C2×C4C22×#4111yescharacteristic
597C2×C4C2×H(5,𝔽2)12yes—
598C2×C4C2×2−1+412yes—
599C2×C4C2×2−1+411yescharacteristic
600C8—24no—
601C8—44no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22#334214yes—
610C22C2×#53511yescharacteristic
611C22C2×#54711yescharacteristic
612C22#348611yescharacteristic
613C22C22×#13511yescharacteristic
614C22C2×#57511yescharacteristic
615C22C2×#57811yescharacteristic
616C22#358611yescharacteristic
617C4—22no—
618C4—22no—
619C4—22no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—81no—
627C4—81no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4—81no—
634C4#321114yes—
635C4#352012yes—
636C4#353812yes—
637C2—24no—
638C2#3891711yescharacteristic
639C2C2×#321111yescharacteristic
640C2#6081411yescharacteristic
641C2C2×#334211yescharacteristic
642C2#6125711yescharacteristic
643C2#6182211yescharacteristic
644C2#6184211yescharacteristic
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
4. #32439C2semidirect
5. #32355C2semidirect
7. #28532C2semidirect
8. #27730C2semidirect
11. #25456C2semidirect
12. #24683C2semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#25682
#329542
#324392
#323552
#299132
#285322
#277302
#275892
#258132
#254562
#246832
#245022
#243012
#225992
#219832
C4×#4512