Finite Group Databasebeta

← #6802859

Subgroups — #6802859

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33109C211yesmaximal, characteristic, complemented
4#32396C214yesmaximal, complemented
5#31592C214yesmaximal, complemented
6#29761C212yesmaximal
7#28835C212yesmaximal, complemented
8#27913C212yesmaximal
9#27910C211yesmaximal, characteristic, complemented
10#27756C212yesmaximal, complemented
11#27648C212yesmaximal, complemented
12#27645C211yesmaximal, characteristic
13#26528C212yesmaximal, complemented
14#26086C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22085C212yesmaximal, complemented
17C4×#481C211yesmaximal, characteristic
18C22×#125—21no—
19C2×#484C2212yes—
20C2×#484C2212yes—
21C2×#484C2212yes—
22C2×#481C2211yescharacteristic
23C2×#481C2212yes—
24#2857C2211yescharacteristic
25#2850C2212yes—
26#2596—22no—
27#2571C2211yescharacteristic
28#2552—22no—
29#2552—21no—
30#2552—21no—
31#2552C2214yes—
32#2552C2212yes—
33#2552C2212yes—
34#2547C2211yescharacteristic
35#2514C2212yes—
36#2510—22no—
37#2494—21no—
38#2487—21no—
39#2448C2212yes—
40C2×#398—22no—
41#2365C2211yescharacteristic
42#2354C2211yescharacteristic
43#2352—21no—
44#2347—21no—
45C4×C22⋊Q8C2212yes—
46C4×#121C2211yescharacteristic
47C2×#389—21no—
48C2×#389C2212yes—
49C2×#384C2212yes—
50C2×#384C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53#2096C2214yes—
54#2095C2214yes—
55#2093C2214yes—
56#2085C2214yes—
57#2084C2212yes—
58#2083C2214yes—
59#2072C2214yes—
60#2071C2212yes—
61#2067C2214yes—
62#2066C2214yes—
63#2042C2214yes—
64#1981C2212yes—
65#1959C2214yes—
66#1956C2212yes—
67#1948C2212yes—
68#1947C2214yes—
69#1947C2212yes—
70#1943C2214yes—
71#1939C2212yes—
72#1935C2214yes—
73#1928C2214yes—
74#1926C2212yes—
75#1919C2212yes—
76#1918C2214yes—
77#1917C2214yes—
78#1874C2211yescharacteristic
79#1872C2212yes—
80#1865C2214yes—
81#1853C2211yescharacteristic
82#1847C2214yes—
83#1845C2212yes—
84#1824C2211yescharacteristic
85#1824C2212yes—
86#1803C2214yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—22no—
97C2×#125—24no—
98C2×#125—22no—
99C2×#125—48no—
100C2×#124C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×#121C2311yescharacteristic
103C2×C4×Q8C2312yes—
104C2×C4×D4—21no—
105C2×C4×D4C2311yescharacteristic
106C2×#117—21no—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#484—28no—
121#484—44no—
122#484—44no—
123#481—24no—
124#481—44no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2311yescharacteristic
142C2×C22⋊C8C2311yescharacteristic
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#401—22no—
146#400—22no—
147#400—22no—
148#399—22no—
149#398—24no—
150#398—24no—
151#398—24no—
152#398—24no—
153#397—22no—
154#397—22no—
155#391C2312yes—
156#391C2312yes—
157#391C2312yes—
158#390C2312yes—
159#390C2311yescharacteristic
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—24no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2314yes—
171#389C2311yescharacteristic
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—21no—
177#388—22no—
178#388—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—21no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384—22no—
189#384—22no—
190#384C2312yes—
191#384C2314yes—
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2314yes—
195#383—22no—
196#382—21no—
197#382C2312yes—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380C2311yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C2×#95—21no—
221C2×#95—22no—
222C2×#95D412yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8C2412yes—
235C22×D4—21no—
236C22×D4C2411yescharacteristic
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4C2×D412yes—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D412yes—
245C23×C4C2411yescharacteristic
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250#125—416no—
251#125—48no—
252#125—48no—
253#124—44no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256#121—22no—
257#121—22no—
258C4×Q8—44no—
259C4×D4—24no—
260C4×D4—44no—
261#117—44no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4—84no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—44no—
336#107—44no—
337#106—24no—
338#106—42no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—28no—
343#103—44no—
344#103—44no—
345#103—44no—
346#102—24no—
347#102—42no—
348#102—44no—
349C22⋊C8—24no—
350C22⋊C8—22no—
351C22⋊C8#4114yes—
352#97—44no—
353C4×C8—24no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374#95C2×D412yes—
375C24—21no—
376C24C22×D411yescharacteristic
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—24no—
380C2×Q8—24no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4#13714yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4C2511yescharacteristic
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8C2×#4114yes—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4AGL(1,ℤ/8)14yes—
482C22⋊C4—44no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C22⋊C4—82no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—24no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—42no—
503C42—82no—
504C422−1+414yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23#55012yes—
517C23C2×#13712yes—
518C23#57812yes—
519C23C23×D411yescharacteristic
520Q8—44no—
521Q8—44no—
522D4—42no—
523D4—42no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—48no—
549C2×C4—44no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—88no—
578C2×C4—84no—
579C2×C4#53612yes—
580C2×C4#53612yes—
581C2×C4#55012yes—
582C2×C4C2×AGL(1,ℤ/8)14yes—
583C2×C4C2×AGL(1,ℤ/8)12yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×2−1+412yes—
589C2×C4#58512yes—
590C8—24no—
591C8—44no—
592C8—42no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22#339814yes—
607C22C2×#53611yescharacteristic
608C22C2×#55011yescharacteristic
609C22#348511yescharacteristic
610C22C22×AGL(1,ℤ/8)11yescharacteristic
611C22C22×#13711yescharacteristic
612C22C2×#57811yescharacteristic
613C22#358511yescharacteristic
614C4—24no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—81no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—82no—
628C4—84no—
629C4—82no—
630C4#323914yes—
631C4#351512yes—
632C4#355912yes—
633C2—24no—
634C2—161no—
635C2#3913411yescharacteristic
636C2C2×#323911yescharacteristic
637C2#6087411yescharacteristic
638C2C2×#339811yescharacteristic
639C2#6140211yescharacteristic
640C2#6171911yescharacteristic
641C2#6173711yescharacteristic
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×#2552C2semidirect
3. #33109C2semidirect
4. #32396C2semidirect
5. #31592C2semidirect
7. #28835C2semidirect
9. #27910C2semidirect
10. #27756C2semidirect
11. #27648C2semidirect
13. #26528C2semidirect
14. #26086C2semidirect
15. #22629C2semidirect
16. #22085C2semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25522
#331092
#323962
#315922
#297612
#288352
#279132
#279102
#277562
#276482
#276452
#265282
#260862
#226292
#220852
C4×#4812