Finite Group Databasebeta

← #6792914

Subgroups — #6792914

Order 512.

1591 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33108C211yesmaximal, characteristic, complemented
4#31586C214yesmaximal, complemented
5#31472C214yesmaximal, complemented
6#29909C212yesmaximal, complemented
7#28798C212yesmaximal, complemented
8#27972C211yesmaximal, characteristic, complemented
9#27756C212yesmaximal, complemented
10#27677C211yesmaximal, characteristic
11#26532C212yesmaximal, complemented
12#26110C212yesmaximal, complemented
13#24609C212yesmaximal, complemented
14#24066C212yesmaximal
15#22603C211yesmaximal, characteristic, complemented
16#21867C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C2×#529C2212yes—
19C22×#125—21no—
20C2×#483C2212yes—
21C2×#480C2211yescharacteristic
22C2×#432C2212yes—
23#2857C2211yescharacteristic
24#2855C2212yes—
25#2552—21no—
26#2552—22no—
27#2552—21no—
28#2552C2214yes—
29#2552C2212yes—
30#2552C2212yes—
31#2552C2211yescharacteristic
32#2547C2211yescharacteristic
33#2496—21no—
34#2486—21no—
35#2478—22no—
36#2453C2212yes—
37#2447C2212yes—
38#2419—22no—
39C2×#398—22no—
40#2365C2211yescharacteristic
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2211yescharacteristic
43#2344—21no—
44C4×#121C2211yescharacteristic
45C2×#389—21no—
46C2×#389C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2214yes—
52#2099C2212yes—
53#2095C2214yes—
54#2093C2212yes—
55#2088C2214yes—
56#2083C2214yes—
57#2072C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2059C2214yes—
61#2042C2214yes—
62#1992C2214yes—
63#1981C2212yes—
64#1959C2214yes—
65#1956C2212yes—
66#1948C2212yes—
67#1947C2212yes—
68#1943C2214yes—
69#1939C2212yes—
70#1935C2214yes—
71#1927C2214yes—
72#1925C2212yes—
73#1918C2212yes—
74#1917C2214yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1872C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1864C2214yes—
81#1863C2214yes—
82#1852C2212yes—
83#1852C2211yescharacteristic
84#1834C2212yes—
85#1823C2211yescharacteristic
86#1817C2212yes—
87#1803C2214yes—
88#1802C2212yes—
89#1802C2214yes—
90#1797C2211yescharacteristic
91C4×#107C2212yes—
92#1770C2212yes—
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96#529—28no—
97#529—44no—
98C2×#126—22no—
99C2×#125—22no—
100C2×#125—22no—
101C2×#125—24no—
102C2×#125—48no—
103C2×#125C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×C22⋊Q8C2312yes—
107C2×#121C2311yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C22×C4⋊C4—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#483—44no—
129#480—24no—
130#432—44no—
131C2×#107C2312yes—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×#97C2314yes—
148C2×C4×C8C2312yes—
149#400—22no—
150#398—22no—
151#398—24no—
152#398—24no—
153#398—24no—
154#398—24no—
155#398—22no—
156#397—22no—
157#397—22no—
158#397—22no—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#390C2311yescharacteristic
163#389—21no—
164#389—22no—
165#389—21no—
166#389—22no—
167#389—24no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389C2314yes—
174#389C2314yes—
175#389C2312yes—
176#389C2312yes—
177#389C2311yescharacteristic
178#388—21no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—21no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—22no—
189#384—22no—
190#384—21no—
191#384—21no—
192#384C2314yes—
193#384C2312yes—
194#384C2312yes—
195#384C2314yes—
196#384C2311yescharacteristic
197#381—22no—
198#381—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380C2311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C2×#95—21no—
219C2×#95—22no—
220C2×#95D412yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8C2412yes—
235C22×D4—21no—
236C22×D4C2411yescharacteristic
237C23×C4—22no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251#126—48no—
252#125—24no—
253#125—42no—
254#125—416no—
255#125—48no—
256#125—48no—
257#124—24no—
258#124—42no—
259#123—48no—
260C22⋊Q8—24no—
261C22⋊Q8—24no—
262#121—22no—
263#121#4114yes—
264C4×Q8—24no—
265C4×Q8—42no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—42no—
269C4×D4—48no—
270C4×D4—44no—
271#117—48no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—24no—
347#107—42no—
348#107—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355#103—44no—
356#102—24no—
357#102—44no—
358#102—42no—
359C22⋊C8—22no—
360C22⋊C8—22no—
361C22⋊C8—24no—
362#97—44no—
363C4×C8—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—21no—
384C24C22×D411yescharacteristic
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4C2×#4112yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—41no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—42no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4AGL(1,ℤ/8)14yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4C2511yescharacteristic
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4#13714yes—
495C4⋊C4C2×#4112yes—
496C22⋊C4—24no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C42—22no—
506C42—22no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—44no—
515C42—44no—
516C42—44no—
517C42—42no—
518C422−1+414yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23#54612yes—
531C23C2×AGL(1,ℤ/8)12yes—
532C23#57812yes—
533C23C23×D411yescharacteristic
534Q8—44no—
535Q8—44no—
536D4—24no—
537D4—48no—
538D4—42no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—88no—
590C2×C4—84no—
591C2×C4—88no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4#53612yes—
595C2×C4#53612yes—
596C2×C4#54612yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×#13714yes—
600C2×C4C2×#13712yes—
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+412yes—
604C2×C4#58512yes—
605C8—42no—
606C8—42no—
607C8—44no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#333014yes—
622C22C2×#53611yescharacteristic
623C22C2×#54611yescharacteristic
624C22#348711yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22C22×#13711yescharacteristic
627C22C2×#57811yescharacteristic
628C22#358511yescharacteristic
629C4—22no—
630C4—22no—
631C4—24no—
632C4—22no—
633C4—41no—
634C4—42no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—81no—
639C4—81no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#323814yes—
646C4#351512yes—
647C4#355912yes—
648C2—24no—
649C2—82no—
650C2#3890711yescharacteristic
651C2C2×#323811yescharacteristic
652C2#6087411yescharacteristic
653C2C2×#333011yescharacteristic
654C2#6116211yescharacteristic
655C2#6189011yescharacteristic
656C2#6190011yescharacteristic
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. #33108C2semidirect
4. #31586C2semidirect
5. #31472C2semidirect
6. #29909C2semidirect
7. #28798C2semidirect
8. #27972C2semidirect
9. #27756C2semidirect
11. #26532C2semidirect
12. #26110C2semidirect
13. #24609C2semidirect
15. #22603C2semidirect

Maximal subgroups

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

subgroupindex
#361162
C2×#25522
#331082
#315862
#314722
#299092
#287982
#279722
#277562
#276772
#265322
#261102
#246092
#240662
#226032
#218672
C4×#4802