Finite Group Databasebeta

← #6567623

Subgroups — #6567623

Order 512.

1593 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34311C211yesmaximal, characteristic
3#30060C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic
5#29554C212yesmaximal
6#29392C212yesmaximal
7#29206C211yesmaximal, characteristic, complemented
8#28205C211yesmaximal, characteristic, complemented
9#27488C214yesmaximal
10#27128C214yesmaximal, complemented
11#25060C212yesmaximal, complemented
12#24330C212yesmaximal
13#23837C212yesmaximal, complemented
14#23677C211yesmaximal, characteristic
15#23062C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22805C212yesmaximal
18C2×D4×Q8C2211yescharacteristic
19C2×#519—21no—
20C2×#475C2212yes—
21C2×C4×SD16C2212yes—
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2582C2212yes—
25#2571—21no—
26#2570—22no—
27#2568—21no—
28#2511C2214yes—
29#2446—22no—
30#2365—21no—
31#2365C2211yescharacteristic
32#2365C2212yes—
33#2362—21no—
34#2362—22no—
35#2355—21no—
36#2355C2212yes—
37#2354C2211yescharacteristic
38#2354C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C22⋊C4—21no—
41#2323—22no—
42#2323—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47#2293C2211yescharacteristic
48#2293C2212yes—
49C42×Q8C2211yescharacteristic
50#2100C2212yes—
51#2096C2214yes—
52#2094C2212yes—
53#2084C2211yescharacteristic
54#2084C2212yes—
55#2084C2214yes—
56#2071C2211yescharacteristic
57#2066C2212yes—
58#2058C2214yes—
59#2058C2212yes—
60#2058C2212yes—
61#2058C2212yes—
62#2055C2212yes—
63#2043C2212yes—
64#2043C2211yescharacteristic
65#2024C2214yes—
66#2023C2212yes—
67#2013C2211yescharacteristic
68#1990C2212yes—
69#1984C2212yes—
70#1968C2214yes—
71#1948C2212yes—
72#1948C2214yes—
73#1933C2214yes—
74#1927C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1927C2214yes—
78#1926C2214yes—
79#1920C2214yes—
80#1919C2214yes—
81#1916C2214yes—
82#1877C2214yes—
83#1864C2214yes—
84#1844C2212yes—
85#1839C2212yes—
86#1831C2211yescharacteristic
87#1803C2212yes—
88#1802C2212yes—
89C4×#106C2214yes—
90C4×#106C2214yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1744C2212yes—
95C23×Q8—21no—
96C22×SD16C2312yes—
97D4×Q8—24no—
98D4×Q8—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—22no—
107#519—48no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4C2312yes—
119C2×C4×D4C2311yescharacteristic
120C2×#117—21no—
121C2×#117—22no—
122C2×#117—22no—
123C2×#117—21no—
124#475—28no—
125C4×SD16—28no—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2314yes—
142#400—24no—
143#400—24no—
144#398—24no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391—22no—
149#391—21no—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2312yes—
154#391C2314yes—
155#391C2314yes—
156#390C2311yescharacteristic
157#390C2311yescharacteristic
158#390C2312yes—
159#389—22no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389—21no—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2311yescharacteristic
171#389C2312yes—
172#388—22no—
173#388—21no—
174#388—22no—
175#388—21no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—22no—
183#387—22no—
184#387—22no—
185#384—21no—
186#384—21no—
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382—22no—
195#380—21no—
196#380—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C43C2311yescharacteristic
218C43C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C22×Q8—28no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8—44no—
238C22×Q8—44no—
239C22×Q8C2×D411yescharacteristic
240C22×Q8C2412yes—
241C22×D4C2411yescharacteristic
242C23×C4—22no—
243C23×C4—21no—
244C2×SD16—24no—
245C2×SD16—24no—
246C2×SD16—42no—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—44no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8#4114yes—
258C22⋊Q8—28no—
259C22⋊Q8—216no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—42no—
265C4×Q8—44no—
266C4×Q8—48no—
267C4×Q8—44no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8—84no—
271C4×Q8SD1618yescomplemented
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4#4114yes—
276#117—48no—
277#117—44no—
278#117—48no—
279#117—44no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#106—24no—
343#106—28no—
344#106—44no—
345#106—44no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349#103—24no—
350#103—24no—
351#103—28no—
352#103—44no—
353#103—42no—
354#102—24no—
355#102—44no—
356C4×C8—28no—
357C4×C8—24no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—21no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C2×Q8—28no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—432no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×SD1614yes—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4C2×#4112yes—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427SD16—48no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C82−1+414yes—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C2×SD1618yes—
472C4⋊C4C2×SD1614yes—
473C4⋊C4C2×#4114yes—
474C4⋊C4C2×#4112yes—
475C22⋊C4—28no—
476C22⋊C4—24no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—82no—
482C22⋊C4—81no—
483C42—22no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—24no—
489C42—22no—
490C42—22no—
491C42—41no—
492C42—42no—
493C42—42no—
494C42—44no—
495C42—44no—
496C42—44no—
497C42—44no—
498C42—44no—
499C42—42no—
500C42—84no—
501C42C2×SD1618yes—
502C42C2×SD1614yes—
503C42C2×#4112yes—
504C42C2×#4112yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C23×D411yescharacteristic
513Q8—28no—
514Q8—416no—
515Q8—48no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—88no—
521D4—24no—
522D4—42no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—416no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4#52512yes—
571C2×C4#52512yes—
572C2×C4C22×SD1614yes—
573C2×C4C22×SD1612yes—
574C2×C4C22×SD1614yes—
575C2×C4C22×SD1612yes—
576C2×C4C22×SD1612yes—
577C2×C4#57512yes—
578C2×C4#57712yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×2−1+412yes—
582C2×C4#58512yes—
583C8—24no—
584C8—42no—
585C8—44no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22C2×#52511yescharacteristic
594C22#348711yescharacteristic
595C22#349911yescharacteristic
596C22C23×SD1611yescharacteristic
597C22C2×#57511yescharacteristic
598C22C2×#57711yescharacteristic
599C22#358611yescharacteristic
600C4—24no—
601C4—22no—
602C4—24no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—42no—
608C4—44no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—82no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—84no—
617C4#310114yes—
618C4#349512yes—
619C4#351812yes—
620C2—82no—
621C2#3603011yescharacteristic
622C2C2×#310111yescharacteristic
623C2#6059911yescharacteristic
624C2#6188311yescharacteristic
625C2#6190411yescharacteristic
626C2#6246911yescharacteristic
627C2#6250711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #36598C2semidirect
3. #30060C2semidirect
7. #29206C2semidirect
8. #28205C2semidirect
10. #27128C2semidirect
11. #25060C2semidirect
13. #23837C2semidirect
15. #23062C2semidirect
16. #22828C2semidirect
271. C4×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#365982
#343112
#300602
#299812
#295542
#293922
#292062
#282052
#274882
#271282
#250602
#243302
#238372
#236772
#230622
#228282
#228052