Finite Group Databasebeta

← #6856526

Subgroups — #6856526

Order 512.

1529 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36593C211yesmaximal, characteristic, complemented
2#34929C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#32314C214yesmaximal, complemented
5#31458C214yesmaximal
6#29868C212yesmaximal
7#29287C211yesmaximal, characteristic, complemented
8#29273C212yesmaximal, complemented
9#29218C212yesmaximal, complemented
10#29218C211yesmaximal, characteristic, complemented
11#28791C212yesmaximal, complemented
12#27727C212yesmaximal
13#26278C212yesmaximal, complemented
14#24992C212yesmaximal
15#22821C211yesmaximal, characteristic, complemented
16#22111C212yesmaximal
17C4×#476C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#533—21no—
20C2×#476C2212yes—
21C2×#476C2211yescharacteristic
22#2868C2212yes—
23#2868C2211yescharacteristic
24#2556—22no—
25#2552—22no—
26#2552—21no—
27#2552C2214yes—
28#2552C2212yes—
29#2551C2211yescharacteristic
30#2546—21no—
31#2546—22no—
32#2546C2211yescharacteristic
33#2542C2211yescharacteristic
34#2538C2211yescharacteristic
35#2506C2212yes—
36#2492—21no—
37#2452C2212yes—
38#2429—21no—
39#2396—22no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2212yes—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C22⋊Q8—21no—
47#2293C2212yes—
48C42×Q8C2211yescharacteristic
49C42×D4C2211yescharacteristic
50#2100C2214yes—
51#2100C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2071C2214yes—
58#2071C2214yes—
59#2058C2214yes—
60#1990C2212yes—
61#1990C2214yes—
62#1988C2212yes—
63#1987C2212yes—
64#1987C2211yescharacteristic
65#1987C2212yes—
66#1987C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1941C2214yes—
75#1934C2214yes—
76#1926C2214yes—
77#1920C2212yes—
78#1919C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1916C2214yes—
82#1864C2212yes—
83#1842C2211yescharacteristic
84#1834C2214yes—
85#1834C2212yes—
86#1831C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#557—48no—
96#533—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×#125—24no—
109C2×#125—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—22no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—22no—
126C22×C42—21no—
127#476—28no—
128#476—24no—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2314yes—
143#400—22no—
144#399—22no—
145#398—22no—
146#398—22no—
147#398—24no—
148#398—24no—
149#398—24no—
150#396—22no—
151#396—22no—
152#395—22no—
153#393—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—21no—
158#391—22no—
159#390C2312yes—
160#390C2312yes—
161#389—24no—
162#389—22no—
163#389—22no—
164#389—24no—
165#389—21no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#387—22no—
180#385—22no—
181#384—22no—
182#384—21no—
183#384—22no—
184#384—21no—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#384C2314yes—
191#384C2314yes—
192#380—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C43C2312yes—
214C43C2311yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×D4C2411yescharacteristic
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—42no—
252C4⋊Q8—48no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256C4⋊Q8—82no—
257C4⋊Q8—84no—
258#125—416no—
259#125—48no—
260#125—48no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265C4×Q8—24no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8—48no—
271C4×Q8—48no—
272C4×Q8—44no—
273C4×Q8SD1618yes—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—41no—
279C4×D4#4114yes—
280#117—48no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—44no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2412yes—
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#106—24no—
348#106—28no—
349#106—24no—
350#106—44no—
351C4⋊C8—22no—
352C4⋊C8—24no—
353C4⋊C8—24no—
354C4⋊C8—22no—
355C4⋊C8—42no—
356#102—28no—
357#102—24no—
358#102—42no—
359C4×C8—24no—
360C4×C8—44no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—82no—
392C2×Q8C2×SD1614yes—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4C2×#4112yes—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—42no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4C2×SD1618yes—
479C4⋊C4C2×SD1614yes—
480C4⋊C4AGL(1,ℤ/8)14yes—
481C4⋊C4C2×#4112yes—
482C4⋊C42−1+412yes—
483C22⋊C4—24no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—82no—
490C42—24no—
491C42—22no—
492C42—24no—
493C42—22no—
494C42—21no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—21no—
500C42—41no—
501C42—42no—
502C42—48no—
503C42—42no—
504C42—42no—
505C42—42no—
506C42—44no—
507C42—44no—
508C42C2×SD1614yes—
509C42C2×SD1618yes—
510C42C2×#4112yes—
511C422−1+412yes—
512C422−1+412yes—
513C422−1+412yes—
514C23—22no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C23×D411yescharacteristic
522Q8—28no—
523Q8—88no—
524Q8—88no—
525Q8—84no—
526Q8—84no—
527D4—22no—
528D4—22no—
529D4—41no—
530D4—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—48no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4#53612yes—
577C2×C4#53612yes—
578C2×C4#54812yes—
579C2×C4#54812yes—
580C2×C4C22×SD1612yes—
581C2×C4C22×SD1614yes—
582C2×C4C22×SD1614yes—
583C2×C4C22×SD1612yes—
584C2×C4C22×SD1612yes—
585C2×C4C2×AGL(1,ℤ/8)14yes—
586C2×C4C2×AGL(1,ℤ/8)12yes—
587C2×C4#57812yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C2×C4#58511yescharacteristic
594C8—44no—
595C8—42no—
596C8—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22C2×#53611yescharacteristic
605C22C2×#54811yescharacteristic
606C22#348911yescharacteristic
607C22C23×SD1611yescharacteristic
608C22C22×AGL(1,ℤ/8)11yescharacteristic
609C22C2×#57811yescharacteristic
610C22#358511yescharacteristic
611C4—22no—
612C4—22no—
613C4—24no—
614C4—24no—
615C4—24no—
616C4—22no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—81no—
625C4—82no—
626C4—82no—
627C4#322714yes—
628C4#335914yes—
629C4#350711yescharacteristic
630C4#350911yescharacteristic
631C4#355511yescharacteristic
632C4#355611yescharacteristic
633C2—82no—
634C2#3972411yescharacteristic
635C2C2×#322711yescharacteristic
636C2#6089311yescharacteristic
637C2C2×#335911yescharacteristic
638C2#6132111yescharacteristic
639C2#6209611yescharacteristic
640C2#6213211yescharacteristic
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. #36593C2semidirect
4. #32314C2semidirect
7. #29287C2semidirect
8. #29273C2semidirect
9. #29218C2semidirect
10. #29218C2semidirect
11. #28791C2semidirect
13. #26278C2semidirect
15. #22821C2semidirect

Maximal subgroups

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

subgroupindex
#365932
#349292
#331972
#323142
#314582
#298682
#292872
#292732
#292182
#287912
#277272
#262782
#249922
#228212
#221112
C4×#4762