Finite Group Databasebeta

← #6798778

Subgroups — #6798778

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2D4×#125C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#31768C214yesmaximal, complemented
5#31766C214yesmaximal, complemented
6#29851C212yesmaximal, complemented
7#29213C211yesmaximal, characteristic, complemented
8#28724C211yesmaximal, characteristic, complemented
9#27880C212yesmaximal, complemented
10#26025C212yesmaximal, complemented
11#26021C212yesmaximal, complemented
12#25913C212yesmaximal, complemented
13#24967C212yesmaximal, complemented
14#24965C212yesmaximal, complemented
15#21838C212yesmaximal, complemented
16C4×#490C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic, complemented
18C2×#555—21no—
19C22×#125C2212yes—
20C2×#490C2211yescharacteristic
21C2×#490C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2856C2212yes—
26#2851C2212yes—
27#2563—21no—
28#2552C2211yescharacteristic
29#2552C2211yescharacteristic
30#2537—21no—
31#2537C2212yes—
32#2506—21no—
33#2506—21no—
34#2500—22no—
35#2491—22no—
36#2491C2214yes—
37#2473—22no—
38#2473—22no—
39#2442C2212yes—
40#2442C2212yes—
41#2354—21no—
42#2354C2212yes—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C4×#125C2211yescharacteristic
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2077C2214yes—
52#2075C2214yes—
53#2073C2214yes—
54#2067C2214yes—
55#2065C2214yes—
56#2058C2212yes—
57#2057C2212yes—
58#2034C2214yes—
59#2033C2214yes—
60#2025C2214yes—
61#2025C2214yes—
62#1993C2214yes—
63#1992C2212yes—
64#1992C2212yes—
65#1992C2212yes—
66#1992C2212yes—
67#1992C2211yescharacteristic
68#1989C2214yes—
69#1989C2214yes—
70#1961C2211yescharacteristic
71#1954C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1936C2212yes—
76#1936C2212yes—
77#1924C2214yes—
78#1921C2214yes—
79#1919C2214yes—
80#1874C2214yes—
81#1874C2214yes—
82#1847C2212yes—
83#1836C2211yescharacteristic
84#1834C2212yes—
85#1834C2212yes—
86#1824C2214yes—
87#1823C2214yes—
88#1802C2212yes—
89#1802C2212yes—
90#1767C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95#555—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2312yes—
99C2×#127C2311yescharacteristic
100C2×#125—28no—
101C2×#125—22no—
102C2×#125—22no—
103C2×#125—44no—
104C2×#125C2312yes—
105C2×#125C2312yes—
106C2×#125C2312yes—
107C2×#125C2311yescharacteristic
108C2×#124—24no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121C2314yes—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128#490—24no—
129#490—44no—
130#432—44no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2311yescharacteristic
144C2×#97C2312yes—
145C2×#97C2311yescharacteristic
146C2×#97C2312yes—
147C2×C4×C8C2312yes—
148#400—22no—
149#400—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#398—22no—
154#394—24no—
155#394—24no—
156#393—24no—
157#390—21no—
158#390—21no—
159#390—22no—
160#390—22no—
161#390C2312yes—
162#390C2314yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389C2311yescharacteristic
172#389C2311yescharacteristic
173#389C2312yes—
174#389C2312yes—
175#388—24no—
176#388—24no—
177#387—21no—
178#387—21no—
179#387—22no—
180#387—22no—
181#385—22no—
182#385—22no—
183#385—22no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382C2311yescharacteristic
198#382C2312yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—22no—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2412yes—
240C23×C4—22no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251C4⋊Q8—82no—
252#127—22no—
253#127—22no—
254#125—24no—
255#125—24no—
256#125—24no—
257#125—22no—
258#125—24no—
259#125—42no—
260#125—42no—
261#125—48no—
262#125—48no—
263#125#4114yes—
264#124—416no—
265#121—48no—
266#121—48no—
267#121—44no—
268#121—44no—
269#121—84no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—42no—
275C4×D4—44no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—44no—
279#117—48no—
280#117—48no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—48no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2411yescharacteristic
350#107—44no—
351#106—44no—
352C4⋊C8—24no—
353C4⋊C8—44no—
354#102—24no—
355#102—24no—
356#102—42no—
357#102—42no—
358#102—44no—
359#102—44no—
360C22⋊C8—44no—
361C22⋊C8—44no—
362#97—22no—
363#97—22no—
364#97—24no—
365#97—42no—
366C4×C8—24no—
367#95—24no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—24no—
373#95—21no—
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—
384C24—22no—
385C24C22×D412yes—
386C2×Q8—44no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—48no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—84no—
405C2×D4—84no—
406C2×D4AGL(1,ℤ/8)14yes—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—82no—
433C22×C4—82no—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—24no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—82no—
492C4⋊C4—82no—
493C4⋊C4C2×#4112yes—
494C4⋊C4C2×#4112yes—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—84no—
506C22⋊C4—84no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—48no—
517C42—42no—
518C42—42no—
519C42C2×#4112yes—
520C42H(5,𝔽2)12yes—
521C422−1+414yes—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23C2×AGL(1,ℤ/8)14yes—
536C23C23×D411yescharacteristic
537Q8—164no—
538D4—28no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—84no—
544D4—84no—
545D4—168no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—48no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—88no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4#53612yes—
599C2×C4#53612yes—
600C2×C4#54612yes—
601C2×C4#54612yes—
602C2×C4C2×AGL(1,ℤ/8)12yes—
603C2×C4#57512yes—
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×2−1+412yes—
607C2×C4#58512yes—
608C8—42no—
609C8—42no—
610C8—44no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22C2×#53611yescharacteristic
626C22C2×#54611yescharacteristic
627C22#350011yescharacteristic
628C22C22×AGL(1,ℤ/8)11yescharacteristic
629C22C2×#57511yescharacteristic
630C22#358511yescharacteristic
631C22#358611yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—84no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4#323414yes—
649C4#351812yes—
650C4#354712yes—
651C2—44no—
652C2—162no—
653C2#3895811yescharacteristic
654C2C2×#323411yescharacteristic
655C2#6088611yescharacteristic
656C2#6116111yescharacteristic
657C2#6116611yescharacteristic
658C2#6257611yescharacteristic
659C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×#125C2semidirect
3. #33297C2semidirect
4. #31768C2semidirect
5. #31766C2semidirect
6. #29851C2semidirect
7. #29213C2semidirect
8. #28724C2semidirect
9. #27880C2semidirect
10. #26025C2semidirect
11. #26021C2semidirect
12. #25913C2semidirect
13. #24967C2semidirect
14. #24965C2semidirect
15. #21838C2semidirect
16. C4×#490C2semidirect
17. #21270C2semidirect

Maximal subgroups

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

subgroupindex
#361872
D4×#1252
#332972
#317682
#317662
#298512
#292132
#287242
#278802
#260252
#260212
#259132
#249672
#249652
#218382
C4×#4902
#212702