Finite Group Databasebeta

← #6644657

Subgroups — #6644657

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35015C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3#33007C212yesmaximal
4#32428C214yesmaximal, complemented
5#32344C214yesmaximal, complemented
6#31636C214yesmaximal, complemented
7#31520C214yesmaximal, complemented
8#29641C212yesmaximal, complemented
9#29103C211yesmaximal, characteristic
10#28302C211yesmaximal, characteristic
11#28120C212yesmaximal
12#27960C211yesmaximal, characteristic, complemented
13#27905C211yesmaximal, characteristic, complemented
14#27647C212yesmaximal, complemented
15#21912C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8—21no—
18C22×C22⋊Q8—21no—
19C22×C4×D4C2211yescharacteristic
20C2×#483C2212yes—
21C2×#483C2212yes—
22C2×#483C2212yes—
23C2×#450C2212yes—
24#2857C2211yescharacteristic
25#2857C2211yescharacteristic
26#2855C2212yes—
27#2646—21no—
28#2627—21no—
29#2590—22no—
30#2588—22no—
31#2542—22no—
32#2542—22no—
33#2542C2214yes—
34#2542C2214yes—
35#2511C2212yescomplemented
36#2507C2212yescomplemented
37#2451C2212yescomplemented
38#2446C2212yescomplemented
39C2×#399—21no—
40C2×#399—21no—
41#2329—21no—
42#2313—21no—
43C4×C22⋊Q8C2212yes—
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2099C2214yescomplemented
50#2096C2214yescomplemented
51#2094C2212yes—
52#2084C2214yescomplemented
53#2072C2214yescomplemented
54#2071C2212yes—
55#2065C2214yes—
56#2058C2214yescomplemented
57#2058C2214yescomplemented
58#2055C2214yes—
59#2024C2214yescomplemented
60#2024C2214yescomplemented
61#2023C2212yes—
62#2013C2212yes—
63#1990C2212yescomplemented
64#1968C2214yes—
65#1956C2212yescomplemented
66#1948C2214yescomplemented
67#1934C2214yes—
68#1926C2214yes—
69#1916C2214yescomplemented
70#1916C2214yescomplemented
71#1880C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1864C2214yes—
75#1864C2214yes—
76#1844C2212yescomplemented
77#1834C2212yescomplemented
78#1823C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1817C2212yes—
82#1803C2214yescomplemented
83#1797C2212yes—
84#1786C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
102C2×C22⋊Q8—48no—
103C2×C22⋊Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—44no—
107C2×C4×D4C2314yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#483—44no—
127#483—44no—
128#483—44no—
129#450—44no—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—22no—
146#400—22no—
147#399—24no—
148#399—24no—
149#399—24no—
150#399—24no—
151#398—22no—
152#398—22no—
153#398—22no—
154#398—22no—
155#395—22no—
156#391—21no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2314yes—
166#390C2312yes—
167#390C2312yes—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384C2314yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2312yes—
194#384C2312yes—
195#381—22no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218#336C2312yes—
219#336C2312yes—
220C25—21no—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×SD16—28no—
247C2×SD16—44no—
248C2×SD16—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—48no—
252C4⋊Q8—42no—
253C4⋊Q8—48no—
254C4⋊Q8—42no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—416no—
258C22⋊Q8—416no—
259C22⋊Q8—44no—
260C22⋊Q8—416no—
261C22⋊Q8—416no—
262C4×Q8—44no—
263C4×D4—48no—
264C4×D4—48no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4SD1618yescomplemented
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#106—44no—
336#106—44no—
337#106—44no—
338C4⋊C8—44no—
339#103—44no—
340#103—44no—
341#103—44no—
342#102—28no—
343#102—44no—
344#102—44no—
345C22⋊C8—24no—
346C22⋊C8—24no—
347C4×C8—42no—
348C4×C8—42no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24C22×D411yescharacteristic
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—42no—
384C2×Q8—81no—
385C2×Q8—81no—
386C2×Q8—816no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—24no—
391C2×D4—416no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D42−1+414yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—48no—
430C22×C4—816no—
431C22×C4C2×SD1614yes—
432C22×C4C2×SD1614yes—
433C22×C4C2×SD1614yes—
434C22×C4AGL(1,ℤ/8)14yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4H(5,𝔽2)14yes—
445C22×C4C2511yescharacteristic
446SD16—88no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4C2×SD1618yes—
486C4⋊C4C2×SD1618yes—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—42no—
502C42—41no—
503C42—41no—
504C42—82no—
505C42—82no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—24no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23#53512yes—
525C23C22×SD1612yes—
526C23C2×AGL(1,ℤ/8)12yes—
527C23C23×D411yescharacteristic
528Q8—44no—
529Q8—44no—
530Q8—816no—
531Q8—162no—
532D4—28no—
533D4—48no—
534D4—44no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—816no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4#53512yes—
588C2×C4C22×SD1614yes—
589C2×C4C22×SD1612yes—
590C2×C4C22×SD1612yes—
591C2×C4C22×SD1614yes—
592C2×C4C2×AGL(1,ℤ/8)12yes—
593C2×C4C2×AGL(1,ℤ/8)12yes—
594C2×C4C2×H(5,𝔽2)12yes—
595C2×C4C2×2−1+412yes—
596C2×C4#58512yes—
597C8—44no—
598C8—44no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—816no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22#320814yes—
614C22C2×#53511yescharacteristic
615C22#348511yescharacteristic
616C22#348611yescharacteristic
617C22C23×SD1611yescharacteristic
618C22C22×AGL(1,ℤ/8)11yescharacteristic
619C22#358611yescharacteristic
620C22#358611yescharacteristic
621C4—22no—
622C4—22no—
623C4—24no—
624C4—22no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—42no—
629C4—42no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4#353514yes—
638C2—24no—
639C2—84no—
640C2#3711811yescharacteristic
641C2C2×#320811yescharacteristic
642C2#6080711yescharacteristic
643C2#6169511yescharacteristic
644C2#6170611yescharacteristic
645C2#6181811yescharacteristic
646C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
1. #35015C2semidirect
4. #32428C2semidirect
5. #32344C2semidirect
6. #31636C2semidirect
7. #31520C2semidirect
8. #29641C2semidirect
12. #27960C2semidirect
13. #27905C2semidirect
14. #27647C2semidirect
15. #21912C2semidirect
35. #2511C22semidirect
36. #2507C22semidirect
37. #2451C22semidirect
38. #2446C22semidirect
49. #2099C22semidirect
50. #2096C22semidirect
52. #2084C22semidirect
53. #2072C22semidirect
56. #2058C22semidirect
57. #2058C22semidirect
59. #2024C22semidirect
60. #2024C22semidirect
63. #1990C22semidirect
65. #1956C22semidirect
66. #1948C22semidirect
69. #1916C22semidirect
70. #1916C22semidirect
76. #1844C22semidirect
77. #1834C22semidirect
82. #1803C22semidirect
297. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350152
C2×#25422
#330072
#324282
#323442
#316362
#315202
#296412
#291032
#283022
#281202
#279602
#279052
#276472
#219122