Finite Group Databasebeta

← #6793067

Subgroups — #6793067

Order 512.

1523 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#36134C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#31628C214yesmaximal, complemented
5#31568C214yesmaximal, complemented
6#29936C212yesmaximal, complemented
7#29219C211yesmaximal, characteristic, complemented
8#28284C211yesmaximal, characteristic, complemented
9#28151C212yesmaximal
10#26108C212yesmaximal, complemented
11#25349C212yesmaximal
12#24845C212yesmaximal, complemented
13#24817C212yesmaximal, complemented
14#24157C212yesmaximal, complemented
15#22650C211yesmaximal, characteristic
16#21894C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#477C2212yes—
20C2×#475C2212yes—
21C2×#462C2211yescharacteristic
22C2×C4×Q16C2212yes—
23C2×#436C2212yes—
24C2×#423C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2574—22no—
28#2567C2211yescharacteristic
29#2563C2211yescharacteristic
30#2552—21no—
31#2552C2211yescharacteristic
32#2542—21no—
33#2538C2212yes—
34#2538C2211yescharacteristic
35#2508—21no—
36#2497C2212yes—
37#2454—21no—
38#2441—22no—
39#2429—22no—
40#2423C2214yes—
41#2423C2212yes—
42#2396—22no—
43#2365—21no—
44#2354—21no—
45Q8×C4⋊C4C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#125—21no—
49C4×C22⋊Q8C2212yes—
50#2293—21no—
51#2285C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2099C2214yes—
56#2088C2214yes—
57#2083C2214yes—
58#2075C2214yes—
59#2075C2214yes—
60#2068C2214yes—
61#2068C2214yes—
62#2035C2214yes—
63#2001C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1988C2212yes—
68#1981C2212yes—
69#1981C2211yescharacteristic
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1928C2212yes—
76#1894C2214yes—
77#1894C2214yes—
78#1894C2214yes—
79#1878C2212yes—
80#1878C2214yes—
81#1865C2214yes—
82#1851C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1809C2214yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2214yes—
89#1803C2212yes—
90#1803C2214yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C22×M16C2314yes—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—21no—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2314yes—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—22no—
122C22×C42C2312yes—
123#477—44no—
124#475—44no—
125#462—24no—
126C4×Q16—44no—
127#436—44no—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131#423—44no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×C22⋊C8C2314yes—
146C2×C4×C8C2312yes—
147#400—22no—
148#399—22no—
149#399—24no—
150#398—22no—
151#398—22no—
152#398—24no—
153#396—22no—
154#396—22no—
155#396—24no—
156#395—22no—
157#393—22no—
158#391—21no—
159#391—22no—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390—22no—
164#390—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—24no—
175#389C2312yes—
176#389C2314yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#388—24no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—24no—
185#385—22no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#382—21no—
196#382—22no—
197#380—22no—
198#380—22no—
199#380C2312yes—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223C43—21no—
224C43C2311yescharacteristic
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C2×Q16—24no—
245C2×Q16—24no—
246C2×Q16—42no—
247C2×M16—44no—
248C2×M16—44no—
249C2×M16—44no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#125—48no—
262#125—44no—
263C22⋊Q8—28no—
264C22⋊Q8—24no—
265C22⋊Q8—42no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—84no—
269C4×Q8—28no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—41no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—82no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285#117—48no—
286#117—84no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353#107—42no—
354#107—44no—
355#106—44no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360C4⋊C8—42no—
361C4⋊C8—44no—
362#103—24no—
363#103—42no—
364#103—44no—
365#103—42no—
366#103—42no—
367#103—44no—
368#103—44no—
369C22⋊C8—44no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24C22×D412yes—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×Q8C2×#4114yes—
407C2×Q82−1+412yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—42no—
432C22×C4C22×D412yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442Q16—48no—
443M16—88no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—24no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4#13714yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—48no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—84no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—21no—
512C42—21no—
513C42—41no—
514C42—48no—
515C42—42no—
516C42—42no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—81no—
523C42#13714yes—
524C42C2×#4112yes—
525C42H(5,𝔽2)12yes—
526C42H(5,𝔽2)12yes—
527C422−1+412yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—22no—
538Q8—24no—
539Q8—22no—
540Q8—41no—
541Q8—42no—
542Q8—41no—
543Q8—42no—
544Q8—84no—
545Q8—88no—
546D4—44no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4#53712yes—
598C2×C4#53712yes—
599C2×C4#54612yes—
600C2×C4#54612yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13714yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4C2×#13712yes—
606C2×C4#57612yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×2−1+411yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4#58511yescharacteristic
613C8—42no—
614C8—44no—
615C8—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22C2×#53711yescharacteristic
624C22C2×#54611yescharacteristic
625C22#348911yescharacteristic
626C22C22×#13711yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57611yescharacteristic
629C22#358611yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—24no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—42no—
639C4—81no—
640C4—82no—
641C4—82no—
642C4—81no—
643C4—82no—
644C4—84no—
645C4—81no—
646C4#325514yes—
647C4#333314yes—
648C4#351511yescharacteristic
649C4#352011yescharacteristic
650C4#354611yescharacteristic
651C4#355211yescharacteristic
652C2—82no—
653C2#3890811yescharacteristic
654C2C2×#325511yescharacteristic
655C2#6092811yescharacteristic
656C2C2×#333311yescharacteristic
657C2#6115711yescharacteristic
658C2#6211311yescharacteristic
659C2#6214911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33196C2semidirect
4. #31628C2semidirect
5. #31568C2semidirect
6. #29936C2semidirect
7. #29219C2semidirect
8. #28284C2semidirect
10. #26108C2semidirect
12. #24845C2semidirect
13. #24817C2semidirect
14. #24157C2semidirect
16. #21894C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349412
#331962
#316282
#315682
#299362
#292192
#282842
#281512
#261082
#253492
#248452
#248172
#241572
#226502
#218942
C4×#4622