Finite Group Databasebeta

← #6795538

Subgroups — #6795538

Order 512.

1519 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic, complemented
4#32302C214yesmaximal, complemented
5#31956C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#28687C211yesmaximal, characteristic, complemented
8#28542C212yesmaximal, complemented
9#28283C211yesmaximal, characteristic, complemented
10#27738C212yesmaximal, complemented
11#27680C212yesmaximal, complemented
12#25702C212yesmaximal
13#24660C212yesmaximal, complemented
14#24122C212yesmaximal
15#22799C211yesmaximal, characteristic
16#21987C212yesmaximal, complemented
17C4×#462C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#498C2212yes—
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#462C2211yescharacteristic
23C2×C4×Q16C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2573—22no—
28#2571C2211yescharacteristic
29#2568C2211yescharacteristic
30#2567C2212yes—
31#2552—21no—
32#2552—21no—
33#2547C2211yescharacteristic
34#2538C2211yescharacteristic
35#2514—21no—
36#2502C2214yes—
37#2494C2212yes—
38#2447—21no—
39#2440—22no—
40#2440—22no—
41#2421C2212yes—
42#2401—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#124—21no—
48C4×C22⋊Q8C2212yes—
49#2295—21no—
50C2×#384C2212yes—
51#2283C2212yes—
52C42×Q8C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2097C2214yes—
56#2096C2214yes—
57#2084C2214yes—
58#2074C2214yes—
59#2066C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2001C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1971C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1947C2211yescharacteristic
76#1935C2214yes—
77#1928C2212yes—
78#1924C2214yes—
79#1878C2212yes—
80#1872C2214yes—
81#1853C2214yes—
82#1853C2214yes—
83#1852C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1836C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#554—44no—
97#554—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—22no—
104C2×#124—21no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2314yes—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—22no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4C2314yes—
120C22×C42C2312yes—
121#498—44no—
122#477—44no—
123#475—44no—
124#462—24no—
125C4×Q16—44no—
126#432—28no—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#401—22no—
149#400—22no—
150#400—22no—
151#400—22no—
152#398—24no—
153#398—24no—
154#397—22no—
155#397—22no—
156#397—22no—
157#396—22no—
158#395—24no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2314yes—
165#391C2312yes—
166#390—21no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2312yes—
178#388—22no—
179#388—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#385—22no—
184#385—24no—
185#385—24no—
186#384—21no—
187#384—21no—
188#384—22no—
189#384—22no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2314yes—
193#384C2312yes—
194#384C2312yes—
195#383—22no—
196#382—21no—
197#382C2312yes—
198#382C2312yes—
199#381—22no—
200#380—22no—
201#380—22no—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
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⋊C4—22no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225C43—21no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×Q8C2411yescharacteristic
240C22×D4C2×D411yescharacteristic
241C23×C4—24no—
242C23×C4C2×D412yes—
243C23×C4C2×D412yes—
244C23×C4C2412yes—
245C2×Q16—24no—
246C2×Q16—24no—
247C2×Q16—42no—
248C22×C8C2412yes—
249C22×C8C2414yes—
250C22×C8C2412yes—
251C4⋊Q8—22no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—82no—
258C4⋊Q8—82no—
259#125—48no—
260#124—44no—
261C22⋊Q8—24no—
262C22⋊Q8—416no—
263C22⋊Q8—42no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—41no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—48no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—84no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—42no—
346#107—44no—
347#106—44no—
348#106—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—42no—
354C4⋊C8#4114yes—
355#103—24no—
356#103—42no—
357#103—44no—
358#103—42no—
359#103—42no—
360#103—44no—
361#103—44no—
362C22⋊C8—28no—
363C22⋊C8—44no—
364#97—28no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24C22×D412yes—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q82−1+412yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4#13714yes—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444Q16—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8C2×#4114yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—82no—
490C4⋊C4—84no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—42no—
512C42—44no—
513C42—42no—
514C42—42no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—81no—
519C42#13714yes—
520C42C2×#4112yes—
521C42H(5,𝔽2)12yes—
522C42H(5,𝔽2)12yes—
523C422−1+412yes—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23C2×#13714yes—
531C23C23×D411yescharacteristic
532Q8—22no—
533Q8—22no—
534Q8—41no—
535Q8—41no—
536Q8—42no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—84no—
541Q8—164no—
542D4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—48no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—168no—
595C2×C4#53512yes—
596C2×C4#53512yes—
597C2×C4#54712yes—
598C2×C4#54712yes—
599C2×C4C2×#13712yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4#57712yes—
604C2×C4C22×#4111yescharacteristic
605C2×C4C2×H(5,𝔽2)11yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C2×C4C2×2−1+411yescharacteristic
609C2×C4#58511yescharacteristic
610C8—24no—
611C8—44no—
612C8—42no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22C2×#53511yescharacteristic
621C22C2×#54711yescharacteristic
622C22#348911yescharacteristic
623C22C22×#13711yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57711yescharacteristic
626C22#358511yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—81no—
640C4—82no—
641C4—84no—
642C4—81no—
643C4#321814yes—
644C4#335614yes—
645C4#351711yescharacteristic
646C4#352011yescharacteristic
647C4#354711yescharacteristic
648C4#355211yescharacteristic
649C2—44no—
650C2#3892011yescharacteristic
651C2C2×#321811yescharacteristic
652C2#6083311yescharacteristic
653C2C2×#335611yescharacteristic
654C2#6125411yescharacteristic
655C2#6211211yescharacteristic
656C2#6213111yescharacteristic
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. #33198C2semidirect
4. #32302C2semidirect
5. #31956C2semidirect
6. #29926C2semidirect
7. #28687C2semidirect
8. #28542C2semidirect
9. #28283C2semidirect
10. #27738C2semidirect
11. #27680C2semidirect
13. #24660C2semidirect
16. #21987C2semidirect
17. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349452
#331982
#323022
#319562
#299262
#286872
#285422
#282832
#277382
#276802
#257022
#246602
#241222
#227992
#219872
C4×#4622