Finite Group Databasebeta

← #6627951

Subgroups — #6627951

Order 512.

1767 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36599C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#32906C211yesmaximal, characteristic
4#32906C212yesmaximal
5#29335C212yesmaximal
6#29243C212yesmaximal
7#27972C211yesmaximal, characteristic
8#27680C211yesmaximal, characteristic
9#27569C214yesmaximal, complemented
10#26187C214yesmaximal, complemented
11#26108C212yesmaximal, complemented
12#24732C212yesmaximal, complemented
13#23004C212yesmaximal, complemented
14#22763C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic
16#21455C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C2×#462C2212yes—
21C2×#462C2214yes—
22C2×#451C2211yescharacteristic
23C2×#451C2212yes—
24C2×#441C2214yes—
25C2×C4×Q16C2212yes—
26#2857C2211yescharacteristic
27#2857C2212yes—
28#2585—22no—
29#2569—22no—
30#2552C2211yescharacteristic
31#2546C2211yescharacteristic
32#2511C2212yes—
33#2509—21no—
34#2446C2212yes—
35#2442—21no—
36#2370—21no—
37#2365C2211yescharacteristic
38#2355C2211yescharacteristic
39Q8×C4⋊C4—22no—
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2214yes—
43Q8×C4⋊C4C2212yes—
44Q8×C4⋊C4C2212yes—
45#2345—21no—
46C4×C22⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48C2×#389—22no—
49C2×#389C2212yes—
50C2×#384—21no—
51C2×#384C2211yescharacteristic
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2099C2212yes—
55#2099C2214yes—
56#2096C2212yes—
57#2088C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2043C2214yes—
61#1956C2212yes—
62#1948C2212yes—
63#1939C2212yes—
64#1935C2214yes—
65#1935C2214yes—
66#1918C2214yes—
67#1917C2214yes—
68#1878C2212yes—
69#1878C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1865C2214yes—
79#1865C2214yes—
80#1863C2212yes—
81#1853C2212yes—
82#1852C2211yescharacteristic
83#1834C2212yes—
84#1803C2212yes—
85#1803C2214yes—
86#1803C2212yes—
87#1797C2211yescharacteristic
88#1797C2212yes—
89#1787C2211yescharacteristic
90#1770C2212yes—
91#1763C2214yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C22×Q16C2312yes—
97C22×Q16C2314yes—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#125—21no—
101C2×C22⋊Q8C2312yes—
102C2×C22⋊Q8C2311yescharacteristic
103C2×C4×Q8—28no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—48no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C42—21no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#462—28no—
129#462—48no—
130#451—24no—
131#451—28no—
132#441—48no—
133C4×Q16—28no—
134C2×#107C2312yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2314yes—
138C2×#103C2312yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2311yescharacteristic
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152#400—22no—
153#400—22no—
154#400—22no—
155#398—22no—
156#398—22no—
157#398—22no—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2311yescharacteristic
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—24no—
175#389—24no—
176#389—24no—
177#389C2314yes—
178#389C2312yes—
179#389C2314yes—
180#389C2311yescharacteristic
181#387—22no—
182#387—22no—
183#387—21no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—21no—
190#385—22no—
191#384—22no—
192#384—24no—
193#384—22no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#382—21no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380—21no—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8#4114yes—
241C22×Q8C2411yescharacteristic
242C22×Q8C2412yes—
243C22×Q8C2412yes—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—28no—
255C2×Q16—28no—
256C2×Q16—28no—
257C2×Q16—24no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—44no—
263#125—44no—
264C22⋊Q8—24no—
265C22⋊Q8—22no—
266C22⋊Q8—24no—
267C22⋊Q8—22no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—28no—
274C4×Q8—28no—
275C4×Q8—24no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—48no—
281C4×Q8#4118yes—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—48no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—82no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2414yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—44no—
343C2×C42—48no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—24no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—44no—
356#103—24no—
357#103—24no—
358#103—28no—
359#103—24no—
360#103—24no—
361#103—28no—
362#103—44no—
363#103—44no—
364#103—44no—
365C22⋊C8—22no—
366C22⋊C8—24no—
367C22⋊C8—22no—
368#97—44no—
369C4×C8—24no—
370#95—24no—
371#95—24no—
372#95—21no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8—24no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—21no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8C2×#4114yes—
409C2×Q8C2×#4114yes—
410C2×Q8H(5,𝔽2)14yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4#13714yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4112yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455Q16—48no—
456Q16—416no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C4⋊C4—28no—
467C4⋊C4—24no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4#13714yes—
507C4⋊C4C2×#4114yes—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C42—28no—
512C42—28no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—42no—
519C42—42no—
520C42—48no—
521C42—48no—
522C42—42no—
523C42—42no—
524C42—82no—
525C42—82no—
526C42C2×#4114yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23#54812yes—
532C23C2×#13712yes—
533C23#57812yes—
534C23C23×D411yescharacteristic
535Q8—216no—
536Q8—24no—
537Q8—24no—
538Q8—24no—
539Q8—42no—
540Q8—44no—
541Q8—42no—
542Q8—44no—
543Q8#55318yes—
544C2×C4—216no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—21no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—88no—
600C2×C4#52512yes—
601C2×C4#52512yes—
602C2×C4#54812yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13714yes—
605C2×C4C2×#13712yes—
606C2×C4C2×#13712yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4112yes—
609C2×C4C2×H(5,𝔽2)12yes—
610C2×C4C2×2−1+411yescharacteristic
611C8—42no—
612C8—44no—
613C8—42no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22#337414yes—
621C22C2×#52511yescharacteristic
622C22C2×#54811yescharacteristic
623C22#348411yescharacteristic
624C22C22×#13711yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57811yescharacteristic
627C22#358511yescharacteristic
628C4—22no—
629C4—24no—
630C4—22no—
631C4—28no—
632C4—22no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—41no—
637C4—44no—
638C4—82no—
639C4—81no—
640C4—84no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—82no—
645C4#311314yes—
646C4C2×#55312yes—
647C4#349212yes—
648C2—24no—
649C2#3658111yescharacteristic
650C2C2×#311311yescharacteristic
651C2#6058911yescharacteristic
652C2C2×#337411yescharacteristic
653C2#6131911yescharacteristic
654C2#6158811yescharacteristic
655C2#6160211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
9. #27569C2semidirect
10. #26187C2semidirect
11. #26108C2semidirect
12. #24732C2semidirect
13. #23004C2semidirect
14. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×Q8×C4⋊C42
#329062
#293352
#292432
#279722
#276802
#275692
#261872
#261082
#247322
#230042
#227632
#226292
#214552
C4×#4512