Finite Group Databasebeta

← #6506773

Subgroups — #6506773

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2441C211yesmaximal, characteristic
3#29303C212yesmaximal, complemented
4#29298C212yesmaximal, complemented
5#28788C214yesmaximal, complemented
6#28786C214yesmaximal, complemented
7#27953C212yesmaximal, complemented
8#26075C214yesmaximal, complemented
9#26073C214yesmaximal, complemented
10#25815C212yesmaximal, complemented
11#21942C211yesmaximal, characteristic
12#21942C211yesmaximal, characteristic
13#20846C211yesmaximal, characteristic
14#20846C211yesmaximal, characteristic
15#20524C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#499C2214yescomplemented
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#407C2211yescharacteristic
22C2×#407C2211yescharacteristic
23#2851C2212yes—
24#2851C2212yes—
25#2646—21no—
26#2646—21no—
27#2569—22no—
28#2569—22no—
29#2442C2212yes—
30#2442C2212yes—
31#2442C2212yes—
32#2442C2212yes—
33#2441—22no—
34#2441—22no—
35#2441C2214yes—
36#2441C2214yes—
37C2×#399—21no—
38C2×#396—21no—
39#2315—21no—
40#2315—21no—
41C2×#389—21no—
42C2×#389C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#387—21no—
45C2×#384C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2085C2214yescomplemented
51#2085C2214yescomplemented
52#2084C2214yescomplemented
53#2083C2214yescomplemented
54#2059C2214yescomplemented
55#2059C2214yescomplemented
56#2059C2214yescomplemented
57#2059C2214yescomplemented
58#1992C2214yescomplemented
59#1991C2214yescomplemented
60#1944C2214yescomplemented
61#1943C2214yescomplemented
62#1942C2214yescomplemented
63#1941C2214yescomplemented
64#1854C2214yes—
65#1854C2214yes—
66#1853C2212yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2212yes—
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1852C2212yes—
78#1847C2214yescomplemented
79#1809C2214yes—
80#1809C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87C24×C4C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90C2×#125—22no—
91C2×#125—44no—
92C2×#125C2314yes—
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×#117—21no—
96C2×#117—21no—
97C2×#117—44no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4D411yescharacteristic
119C22×C22⋊C4D411yescharacteristic
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#499—48no—
124#428—44no—
125#428—44no—
126C2×#107C2314yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134#407Q818yes—
135#407Q818yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2314yes—
142#400—22no—
143#400—22no—
144#400—22no—
145#400—22no—
146#399—24no—
147#399—24no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#396—24no—
153#396—24no—
154#389—22no—
155#389—22no—
156#389—24no—
157#389—24no—
158#389C2312yes—
159#389C2312yes—
160#389C2314yes—
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—24no—
169#387—24no—
170#385—22no—
171#385—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—22no—
179#384C2314yes—
180#382C2312yes—
181#382C2312yes—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4C2312yes—
201C4×C22⋊C4C2312yes—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25C2×D411yescharacteristic
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×Q814yes—
238C23×C4C2×Q814yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×M16—42no—
243C2×M16—42no—
244C2×M16—42no—
245C2×M16—42no—
246C2×M16C2×Q818yes—
247C2×M16C2×Q818yes—
248C22×C8C2414yes—
249C22×C8C2414yes—
250#125—28no—
251#125—44no—
252#125—44no—
253#125—44no—
254#125—48no—
255#117—24no—
256#117—24no—
257#117—42no—
258#117—42no—
259#117—44no—
260#117—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—82no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2412yes—
332#107—28no—
333#107—44no—
334#107—44no—
335#107—44no—
336#106—28no—
337#106—44no—
338#106—44no—
339#106—44no—
340C22⋊C8C2×Q818yes—
341C22⋊C8C2×Q818yes—
342C22⋊C8C2×Q818yes—
343C22⋊C8C2×Q818yes—
344#97—44no—
345C4×C8—44no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—24no—
353#95—24no—
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#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—22no—
367C24—22no—
368C24—41no—
369C24—41no—
370C24—41no—
371C24—41no—
372C24C4⋊Q814yes—
373C24C22×D411yescharacteristic
374C24C22×D411yescharacteristic
375C24C22×D411yescharacteristic
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—81no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4C4⋊Q814yes—
435C22×C4C4⋊Q814yes—
436C22×C4C4⋊Q814yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q812yes—
450C22×C4C2511yescharacteristic
451M16—48no—
452M16—48no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8C22×Q818yes—
458C2×C8C22×Q818yes—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4H(5,𝔽2)14yes—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—44no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—28no—
526C23—44no—
527C23—44no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23C2×C4⋊Q812yes—
533C23C2×C4⋊Q812yes—
534C23C2×C4⋊Q812yes—
535C23#57512yes—
536C23#57512yes—
537C23C23×D411yescharacteristic
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—48no—
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—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—162no—
598C2×C4—162no—
599C2×C4—162no—
600C2×C4—162no—
601C2×C4C2×C4⋊Q812yes—
602C2×C4C2×C4⋊Q812yes—
603C2×C4C2×C4⋊Q812yes—
604C2×C4C23×Q811yescharacteristic
605C2×C4C23×Q811yescharacteristic
606C2×C4C2×H(5,𝔽2)12yes—
607C2×C4#58512yes—
608C8—28no—
609C8—28no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—41no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—88no—
631C22#316614yes—
632C22C22×C4⋊Q811yescharacteristic
633C22#348411yescharacteristic
634C22#348511yescharacteristic
635C22C2×#57511yescharacteristic
636C22C2×#57511yescharacteristic
637C22#358511yescharacteristic
638C22#358611yescharacteristic
639C4—22no—
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—44no—
646C4—42no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4#350112yes—
655C4#350112yes—
656C2—24no—
657C2—42no—
658C2—42no—
659C2#3539111yescharacteristic
660C2C2×#316611yescharacteristic
661C2#6066411yescharacteristic
662C2#6157511yescharacteristic
663C2#6157711yescharacteristic
664C2#6171611yescharacteristic
665C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29303C2semidirect
4. #29298C2semidirect
5. #28788C2semidirect
6. #28786C2semidirect
7. #27953C2semidirect
8. #26075C2semidirect
9. #26073C2semidirect
10. #25815C2semidirect
18. C2×#499C22semidirect
50. #2085C22semidirect
51. #2085C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
54. #2059C22semidirect
55. #2059C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #1992C22semidirect
59. #1991C22semidirect
60. #1944C22semidirect
61. #1943C22semidirect
62. #1942C22semidirect
63. #1941C22semidirect
78. #1847C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#24412
#293032
#292982
#287882
#287862
#279532
#260752
#260732
#258152
#219422
#208462
#205242