Finite Group Databasebeta

← #6500827

Subgroups — #6500827

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2417C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic
3#27614C211yesmaximal, characteristic, complemented
4#27611C211yesmaximal, characteristic, complemented
5#27604C212yesmaximal, complemented
6#27586C211yesmaximal, characteristic, complemented
7#27583C211yesmaximal, characteristic, complemented
8#26227C214yesmaximal, complemented
9#26223C214yesmaximal, complemented
10#26155C214yesmaximal, complemented
11#26151C214yesmaximal, complemented
12#24010C212yesmaximal, complemented
13#21768C212yesmaximal, complemented
14#21742C211yesmaximal, characteristic
15#20844C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#485C2214yescomplemented
21C2×#428C2214yescomplemented
22C2×#407C2212yes—
23#2417—22no—
24#2417—22no—
25#2417C2214yes—
26#2417C2214yes—
27C2×#398—21no—
28C2×#396—21no—
29#2357—22no—
30#2357—22no—
31#2354C2212yes—
32#2354C2212yes—
33#2354C2212yes—
34#2354C2212yes—
35#2312—21no—
36#2312—21no—
37#2309—21no—
38C2×#391C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#386—21no—
41#2285C2211yescharacteristic
42#2285C2211yescharacteristic
43#2280C2212yes—
44#2280C2212yes—
45C2×#380—21no—
46C2×#380C2211yescharacteristic
47#2274—21no—
48#2100C2212yescomplemented
49#2099C2212yescomplemented
50#2096C2212yescomplemented
51#2095C2212yescomplemented
52#2084C2212yescomplemented
53#2083C2212yescomplemented
54#2077C2214yescomplemented
55#2075C2214yescomplemented
56#2072C2212yescomplemented
57#2072C2212yescomplemented
58#2036C2214yescomplemented
59#2035C2214yescomplemented
60#1928C2214yescomplemented
61#1927C2214yescomplemented
62#1921C2214yescomplemented
63#1920C2214yescomplemented
64#1894C2214yescomplemented
65#1894C2214yescomplemented
66#1873C2214yescomplemented
67#1873C2214yescomplemented
68#1873C2214yescomplemented
69#1870C2214yes—
70#1870C2214yes—
71#1854C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1853C2214yescomplemented
75#1852C2212yes—
76#1852C2212yes—
77#1852C2214yescomplemented
78#1825C2214yescomplemented
79#1824C2212yes—
80#1824C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1787C2212yes—
84#1787C2212yes—
85#1779C2212yes—
86#1736C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2314yes—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8C2314yes—
98C2×C22⋊Q8C2314yes—
99C2×C4×D4—28no—
100C2×C4×D4D412yes—
101C2×C4×D4D412yes—
102C2×C4×D4C2312yes—
103C2×C4×D4C2312yes—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117C2312yes—
107C2×#117C2312yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—44no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42C2311yescharacteristic
127#485—48no—
128#428—48no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137#407—28no—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143#398—24no—
144#398—24no—
145#396—24no—
146#396—24no—
147#391—21no—
148#391—21no—
149#391C2312yes—
150#391C2312yes—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#389—21no—
156#389—21no—
157#389C2312yes—
158#389C2312yes—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#386—24no—
168#386—24no—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#382—21no—
174#382—21no—
175#381—22no—
176#381—22no—
177#380—21no—
178#380—21no—
179#380—22no—
180#380—24no—
181#380—24no—
182#380C2312yes—
183#380C2314yes—
184#380C2312yes—
185#380C2312yes—
186#380C2312yes—
187#380C2312yes—
188#380C2312yes—
189#379—22no—
190#379—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219C25C2×D411yescharacteristic
220C25C2×D411yescharacteristic
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×D4—24no—
227C22×D4—24no—
228C22×D4—42no—
229C22×D4C2×D412yes—
230C22×D4C2×D412yes—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D412yes—
250C23×C4#4118yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C2×M16—28no—
255C2×M16—44no—
256C2×M16—44no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C22⋊Q8—28no—
260C22⋊Q8—28no—
261C22⋊Q8—28no—
262C22⋊Q8—44no—
263C22⋊Q8—48no—
264C22⋊Q8—48no—
265C4×D4—28no—
266C4×D4—28no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271#117—44no—
272#117—44no—
273#117—44no—
274#117—44no—
275#117—44no—
276#117—44no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—82no—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—28no—
347#107—44no—
348#106—28no—
349#106—44no—
350#103—44no—
351#103—44no—
352#103—44no—
353#103—44no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—28no—
357C22⋊C8—44no—
358C22⋊C8—44no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24#12314yes—
388C24C22×D412yes—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—88no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—416no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—416no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—48no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—88no—
453C22×C4—84no—
454C22×C4—816no—
455C22×C4#12314yes—
456C22×C4#12318yes—
457C22×C4#13714yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2511yescharacteristic
469M16—416no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—48no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4H(5,𝔽2)14yes—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—84no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—44no—
524C42—44no—
525C42—82no—
526C42—82no—
527C42—82no—
528C42—82no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—82no—
557C23—82no—
558C23—82no—
559C23—82no—
560C23C2×#12314yes—
561C23C2×#12312yes—
562C23C2×#13712yes—
563C23#57512yes—
564C23C23×D411yescharacteristic
565Q8—44no—
566Q8—44no—
567Q8—84no—
568D4—48no—
569D4—48no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—816no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4C2×#12312yes—
628C2×C4C2×#12314yes—
629C2×C4C2×#13712yes—
630C2×C4C2×#13712yes—
631C2×C4C22×#4112yes—
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4#58512yes—
634C8—44no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—416no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—88no—
658C22#309514yes—
659C22C22×#12311yescharacteristic
660C22#348411yescharacteristic
661C22#348511yescharacteristic
662C22C22×#13711yescharacteristic
663C22C2×#57511yescharacteristic
664C22#358511yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—24no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—41no—
672C4—41no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#350514yes—
682C2—24no—
683C2—44no—
684C2—82no—
685C2#3535111yescharacteristic
686C2C2×#309511yescharacteristic
687C2#6052011yescharacteristic
688C2#6157611yescharacteristic
689C2#6158411yescharacteristic
690C2#6171111yescharacteristic
691C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27614C2semidirect
4. #27611C2semidirect
5. #27604C2semidirect
6. #27586C2semidirect
7. #27583C2semidirect
8. #26227C2semidirect
9. #26223C2semidirect
10. #26155C2semidirect
11. #26151C2semidirect
12. #24010C2semidirect
13. #21768C2semidirect
15. #20844C2semidirect
20. C2×#485C22semidirect
21. C2×#428C22semidirect
48. #2100C22semidirect
49. #2099C22semidirect
50. #2096C22semidirect
51. #2095C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
54. #2077C22semidirect
55. #2075C22semidirect
56. #2072C22semidirect
57. #2072C22semidirect
58. #2036C22semidirect
59. #2035C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
62. #1921C22semidirect
63. #1920C22semidirect
64. #1894C22semidirect
65. #1894C22semidirect
66. #1873C22semidirect
67. #1873C22semidirect
68. #1873C22semidirect
71. #1854C22semidirect
74. #1853C22semidirect
77. #1852C22semidirect
78. #1825C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24172
#341862
#276142
#276112
#276042
#275862
#275832
#262272
#262232
#261552
#261512
#240102
#217682
#217422
#208442