Finite Group Databasebeta

← #6498110

Subgroups — #6498110

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#33150C212yesmaximal, complemented
4#32959C212yesmaximal
5#29982C212yesmaximal, complemented
6#29870C212yesmaximal, complemented
7#29582C214yesmaximal, complemented
8#29392C212yesmaximal, complemented
9#29388C212yesmaximal, complemented
10#29302C211yesmaximal, characteristic, complemented
11#29301C211yesmaximal, characteristic, complemented
12#29109C211yesmaximal, characteristic
13#29109C211yesmaximal, characteristic
14#27617C212yesmaximal, complemented
15#27588C212yesmaximal
16#22120C214yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C22×C4×D4C2211yescharacteristic
19C22×#117—21no—
20C2×#478C2214yes—
21#2864C2212yescomplemented
22#2863—21no—
23#2863—21no—
24#2862C2211yescharacteristic, complemented
25#2859C2211yescharacteristic, complemented
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2855—21no—
29#2855—21no—
30#2851C2212yes—
31C4×C4⋊Q8C2211yescharacteristic, complemented
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×#125C2212yescomplemented
34C4×C22⋊Q8—21no—
35C4×C22⋊Q8—21no—
36#2295—21no—
37#2295—21no—
38#2293—21no—
39#2293—21no—
40#2293—21no—
41#2293—21no—
42#2293C2212yes—
43#2293C2212yes—
44#2293C2212yes—
45#2293C2212yes—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#389C2211yescharacteristic
50C2×#388—21no—
51C2×#388—21no—
52#2283C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4—21no—
55C2×C43C2211yescharacteristic
56#2101C2214yescomplemented
57#2092C2214yes—
58#2085C2214yescomplemented
59#2084C2212yescomplemented
60#2083C2212yescomplemented
61#2073C2214yes—
62#2071C2212yes—
63#2071C2212yes—
64#2058C2212yescomplemented
65#2057C2212yescomplemented
66#2023C2212yes—
67#2023C2212yes—
68#1989C2214yescomplemented
69#1949C2214yescomplemented
70#1944C2214yescomplemented
71#1943C2214yescomplemented
72#1936C2214yes—
73#1936C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1920C2214yescomplemented
77#1919C2214yescomplemented
78#1918C2214yescomplemented
79#1918C2214yescomplemented
80#1864C2214yes—
81#1862C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1852C2212yes—
85#1852C2212yes—
86#1839C2212yescomplemented
87#1831C2212yescomplemented
88#1824C2212yes—
89#1824C2212yes—
90#1817C2212yes—
91#1804C2214yes—
92#1797C2212yes—
93#1786C2212yes—
94#1786C2212yes—
95#1767C2212yescomplemented
96#1749C2211yescharacteristic
97#1749C2211yescharacteristic
98#1745C2212yescomplemented
99C23×D4C2311yescharacteristic
100C24×C4—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×D4—28no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117—22no—
117C2×#117—48no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#478—216no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#383—22no—
207#382C2312yes—
208#382C2312yes—
209#382C2312yes—
210#381—22no—
211#380—21no—
212#380—21no—
213#380C2312yes—
214#380C2312yes—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230#376—22no—
231C2×#95—21no—
232C2×#95—21no—
233C2×#95—21no—
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43D411yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241C43C2311yescharacteristic
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#336C2314yes—
251#336C2312yes—
252#336C2312yes—
253#336C2314yes—
254#336C2314yes—
255C25—21no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×D4—24no—
259C22×D4—21no—
260C22×D4—24no—
261C22×D4—42no—
262C22×D4C2412yes—
263C23×C4—28no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4—22no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—44no—
270C23×C4—44no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C22×C8C2414yes—
279C22×C8C2414yes—
280C4⋊Q8—42no—
281C4⋊Q8—42no—
282#125—28no—
283C22⋊Q8—44no—
284C22⋊Q8—44no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4#4118yes—
293#117—48no—
294#117—48no—
295#117—48no—
296#117—48no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4—44no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—42no—
353C2×C42—42no—
354C2×C42—42no—
355C2×C42—42no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—44no—
387#106—28no—
388#106—44no—
389C4⋊C8—28no—
390#102—28no—
391#102—28no—
392#102—44no—
393#102—44no—
394C22⋊C8—24no—
395C22⋊C8—24no—
396#97—42no—
397#97—42no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418C24—24no—
419C24—22no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24C22×D411yescharacteristic
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—416no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C2×#4114yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—44no—
472C22×C4—432no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C42−1+414yes—
498C22×C4C2511yescharacteristic
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4C2×#4114yes—
532C22⋊C4—28no—
533C22⋊C4—28no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—41no—
559C42—41no—
560C42—41no—
561C42—41no—
562C42—41no—
563C42—41no—
564C42—41no—
565C42—41no—
566C42—41no—
567C42—41no—
568C42—41no—
569C42—41no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42AGL(1,ℤ/8)14yes—
575C42C2×#4114yes—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—24no—
581C23—48no—
582C23—48no—
583C23—48no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23#53612yes—
595C23#57512yes—
596C23#57512yes—
597C23C23×D411yescharacteristic
598Q8—84no—
599Q8—84no—
600D4—28no—
601D4—48no—
602D4—44no—
603C2×C4—28no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—28no—
619C2×C4—21no—
620C2×C4—24no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—416no—
643C2×C4—416no—
644C2×C4—48no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4#52512yes—
669C2×C4#52512yes—
670C2×C4#53612yes—
671C2×C4C2×AGL(1,ℤ/8)12yes—
672C2×C4C2×AGL(1,ℤ/8)12yes—
673C2×C4C2×AGL(1,ℤ/8)12yes—
674C2×C4#57512yes—
675C2×C4C22×#4112yes—
676C2×C4C2×2−1+412yes—
677C2×C4#58512yes—
678C8—44no—
679C8—44no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—816no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22#324014yes—
695C22C2×#52511yescharacteristic
696C22C2×#53611yescharacteristic
697C22#344911yescharacteristic
698C22C22×AGL(1,ℤ/8)11yescharacteristic
699C22C2×#57511yescharacteristic
700C22C2×#57511yescharacteristic
701C22C2×#57511yescharacteristic
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—44no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4#349514yes—
723C2—24no—
724C2—84no—
725C2#3532511yescharacteristic
726C2#6054611yescharacteristic
727C2#6055811yescharacteristic
728C2C2×#324011yescharacteristic
729C2#6086411yescharacteristic
730C2#6094911yescharacteristic
731C2#6096411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #33150C2semidirect
5. #29982C2semidirect
6. #29870C2semidirect
7. #29582C2semidirect
8. #29392C2semidirect
9. #29388C2semidirect
10. #29302C2semidirect
11. #29301C2semidirect
14. #27617C2semidirect
16. #22120C2semidirect
17. #20778C2semidirect
21. #2864C22semidirect
24. #2862C22semidirect
25. #2859C22semidirect
31. C4×C4⋊Q8C22semidirect
32. C4×C4⋊Q8C22semidirect
33. C4×#125C22semidirect
56. #2101C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2083C22semidirect
64. #2058C22semidirect
65. #2057C22semidirect
68. #1989C22semidirect
69. #1949C22semidirect
70. #1944C22semidirect
71. #1943C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1918C22semidirect
86. #1839C22semidirect
87. #1831C22semidirect
95. #1767C22semidirect
98. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#22932
#331502
#329592
#299822
#298702
#295822
#293922
#293882
#293022
#293012
#291092
#276172
#275882
#221202
#207782