Finite Group Databasebeta

← #6446027

Subgroups — #6446027

Order 512.

1942 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic, complemented
3#25073C214yesmaximal, complemented
4#25072C214yesmaximal, complemented
5#25059C212yesmaximal, complemented
6#25058C212yesmaximal, complemented
7#25057C212yesmaximal, complemented
8#25056C212yesmaximal, complemented
9#23731C212yesmaximal
10#23657C212yesmaximal, complemented
11#21303C211yesmaximal, characteristic, complemented
12#21171C211yesmaximal, characteristic, complemented
13#21060C212yesmaximal
14#21034C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16#20646C211yesmaximal, characteristic
17C4×#418C211yesmaximal, characteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#428C2212yes—
24C2×#423C2211yescharacteristic
25C2×#418C2211yescharacteristic
26C2×#418C2212yes—
27C2×#418C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2538—21no—
30#2538—21no—
31#2538—22no—
32#2538C2212yes—
33#2538C2212yes—
34#2538C2214yes—
35C2×#396—22no—
36#2355C2211yescharacteristic, complemented
37#2355C2211yescharacteristic, complemented
38#2355C2212yescomplemented
39#2355C2212yescomplemented
40Q8×C4⋊C4C2211yescharacteristic, complemented
41Q8×C4⋊C4C2211yescharacteristic, complemented
42#2345—22no—
43#2345—21no—
44Q8×C22⋊C4—21no—
45#2295—22no—
46#2295—21no—
47C2×#390C2212yes—
48C2×#389C2212yes—
49#2285C2211yescharacteristic
50#2285C2212yes—
51C2×#384C2211yescharacteristic
52C2×#380—21no—
53C42×Q8—21no—
54C4×#117C2212yes—
55C4×#117C2211yescharacteristic
56#2094C2214yes—
57#2093C2214yes—
58#2092C2214yes—
59#2092C2214yes—
60#2085C2214yescomplemented
61#2085C2214yescomplemented
62#2084C2214yescomplemented
63#2083C2214yescomplemented
64#1985C2214yescomplemented
65#1984C2212yescomplemented
66#1984C2212yescomplemented
67#1948C2212yescomplemented
68#1948C2212yescomplemented
69#1947C2212yescomplemented
70#1947C2212yescomplemented
71#1926C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1925C2214yes—
75#1921C2214yescomplemented
76#1920C2214yescomplemented
77#1919C2214yescomplemented
78#1918C2214yescomplemented
79#1847C2214yescomplemented
80#1845C2212yescomplemented
81#1844C2212yescomplemented
82#1804C2214yes—
83#1802C2212yes—
84#1802C2212yes—
85#1762C2212yes—
86#1762C2212yes—
87C4×#102C2212yes—
88C4×#102C2212yes—
89#1736C2211yescharacteristic
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C23×Q8D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×M16C2312yes—
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—48no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D412yes—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2312yes—
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D412yes—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#428—28no—
137#428—44no—
138#428—44no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145#423—24no—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148#418—24no—
149#418—44no—
150#418Q818yescomplemented
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#102C2312yes—
157C4×M16—24no—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#396—24no—
161#396—24no—
162#396—24no—
163#396—24no—
164#391—21no—
165#391—21no—
166#391—22no—
167#391—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#384C2312yes—
194#382C2312yes—
195#382C2312yes—
196#382C2312yes—
197#380—22no—
198#380—21no—
199#380—22no—
200#380—21no—
201#380—22no—
202#380—24no—
203#380—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—21no—
224#376—22no—
225#376—21no—
226C2×#95—22no—
227C43—22no—
228C43—21no—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240#336C2314yes—
241C25—21no—
242C22×Q8—24no—
243C22×Q8—24no—
244C22×Q8—42no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×D412yes—
247C22×Q8C2×D411yescharacteristic
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4C2×Q814yes—
255C22×D4C2412yes—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—24no—
273C2×M16—24no—
274C2×M16—24no—
275C2×M16C2×Q814yes—
276C2×M16#4114yes—
277C22×C8C2414yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22⋊Q8—48no—
281C22⋊Q8—416no—
282C22⋊Q8—48no—
283C4×Q8—24no—
284C4×Q8—28no—
285C4×Q8—24no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—24no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—44no—
300#117—22no—
301#117—22no—
302#117—24no—
303#117—24no—
304#117—22no—
305#117C2×Q814yes—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—48no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—48no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—84no—
358C2×C42—21no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42#4114yes—
376C2×C42#4114yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—28no—
382#107—44no—
383#107—44no—
384#106—28no—
385#106—44no—
386#106—44no—
387C4⋊C8—24no—
388C4⋊C8—24no—
389#102—24no—
390#102—24no—
391#102—44no—
392#102—44no—
393#102C2×Q818yescomplemented
394#102C2×Q818yescomplemented
395#97—24no—
396C4×C8—24no—
397#95—24no—
398#95—24no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413C24—24no—
414C24—22no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24C22×D411yescharacteristic
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—42no—
430C2×Q8—416no—
431C2×Q8AGL(1,ℤ/8)14yes—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—28no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4C22×Q814yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—816no—
476C22×C4#12314yes—
477C22×C4#12314yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×Q812yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494M16—28no—
495M16—28no—
496M16—48no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8C22×Q814yes—
505C2×C8C2×#4114yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—81no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4C22×Q812yes—
544C4⋊C4C22×Q812yes—
545C22⋊C4—24no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—44no—
562C42—42no—
563C42—44no—
564C42—44no—
565C42—42no—
566C42—42no—
567C42—81no—
568C42—81no—
569C42—82no—
570C42—82no—
571C42#12314yes—
572C42#12314yes—
573C42C2×#4112yes—
574C42C2×#4112yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—24no—
582C23—48no—
583C23—48no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—48no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—84no—
592C23—84no—
593C23—84no—
594C23—84no—
595C23C2×#12312yes—
596C23D4×Q812yes—
597C23#55012yes—
598C23C23×D411yescharacteristic
599Q8—28no—
600Q8—48no—
601Q8—44no—
602D4—24no—
603D4—24no—
604D4—48no—
605D4—42no—
606D4—42no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—48no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—88no—
652C2×C4—84no—
653C2×C4—88no—
654C2×C4—84no—
655C2×C4—88no—
656C2×C4C2×#12312yes—
657C2×C4C2×#12312yes—
658C2×C4C2×#12312yes—
659C2×C4C2×#12312yes—
660C2×C4C2×#12312yes—
661C2×C4#53314yes—
662C2×C4#54214yes—
663C2×C4D4×Q812yes—
664C2×C4#55012yes—
665C2×C4#55214yes—
666C2×C4#55414yes—
667C2×C4C2×AGL(1,ℤ/8)14yes—
668C2×C4C2×AGL(1,ℤ/8)12yes—
669C2×C4#57512yes—
670C2×C4C23×Q811yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C22×#4111yescharacteristic
674C2×C4C2×H(5,𝔽2)11yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C8—24no—
677C8—24no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—816no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22#242314yes—
694C22C22×#12311yescharacteristic
695C22C2×D4×Q811yescharacteristic
696C22C2×#55011yescharacteristic
697C22C22×AGL(1,ℤ/8)11yescharacteristic
698C22C2×#57511yescharacteristic
699C22#358511yescharacteristic
700C22#358611yescharacteristic
701C4—22no—
702C4—24no—
703C4—22no—
704C4—22no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—82no—
711C4—81no—
712C4—82no—
713C4—84no—
714C4—81no—
715C4—82no—
716C4#242314yes—
717C4#309414yes—
718C4C2×#53311yescharacteristic
719C4C2×#54211yescharacteristic
720C4C2×#55211yescharacteristic
721C4C2×#55411yescharacteristic
722C2—24no—
723C2—84no—
724C2C2×#242311yescharacteristic
725C2C2×#309411yescharacteristic
726C2#6052011yescharacteristic
727C2Q8×AGL(1,ℤ/8)11yescharacteristic
728C2#6136711yescharacteristic
729C2#6140211yescharacteristic
730C2#6140311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×#117C2semidirect
3. #25073C2semidirect
4. #25072C2semidirect
5. #25059C2semidirect
6. #25058C2semidirect
7. #25057C2semidirect
8. #25056C2semidirect
10. #23657C2semidirect
11. #21303C2semidirect
12. #21171C2semidirect
14. #21034C2semidirect
15. #20800C2semidirect
36. #2355C22semidirect
37. #2355C22semidirect
38. #2355C22semidirect
39. #2355C22semidirect
40. Q8×C4⋊C4C22semidirect
41. Q8×C4⋊C4C22semidirect
60. #2085C22semidirect
61. #2085C22semidirect
62. #2084C22semidirect
63. #2083C22semidirect
64. #1985C22semidirect
65. #1984C22semidirect
66. #1984C22semidirect
67. #1948C22semidirect
68. #1948C22semidirect
69. #1947C22semidirect
70. #1947C22semidirect
75. #1921C22semidirect
76. #1920C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1847C22semidirect
80. #1845C22semidirect
81. #1844C22semidirect
150. #418Q8semidirect
393. #102C2×Q8semidirect
394. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
Q8×#1172
#250732
#250722
#250592
#250582
#250572
#250562
#237312
#236572
#213032
#211712
#210602
#210342
#208002
#206462
C4×#4182