Finite Group Databasebeta

← #227950

Subgroups — #227950

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22957C211yesmaximal, characteristic, complemented
2D4×C22⋊C8C211yesmaximal, characteristic
3C4×#450C211yesmaximal, characteristic
4C2×#1585C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic
6#9660C211yesmaximal, characteristic, complemented
7#9560C211yesmaximal, characteristic, complemented
8#9541C211yesmaximal, characteristic, complemented
9#9075C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8843C211yesmaximal, characteristic, complemented
13#8832C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic, complemented
15#8149C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#450C411yescharacteristic
18C2×#446C2211yescharacteristic, complemented
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×C22⋊C8—21no—
26C4×C22⋊Q8C2211yescharacteristic
27#1984C2211yescharacteristic, complemented
28#1954C2211yescharacteristic
29#1954C2211yescharacteristic
30#1939—21no—
31#1939C2211yescharacteristic, complemented
32#1917—21no—
33#1877C411yescharacteristic
34#1877C411yescharacteristic
35#1872C411yescharacteristic
36#1869—21no—
37#1869—21no—
38#1864C411yescharacteristic
39#1864C411yescharacteristic
40#1853C411yescharacteristic
41#1851—21no—
42#1797C2211yescharacteristic
43#1786C411yescharacteristic
44#1779—21no—
45C8×C4⋊C4C2211yescharacteristic, complemented
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51#1585—24no—
52C2×#335C2211yescharacteristic, complemented
53C2×#335C2211yescharacteristic, complemented
54#1486—24no—
55C2×Q8⋊C8C2211yescharacteristic, complemented
56C2×Q8⋊C8C2211yescharacteristic, complemented
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic, complemented
62C2×#322C2211yescharacteristic, complemented
63#1279C2211yescharacteristic, complemented
64#1279C2211yescharacteristic, complemented
65#1277C2211yescharacteristic, complemented
66#1277C2211yescharacteristic, complemented
67#1273C2211yescharacteristic, complemented
68#1273C2211yescharacteristic, complemented
69C23×D4C2×C411yescharacteristic
70C24×C4—21no—
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76C2×C22⋊Q8C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—41no—
83C2×C4×D4—41no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C42C2311yescharacteristic
91#450—24no—
92#446—22no—
93#446—41no—
94C4×SD16—22no—
95C4×SD16—22no—
96C4×SD16—41no—
97C4×SD16—41no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—41no—
101C8×D4—41no—
102#431—22no—
103#431—41no—
104C2×#106C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106#423—22no—
107#423—41no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×#103C2×C411yescharacteristic
117C2×#103C2×C411yescharacteristic
118C2×#102C2×C411yescharacteristic
119C2×#102C2×C411yescharacteristic
120C2×#102C2×C411yescharacteristic
121C2×#102C2×C411yescharacteristic
122C2×#102C2×C411yescharacteristic
123C2×#102C2×C411yescharacteristic
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—44no—
133C2×C22⋊C8C2×C411yescharacteristic
134C2×C22⋊C8C2×C411yescharacteristic
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×#97—21no—
138C2×C4×C8—21no—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#391C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#387—21no—
148#385—21no—
149#380C2×C411yescharacteristic
150C4×C4⋊C4—21no—
151C4×C4⋊C4C2311yescharacteristic
152C4×C22⋊C4—21no—
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#336—21no—
158#336—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336C2×C411yescharacteristic
163#336C2×C411yescharacteristic
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—22no—
167#335—41no—
168Q8⋊C8—22no—
169Q8⋊C8—41no—
170D4⋊C8—22no—
171D4⋊C8—22no—
172D4⋊C8—22no—
173D4⋊C8D414yes—
174#322—22no—
175#322—41no—
176C25—21no—
177C22×Q8C22×C411yescharacteristic
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4C22×C411yescharacteristic
186C22×D4C22×C411yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—81no—
197C23×C4—81no—
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4#4111yescharacteristic
201C2×SD16—22no—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—41no—
207C2×M16—42no—
208C2×M16—42no—
209C2×M16—81no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—81no—
221C22×C8—81no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22⋊Q8—22no—
231C22⋊Q8—41no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4C2×D412yes—
251C4×D4#4112yes—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#106—22no—
282#106—41no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—81no—
294C4⋊C8C2×D412yes—
295#103—22no—
296#103—41no—
297#102—22no—
298#102—22no—
299#102—22no—
300#102—22no—
301#102—41no—
302#102M1614yescomplemented
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—41no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—81no—
314C22⋊C8—81no—
315#97—42no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—81no—
328C4×C8C2×D412yes—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334C24—21no—
335C24—24no—
336C24—21no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—81no—
342C24—81no—
343C24C4×D411yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4C4×D412yes—
366C2×D4C2×M1612yes—
367C2×D4#13112yes—
368C2×D4#13112yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—81no—
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4AGL(1,ℤ/8)12yes—
418C22×C4#13712yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422SD16—44no—
423SD16—44no—
424M16—84no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—22no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—84no—
467C2×C8—84no—
468C2×C8C2×M1612yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4C4×D412yes—
483C4⋊C4C2×M1612yes—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—81no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42C22×D411yescharacteristic
503C42C2×#4111yescharacteristic
504C42H(5,𝔽2)11yescharacteristic
505C42H(5,𝔽2)11yescharacteristic
506C23—22no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—21no—
511C23—22no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—48no—
517C23—48no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23#44012yes—
534C23C2×C4×D411yescharacteristic
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
539D4—22no—
540D4—22no—
541D4—22no—
542D4—22no—
543D4—22no—
544D4—48no—
545D4—41no—
546D4—41no—
547D4#43614yes—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4#44012yes—
598C2×C4C2×C4×D411yescharacteristic
599C2×C4#51811yescharacteristic
600C2×C4D4211yescharacteristic
601C2×C4#54611yescharacteristic
602C2×C4#54611yescharacteristic
603C2×C4#55111yescharacteristic
604C2×C4C22×M1611yescharacteristic
605C2×C4C2×#13111yescharacteristic
606C2×C4C2×#13111yescharacteristic
607C2×C4#56811yescharacteristic
608C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
609C2×C4C2×#13711yescharacteristic
610C2×C4#57611yescharacteristic
611C2×C4#57711yescharacteristic
612C8—22no—
613C8—42no—
614C8—41no—
615C8—41no—
616C8—41no—
617C8—41no—
618C8—42no—
619C8—41no—
620C8—41no—
621C8—41no—
622C8—81no—
623C8—81no—
624C8—81no—
625C8—81no—
626C22—24no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—42no—
632C22—41no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—42no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—816no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22#235811yescharacteristic
649C22#292911yescharacteristic
650C22D4×M1611yescharacteristic
651C22C2×#44011yescharacteristic
652C22#294611yescharacteristic
653C22#295511yescharacteristic
654C22#295711yescharacteristic
655C4—22no—
656C4—21no—
657C4—21no—
658C4—41no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4C2×#43611yescharacteristic
667C4#293311yescharacteristic
668C4#297411yescharacteristic
669C4#297611yescharacteristic
670C4#326411yescharacteristic
671C4#326611yescharacteristic
672C4#331011yescharacteristic
673C4#331411yescharacteristic
674C2—42no—
675C2—44no—
676C2—82no—
677C2#1790511yescharacteristic
678C2#1791811yescharacteristic
679C2#1804011yescharacteristic
680C2#1805011yescharacteristic
681C2#2284811yescharacteristic
682C2#2285411yescharacteristic
683C2#2288911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22957C2semidirect
4. C2×#1585C2semidirect
6. #9660C2semidirect
7. #9560C2semidirect
8. #9541C2semidirect
9. #9075C2semidirect
10. #8952C2semidirect
11. #8945C2semidirect
12. #8843C2semidirect
13. #8832C2semidirect
14. #8200C2semidirect
15. #8149C2semidirect
18. C2×#446C22semidirect
27. #1984C22semidirect
31. #1939C22semidirect
45. C8×C4⋊C4C22semidirect
52. C2×#335C22semidirect
53. C2×#335C22semidirect
55. C2×Q8⋊C8C22semidirect
56. C2×Q8⋊C8C22semidirect
61. C2×#322C22semidirect
62. C2×#322C22semidirect
63. #1279C22semidirect
64. #1279C22semidirect
65. #1277C22semidirect
66. #1277C22semidirect
67. #1273C22semidirect
68. #1273C22semidirect
302. #102M16semidirect

Maximal subgroups

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

subgroupindex
#229572
D4×C22⋊C82
C4×#4502
C2×#15852
C2×#14862
#96602
#95602
#95412
#90752
#89522
#89452
#88432
#88322
#82002
#81492