Finite Group Databasebeta

← #208678

Subgroups — #208678

Order 512.

1119 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C8×#41C211yesmaximal, characteristic, complemented
2#21487C211yesmaximal, characteristic, complemented
3C4×#449C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic, complemented
5C2×#1586C211yesmaximal, characteristic, complemented
6#8948C211yesmaximal, characteristic, complemented
7#8947C211yesmaximal, characteristic, complemented
8#8940C211yesmaximal, characteristic, complemented
9#8846C211yesmaximal, characteristic, complemented
10#8843C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic, complemented
12#8312C211yesmaximal, characteristic, complemented
13#8205C211yesmaximal, characteristic, complemented
14#8204C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#449C411yescharacteristic
18C8×#41—24no—
19C8×#41—24no—
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×#436C2211yescharacteristic
25C2×#435—21no—
26C2×C8×D4—21no—
27C2×C8×D4—21no—
28C2×C8×D4C2211yescharacteristic
29C2×C8×D4C2211yescharacteristic
30C2×#431—21no—
31C2×#431C2211yescharacteristic
32C22×C4×C8—21no—
33C22×C4×C8C2211yescharacteristic
34C4×#121C2211yescharacteristic
35#1954C2211yescharacteristic
36#1939C2211yescharacteristic
37#1923—21no—
38#1880C411yescharacteristic
39#1879C411yescharacteristic
40#1875C411yescharacteristic
41#1869—21no—
42#1864C411yescharacteristic
43#1863C411yescharacteristic
44#1854C411yescharacteristic
45#1797C2211yescharacteristic
46#1788C411yescharacteristic
47#1770C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1753—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1588—24no—
54#1586—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×D8C2×C411yescharacteristic
74C22×M16—21no—
75C23×C8—21no—
76C23×C8—21no—
77C23×C8D411yescharacteristic
78C23×C8D411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C4×#41—22no—
81C4×#41—22no—
82C4×#41—22no—
83C4×#41—22no—
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C2×C4×D4C2311yescharacteristic
92C2×#117—21no—
93C2×#117C2×C411yescharacteristic
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#449—24no—
97C8×Q8—22no—
98C8×Q8—22no—
99C8×Q8—22no—
100C8×Q8D414yescomplemented
101C4×SD16—22no—
102C4×SD16—22no—
103C4×D8—22no—
104C4×D8—22no—
105#436—22no—
106#436—22no—
107#435—44no—
108C8×D4—28no—
109C8×D4—28no—
110C8×D4—22no—
111C8×D4—22no—
112C8×D4—22no—
113C8×D4D414yescomplemented
114#431—28no—
115#431—22no—
116#431—22no—
117C2×#107C2×C411yescharacteristic
118C2×#106C2×C411yescharacteristic
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8C2×C411yescharacteristic
143C2×C22⋊C8C2×C411yescharacteristic
144C2×#97—21no—
145C2×C4×C8—28no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8—22no—
150C2×C4×C8—22no—
151C2×C4×C8D411yescharacteristic
152C2×C4×C8D411yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#390C2×C411yescharacteristic
160#389C2×C411yescharacteristic
161#388—21no—
162#386—21no—
163#380C2×C411yescharacteristic
164C4×C4⋊C4C2311yescharacteristic
165C4×C22⋊C4—21no—
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#340C2×C411yescharacteristic
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176#335—22no—
177#334—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184#322—22no—
185#322—22no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—22no—
190C2×#41—22no—
191C2×#41M1614yescomplemented
192C22×Q8C22×C411yescharacteristic
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C22×C411yescharacteristic
196C22×D4C22×C411yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4#4111yescharacteristic
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—22no—
207C2×D8—22no—
208C2×D8—22no—
209C2×D8—22no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16—42no—
213C22×C8—24no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—24no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—41no—
225C22×C8—41no—
226C22×C8—41no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239#121—22no—
240#121—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8C2×D412yes—
246C4×Q8#4112yes—
247C4×D4—24no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4C2×D412yes—
262C4×D4#4112yes—
263#117—24no—
264#117—24no—
265#117—22no—
266#117—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42C2411yescharacteristic
297#107—22no—
298#106—22no—
299C4⋊C8—24no—
300C4⋊C8—24no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—42no—
308C4⋊C8C2×D412yes—
309C4⋊C8C2×D412yes—
310#103—22no—
311#103—22no—
312#103—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316#102—22no—
317#102—22no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321C22⋊C8—22no—
322C22⋊C8—22no—
323C22⋊C8—24no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328#97—42no—
329C4×C8—24no—
330C4×C8—24no—
331C4×C8—22no—
332C4×C8—24no—
333C4×C8—22no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8C2×D412yes—
345C4×C8C2×D412yes—
346#95—21no—
347#95—21no—
348#95—21no—
349C24—21no—
350C24—21no—
351C24C4×D411yescharacteristic
352#41—28no—
353#41—28no—
354#41—28no—
355#41—28no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8C4×D412yes—
360C2×Q8C2×M1612yes—
361C2×Q8#13112yes—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4C4×D412yes—
378C2×D4C2×M1612yes—
379C2×D4#13112yes—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4C4×D411yescharacteristic
419C22×C4C4×D411yescharacteristic
420C22×C4C4×D411yescharacteristic
421C22×C4C2×M1612yes—
422C22×C4#13112yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2×#4111yescharacteristic
426SD16—44no—
427D8—44no—
428M16—84no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—44no—
455C2×C8—44no—
456C2×C8—48no—
457C2×C8—48no—
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—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8#12012yes—
473C2×C8#12012yes—
474C2×C8AGL(1,ℤ/8)12yes—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4C4×D412yes—
488C4⋊C4C4×D412yes—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C42—24no—
500C42—24no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—22no—
505C42—21no—
506C42—22no—
507C42—21no—
508C42—21no—
509C42—41no—
510C42—41no—
511C42C22×D411yescharacteristic
512C42C22×D411yescharacteristic
513C42C2×#4111yescharacteristic
514C42C2×#4111yescharacteristic
515C23—22no—
516C23—21no—
517C23—22no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—22no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23#43512yes—
536C23C2×C4×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—22no—
540Q8#43514yescomplemented
541D4—216no—
542D4—22no—
543D4—22no—
544D4—22no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—84no—
550D4#43514yescomplemented
551C2×C4—216no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4#43512yes—
599C2×C4#44312yes—
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×#12011yescharacteristic
603C2×C4#52511yescharacteristic
604C2×C4#52511yescharacteristic
605C2×C4#52511yescharacteristic
606C2×C4C22×M1611yescharacteristic
607C2×C4C2×#13111yescharacteristic
608C2×C4C2×#13111yescharacteristic
609C2×C4C2×#13111yescharacteristic
610C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
611C2×C4#57511yescharacteristic
612C2×C4#57511yescharacteristic
613C2×C4#57511yescharacteristic
614C8—22no—
615C8—22no—
616C8—22no—
617C8—42no—
618C8—42no—
619C8—41no—
620C8—41no—
621C8—41no—
622C8—41no—
623C8—41no—
624C8—42no—
625C8—42no—
626C8—81no—
627C8—81no—
628C22—22no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—41no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22C4×#12011yescharacteristic
650C22C2×#43511yescharacteristic
651C22#292911yescharacteristic
652C22#294311yescharacteristic
653C22C4×AGL(1,ℤ/8)11yescharacteristic
654C22C2×#44311yescharacteristic
655C22#295511yescharacteristic
656C4—22no—
657C4—24no—
658C4—21no—
659C4—21no—
660C4—22no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4C2×#43511yescharacteristic
668C4C2×#43511yescharacteristic
669C4#292911yescharacteristic
670C4#292911yescharacteristic
671C4#300511yescharacteristic
672C4#300611yescharacteristic
673C4#311111yescharacteristic
674C4#311511yescharacteristic
675C2—42no—
676C2—44no—
677C2—161no—
678C2#1786811yescharacteristic
679C2#1787111yescharacteristic
680C2#1790911yescharacteristic
681C2#1791211yescharacteristic
682C2#2144011yescharacteristic
683C2#2145611yescharacteristic
684C2#2146611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C8×#41C2semidirect
2. #21487C2semidirect
4. C2×#1588C2semidirect
5. C2×#1586C2semidirect
6. #8948C2semidirect
7. #8947C2semidirect
8. #8940C2semidirect
9. #8846C2semidirect
10. #8843C2semidirect
11. #8831C2semidirect
12. #8312C2semidirect
13. #8205C2semidirect
14. #8204C2semidirect
15. #8145C2semidirect
100. C8×Q8D4semidirect
113. C8×D4D4semidirect
191. C2×#41M16semidirect
540. Q8#435semidirect
550. D4#435semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#412
#214872
C4×#4492
C2×#15882
C2×#15862
#89482
#89472
#89402
#88462
#88432
#88312
#83122
#82052
#82042
#81452