Finite Group Databasebeta

← #196173

Subgroups — #196173

Order 512.

1363 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#60276C211yesmaximal, characteristic, complemented
2C4×#575C211yesmaximal, characteristic
3C2×#1477C211yesmaximal, characteristic, complemented
4C2×#1477C211yesmaximal, characteristic, complemented
5#17349C211yesmaximal, characteristic, complemented
6#9096C214yesmaximal, complemented
7#9095C214yesmaximal, complemented
8C2×#1283C211yesmaximal, characteristic, complemented
9#7882C211yesmaximal, characteristic, complemented
10C2×#575C411yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#2988—22no—
14#2988—22no—
15#2982—21no—
16#2972—22no—
17#2972—21no—
18C8×#41—22no—
19C8×#41C2214yes—
20C4×#137—21no—
21C4×#137—21no—
22C4×#137C412yes—
23C4×#137C2212yes—
24C4×AGL(1,ℤ/8)—21no—
25C4×AGL(1,ℤ/8)—21no—
26C4×AGL(1,ℤ/8)C412yes—
27C4×AGL(1,ℤ/8)C2212yes—
28#2943—22no—
29C4×#135—22no—
30C2×#431—22no—
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34#2911C411yescharacteristic
35C22×C4⋊C8C2211yescharacteristic
36#2896C411yescharacteristic
37#2896C411yescharacteristic
38C2×C4×M16C2211yescharacteristic
39#1477—22no—
40#1477—22no—
41#1477—22no—
42#1477C2214yes—
43C2×Q8⋊C8—21no—
44C2×Q8⋊C8C2212yescomplemented
45C2×D4⋊C8—21no—
46C2×D4⋊C8C2212yescomplemented
47#1461C2212yescomplemented
48#1460C2212yescomplemented
49#1459—22no—
50#1458—22no—
51#1283C2214yescomplemented
52#1278C2214yescomplemented
53C22×#41C2×C411yescharacteristic
54C22×#41C2×C411yescharacteristic
55#575—24no—
56#575C818yescomplemented
57C2×#137—21no—
58C2×#137C2×C412yes—
59C2×AGL(1,ℤ/8)—21no—
60C2×AGL(1,ℤ/8)C2×C412yes—
61C2×#135—22no—
62C22×M16C2×C411yescharacteristic
63C22×M16C2×C411yescharacteristic
64C23×C8—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41—21no—
71C4×#41—21no—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41C2×C412yes—
78C4×#41D412yes—
79C4×#41D412yes—
80C4×#41C2312yes—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×C4×D4C2311yescharacteristic
93C2×#117—21no—
94C2×#117—21no—
95C2×#117C2×C411yescharacteristic
96C2×#117C2×C411yescharacteristic
97C22×C42—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100C8×Q8—24no—
101C8×Q8—24no—
102#442—22no—
103#442—22no—
104#441—22no—
105#441—22no—
106#440—22no—
107#440—22no—
108#440—22no—
109#440—22no—
110C4×Q16—22no—
111C4×Q16—22no—
112C4×SD16—22no—
113C4×SD16—22no—
114C4×SD16—22no—
115C4×SD16—22no—
116C4×D8—22no—
117C4×D8—22no—
118C8×D4—28no—
119C8×D4—24no—
120C8×D4—24no—
121C8×D4—28no—
122#431—24no—
123#431—24no—
124#431—21no—
125#431—21no—
126#431—21no—
127#431—21no—
128#431—21no—
129#431—24no—
130#431—21no—
131#431—21no—
132#431D412yes—
133#431D412yes—
134#431D412yes—
135#431C2312yes—
136#431C2312yes—
137#428—21no—
138#428—21no—
139#428C2×C412yes—
140#428C2×C412yes—
141#427—22no—
142C2×C4⋊C8—24no—
143C2×C4⋊C8—21no—
144C2×C4⋊C8—21no—
145C2×C4⋊C8—21no—
146C2×C4⋊C8—21no—
147C2×C4⋊C8—21no—
148C2×C4⋊C8—21no—
149C2×C4⋊C8—21no—
150C2×C4⋊C8—21no—
151C2×C4⋊C8C2312yes—
152#419—21no—
153#419C2×C412yes—
154#418—21no—
155#418C2×C412yes—
156#417—21no—
157#417—21no—
158#417C2×C412yes—
159#417C2×C412yes—
160#416—22no—
161#416—22no—
162C2×C22⋊C8—24no—
163C2×C22⋊C8—21no—
164C2×C22⋊C8—21no—
165C2×C22⋊C8D412yes—
166C4×M16—21no—
167C4×M16—21no—
168C4×M16C2×C412yes—
169C4×M16C2312yes—
170C2×#97—21no—
171C2×C4×C8—21no—
172C2×C4×C8—21no—
173C2×C4×C8—24no—
174C2×C4×C8—21no—
175C2×C4×C8C2312yes—
176Q8⋊C8—24no—
177Q8⋊C8—24no—
178D4⋊C8—24no—
179D4⋊C8—24no—
180#324—24no—
181#323—24no—
182C2×#41—21no—
183C2×#41—21no—
184C2×#41—22no—
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41C22⋊C412yes—
194C2×#41C2×C814yes—
195C2×#41C2×C814yescomplemented
196C2×#41M1614yes—
197C2×#41C22×C412yes—
198C22×Q8C22⋊C411yescharacteristic
199C22×Q8C22×C411yescharacteristic
200C22×D4—21no—
201C22×D4—21no—
202C22×D4C22⋊C411yescharacteristic
203C22×D4C22×C411yescharacteristic
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4#4111yescharacteristic
213#137—24no—
214#137—28no—
215#137C2×C818yescomplemented
216AGL(1,ℤ/8)—24no—
217AGL(1,ℤ/8)—28no—
218AGL(1,ℤ/8)C2×C818yescomplemented
219#135—216no—
220C2×Q16—22no—
221C2×SD16—22no—
222C2×SD16—22no—
223C2×D8—22no—
224C2×M16—21no—
225C2×M16—22no—
226C2×M16—21no—
227C2×M16—22no—
228C2×M16—22no—
229C2×M16C2×C814yescomplemented
230C2×M16C22×C412yes—
231C2×M16C22×C412yes—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—28no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—41no—
239C22×C8—41no—
240C22×C8—41no—
241C22×C8—82no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C22⋊C412yes—
255C4×Q8C22⋊C412yes—
256C4×Q8C2×D412yes—
257C4×Q8C2×D412yes—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4—21no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—41no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—41no—
282C4×D4—41no—
283C4×D4—41no—
284C4×D4—41no—
285C4×D4—41no—
286C4×D4C22⋊C412yes—
287C4×D4C22⋊C412yes—
288C4×D4C2×D412yes—
289C4×D4C2×D412yes—
290#117—21no—
291#117—21no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—41no—
301#117—41no—
302#117—41no—
303#117—41no—
304#117C22⋊C412yes—
305#117C22×C412yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4C22⋊C411yescharacteristic
311C2×C4⋊C4C22⋊C411yescharacteristic
312C2×C4⋊C4C22×C411yescharacteristic
313C2×C4⋊C4C22×C411yescharacteristic
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—41no—
331C2×C42—41no—
332C2×C42—41no—
333C2×C42—41no—
334C2×C42C22×C411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42#4111yescharacteristic
343C2×C42#4111yescharacteristic
344C2×C42#4111yescharacteristic
345C2×C42C2411yescharacteristic
346#107—22no—
347#107—22no—
348#106—22no—
349#106—22no—
350C4⋊C8—28no—
351C4⋊C8—22no—
352C4⋊C8—22no—
353C4⋊C8—22no—
354C4⋊C8—22no—
355C4⋊C8—22no—
356C4⋊C8—22no—
357C4⋊C8—22no—
358C4⋊C8—41no—
359C4⋊C8—41no—
360C4⋊C8—41no—
361#103—22no—
362#103—22no—
363#103—22no—
364#103—22no—
365#102—22no—
366#102—22no—
367#102—22no—
368#102—22no—
369C22⋊C8—28no—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—41no—
375C22⋊C8C2×D414yes—
376#97—22no—
377#97—22no—
378C4×C8—28no—
379C4×C8—22no—
380C4×C8—22no—
381C4×C8—22no—
382C4×C8—22no—
383C4×C8—22no—
384C4×C8—22no—
385C4×C8—41no—
386C4×C8C2×D414yes—
387C24—21no—
388C24—21no—
389C24C4×D411yescharacteristic
390#41—22no—
391#41—22no—
392#41—24no—
393#41—24no—
394#41—24no—
395#41—24no—
396#41—24no—
397#41—24no—
398#41—24no—
399#41—24no—
400#41—24no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—42no—
405#41C22⋊C818yes—
406#41C22×C814yes—
407#41C2×M1614yes—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8C22⋊C814yes—
416C2×Q8C2×C22⋊C412yes—
417C2×Q8C22×C812yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—22no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—42no—
433C2×D4C22⋊C814yes—
434C2×D4C2×C22⋊C412yes—
435C2×D4C22×C812yes—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C2×C22⋊C411yescharacteristic
468C22×C4C2×C22⋊C411yescharacteristic
469C22×C4C4×D411yescharacteristic
470C22×C4C4×D411yescharacteristic
471C22×C4C4×D411yescharacteristic
472C22×C4C4×D411yescharacteristic
473C22×C4C4×D411yescharacteristic
474C22×C4#12011yescharacteristic
475C22×C4#12011yescharacteristic
476C22×C4#12111yescharacteristic
477C22×C4#12111yescharacteristic
478C22×C4#12111yescharacteristic
479C22×C4#12111yescharacteristic
480C22×C4#12311yescharacteristic
481C22×C4#12311yescharacteristic
482C22×C4C22×C812yes—
483C22×C4C2×M1612yes—
484C22×C4#13112yes—
485C22×C4#13112yes—
486C22×C4C23×C411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4111yescharacteristic
489Q16—28no—
490SD16—28no—
491SD16—28no—
492D8—28no—
493M16—22no—
494M16—24no—
495M16—22no—
496M16—22no—
497M16—42no—
498M16C22×C814yes—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—42no—
505C2×C8—42no—
506C2×C8—42no—
507C2×C8—82no—
508C2×C8—82no—
509C2×C8—88no—
510C2×C8—82no—
511C2×C8—82no—
512C2×C8—82no—
513C2×C8—82no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4C2×C22⋊C412yes—
529C4⋊C4C2×C22⋊C412yes—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—41no—
539C22⋊C4—41no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42C2×C22⋊C411yescharacteristic
555C42C2×C22⋊C411yescharacteristic
556C42C4×D411yescharacteristic
557C42C4×D411yescharacteristic
558C42#12011yescharacteristic
559C42#12011yescharacteristic
560C42#12111yescharacteristic
561C42#12111yescharacteristic
562C42#12111yescharacteristic
563C42#12111yescharacteristic
564C42#12311yescharacteristic
565C42#12311yescharacteristic
566C42C22×D411yescharacteristic
567C42C22×D411yescharacteristic
568C42C22×D411yescharacteristic
569C42C2×#4111yescharacteristic
570C23—22no—
571C23—24no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—81no—
578C23—81no—
579C23—81no—
580C23—81no—
581C23#38611yescharacteristic
582C23#38611yescharacteristic
583C23C8×D412yes—
584C23#43512yes—
585C23C2×C4×D411yescharacteristic
586Q8—22no—
587Q8—24no—
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—41no—
592Q8C2×C22⋊C814yes—
593D4—28no—
594D4—24no—
595D4—24no—
596D4—22no—
597D4—22no—
598D4—24no—
599D4—22no—
600D4—44no—
601D4—44no—
602D4—41no—
603D4—41no—
604D4C2×C22⋊C814yes—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—82no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4#38611yescharacteristic
649C2×C4#38611yescharacteristic
650C2×C4#38611yescharacteristic
651C2×C4#38611yescharacteristic
652C2×C4#38611yescharacteristic
653C2×C4#38611yescharacteristic
654C2×C4C2×C22⋊C812yes—
655C2×C4C2×C22⋊C812yes—
656C2×C4#40812yes—
657C2×C4#40812yes—
658C2×C4C8×D412yes—
659C2×C4C8×D412yes—
660C2×C4C8×D412yes—
661C2×C4#43512yes—
662C2×C4#43512yes—
663C2×C4#43512yes—
664C2×C4C22×C22⋊C411yescharacteristic
665C2×C4C2×C4×D411yescharacteristic
666C2×C4C2×#12011yescharacteristic
667C2×C4C2×#12111yescharacteristic
668C2×C4C2×#12111yescharacteristic
669C2×C4C2×#12311yescharacteristic
670C2×C4C23×C811yescharacteristic
671C2×C4C22×M1611yescharacteristic
672C2×C4C2×#13111yescharacteristic
673C2×C4C2×#13111yescharacteristic
674C8—24no—
675C8—44no—
676C8—44no—
677C8—42no—
678C8—82no—
679C8—82no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—24no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—84no—
693C22—82no—
694C22—82no—
695C22—81no—
696C22—81no—
697C22#186912yes—
698C22#186912yes—
699C22C2×#38611yescharacteristic
700C22C2×C8×D411yescharacteristic
701C22C2×#43511yescharacteristic
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—24no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—81no—
714C4—81no—
715C4#186912yes—
716C4#186912yes—
717C4C22×C22⋊C811yescharacteristic
718C4C2×#40811yescharacteristic
719C4C2×C8×D411yescharacteristic
720C4C2×#43511yescharacteristic
721C4#306811yescharacteristic
722C4#306911yescharacteristic
723C2—22no—
724C2—21no—
725C2—21no—
726C2—22no—
727C2—44no—
728C2—82no—
729C2C2×#186911yescharacteristic
730C2#2100011yescharacteristic
731C2#2100411yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#575C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60276C2semidirect
3. C2×#1477C2semidirect
4. C2×#1477C2semidirect
5. #17349C2semidirect
6. #9096C2semidirect
7. #9095C2semidirect
8. C2×#1283C2semidirect
9. #7882C2semidirect
44. C2×Q8⋊C8C22semidirect
46. C2×D4⋊C8C22semidirect
47. #1461C22semidirect
48. #1460C22semidirect
51. #1283C22semidirect
52. #1278C22semidirect
56. #575C8semidirect
195. C2×#41C2×C8semidirect
215. #137C2×C8semidirect
218. AGL(1,ℤ/8)C2×C8semidirect
229. C2×M16C2×C8semidirect

Maximal subgroups

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

subgroupindex
#602762
C4×#5752
C2×#14772
#173492
#90962
#90952
C2×#12832
#78822