Finite Group Databasebeta

← #208708

Subgroups — #208708

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2966C211yesmaximal, characteristic, complemented
2#23569C211yesmaximal, characteristic
3C8×#121C211yesmaximal, characteristic, complemented
4C2×#1582C211yesmaximal, characteristic, complemented
5C2×#1581C211yesmaximal, characteristic, complemented
6#8948C211yesmaximal, characteristic, complemented
7#8947C211yesmaximal, characteristic, complemented
8#8943C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8844C211yesmaximal, characteristic, complemented
11#8834C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic, complemented
13#8211C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2966—24no—
18#2966—24no—
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×#436C2211yescharacteristic
24C2×#435—21no—
25C2×C8×D4—21no—
26C2×C8×D4—21no—
27C2×C8×D4C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#433—21no—
30C2×#432C2211yescharacteristic
31C2×#405—21no—
32C22×C4×C8C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#2059C411yescharacteristic
35#2042C411yescharacteristic
36#2036C411yescharacteristic
37#2034C411yescharacteristic
38#2025C411yescharacteristic
39#2012C411yescharacteristic
40#2000C411yescharacteristic
41#1999C411yescharacteristic
42#1954C2211yescharacteristic
43#1939C2211yescharacteristic
44#1923—21no—
45#1869—21no—
46#1797C2211yescharacteristic
47C8×C4⋊C4C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50C8×C22⋊C4—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1582—24no—
54#1581—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×SD16C2×C411yescharacteristic
72C22×D8C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76C23×C8D411yescharacteristic
77C23×C8D411yescharacteristic
78#529—22no—
79#529—22no—
80#529—22no—
81#529—22no—
82C2×#126—21no—
83C2×#125C2×C411yescharacteristic
84C2×#124C2×C411yescharacteristic
85C2×#123—21no—
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×#121C2×C411yescharacteristic
89C2×C4×Q8C2311yescharacteristic
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4C2311yescharacteristic
93C2×C4×D4C2311yescharacteristic
94C2×#117—21no—
95C22×C42C2311yescharacteristic
96#446—22no—
97#446D414yescomplemented
98C8×Q8—22no—
99C8×Q8—22no—
100C4×SD16—22no—
101C4×SD16—22no—
102C4×D8—22no—
103C4×D8—22no—
104#436—22no—
105#436D414yescomplemented
106#435—28no—
107C8×D4—28no—
108C8×D4—22no—
109C8×D4—22no—
110C8×D4—22no—
111C8×D4—22no—
112C8×D4—44no—
113#433—28no—
114#432—22no—
115#432—22no—
116C2×#107C2×C411yescharacteristic
117C2×#106C2×C411yescharacteristic
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#102C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×C22⋊C8—21no—
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⋊C8C2×C411yescharacteristic
142C2×C22⋊C8C2×C411yescharacteristic
143#405—28no—
144C2×#97—21no—
145C2×#97—21no—
146C2×C4×C8—21no—
147C2×C4×C8—22no—
148C2×C4×C8—22no—
149C2×C4×C8—21no—
150C2×C4×C8D411yescharacteristic
151C2×C4×C8D411yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#390C2×C411yescharacteristic
159#389C2×C411yescharacteristic
160#388—21no—
161#386—21no—
162#380C2×C411yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C22⋊C4—21no—
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#335—22no—
176#334—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184#322—22no—
185C22×Q8C22×C411yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4#4111yescharacteristic
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—41no—
198C2×D8—22no—
199C2×D8—22no—
200C2×D8—41no—
201C2×M16—24no—
202C2×M16—42no—
203C2×M16—42no—
204C22×C8—24no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—41no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—42no—
221C22×C8—42no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230#126—24no—
231#126—24no—
232#125—22no—
233#125—22no—
234#124—22no—
235#124—22no—
236#123—24no—
237#123—24no—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240#121—22no—
241#121—22no—
242#121—22no—
243#121—22no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8C2×D412yes—
247C4×Q8#4112yes—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4C2×D412yes—
259C4×D4#4112yes—
260#117—24no—
261#117—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
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×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42#4111yescharacteristic
291C2×C42C2411yescharacteristic
292#107—22no—
293#106—22no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—42no—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305#103—22no—
306#103—22no—
307#103—41no—
308#102—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312#102—41no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315C22⋊C8—24no—
316C22⋊C8—41no—
317C22⋊C8—41no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323#97—24no—
324#97—24no—
325C4×C8—24no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—42no—
339C4×C8C2×D412yes—
340C4×C8C2×D412yes—
341#95—21no—
342#95—21no—
343#95—21no—
344C24—21no—
345C24—21no—
346C24C4×D411yescharacteristic
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8C4×D412yes—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4C4×D412yes—
370C2×D4#13112yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2×#4111yescharacteristic
411SD16—44no—
412D8—44no—
413M16—48no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—44no—
436C2×C8—48no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—44no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—84no—
455C2×C8#12012yes—
456C2×C8#12012yes—
457C2×C8#13512yes—
458C2×C8#13512yes—
459C2×C8AGL(1,ℤ/8)12yes—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4C4×D412yes—
474C4⋊C4C4×D412yes—
475C4⋊C4#13112yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42C22×D411yescharacteristic
499C42C22×D411yescharacteristic
500C42C2×#4111yescharacteristic
501C42C2×#4111yescharacteristic
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23#44412yes—
523C23C2×C4×D411yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—41no—
527Q8—41no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4—22no—
532D4—48no—
533D4—48no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4C2×C4×D411yescharacteristic
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×#12011yescharacteristic
588C2×C4#52511yescharacteristic
589C2×C4#52511yescharacteristic
590C2×C4#52511yescharacteristic
591C2×C4C2×#13111yescharacteristic
592C2×C4C2×#13111yescharacteristic
593C2×C4#56811yescharacteristic
594C2×C4#56811yescharacteristic
595C2×C4C2×#13511yescharacteristic
596C2×C4C2×#13511yescharacteristic
597C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
598C2×C4#57511yescharacteristic
599C8—22no—
600C8—22no—
601C8—41no—
602C8—41no—
603C8—41no—
604C8—41no—
605C8—41no—
606C8—42no—
607C8—42no—
608C8—81no—
609C8—81no—
610C8—81no—
611C8—81no—
612C8#44914yes—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22C4×#12011yescharacteristic
635C22#292911yescharacteristic
636C22#293211yescharacteristic
637C22C4×#13511yescharacteristic
638C22C4×AGL(1,ℤ/8)11yescharacteristic
639C22C2×#44411yescharacteristic
640C22#295511yescharacteristic
641C4—21no—
642C4—21no—
643C4—41no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4#293311yescharacteristic
653C4#293311yescharacteristic
654C4#293511yescharacteristic
655C4#293511yescharacteristic
656C4C2×#44911yescharacteristic
657C4#300211yescharacteristic
658C4#310211yescharacteristic
659C4#310611yescharacteristic
660C2—42no—
661C2—82no—
662C2—82no—
663C2#1790411yescharacteristic
664C2#1791411yescharacteristic
665C2#1797811yescharacteristic
666C2#1798311yescharacteristic
667C2#2144311yescharacteristic
668C2C4×#44911yescharacteristic
669C2#2146711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radical#1999C4
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×#2966C2semidirect
3. C8×#121C2semidirect
4. C2×#1582C2semidirect
5. C2×#1581C2semidirect
6. #8948C2semidirect
7. #8947C2semidirect
8. #8943C2semidirect
9. #8845C2semidirect
10. #8844C2semidirect
11. #8834C2semidirect
12. #8315C2semidirect
13. #8211C2semidirect
14. #8210C2semidirect
15. #8144C2semidirect
97. #446D4semidirect
105. #436D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29662
#235692
C8×#1212
C2×#15822
C2×#15812
#89482
#89472
#89432
#88452
#88442
#88342
#83152
#82112
#82102
#81442