Finite Group Databasebeta

← #160482

Subgroups — #160482

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C22×#336C211yesmaximal, characteristic, complemented
4#7589C212yesmaximal, complemented
5#7586C212yesmaximal, complemented
6#7579C212yesmaximal, complemented
7#7578C212yesmaximal, complemented
8#7566C212yesmaximal, complemented
9#7565C212yesmaximal, complemented
10C24×C8—21no—
11C2×#525C411yescharacteristic
12C2×C4×#41C411yescharacteristic
13C23×C42C2211yescharacteristic
14C2×#431—21no—
15C4×C22⋊Q8C411yescharacteristic
16C4×C22⋊Q8C411yescharacteristic
17C4×C22⋊Q8C2211yescharacteristic
18C4×C22⋊Q8C2211yescharacteristic
19C4×#123—21no—
20C4×#123—21no—
21C4×#121C411yescharacteristic
22C4×#121C411yescharacteristic
23C4×#121C2211yescharacteristic
24C4×#121C2211yescharacteristic
25C4×#120—21no—
26C4×#120—21no—
27#2297C2211yescharacteristic
28#2296C2211yescharacteristic
29#2290—21no—
30#2290—21no—
31#2280C411yescharacteristic
32C42×D4—21no—
33C42×D4—21no—
34C4×#117C411yescharacteristic
35#1817C2212yes—
36#1797—21no—
37#1797C2212yes—
38#1779—21no—
39C4×C22⋊C8—21no—
40C4×C22⋊C8C2212yes—
41C2×#336—21no—
42C2×#336—21no—
43C2×#336—21no—
44C2×#336C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C23×C8—28no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—44no—
60C23×C8—44no—
61#525—24no—
62#525C818yescomplemented
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#120—21no—
69C4×#41—44no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×#117—21no—
83C2×#117C2×C411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C2×#117C2×C411yescharacteristic
86C22×C42—21no—
87C22×C42—44no—
88C22×C42C2×C411yescharacteristic
89C22×C42C2×C411yescharacteristic
90C22×C42C2×C411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42D411yescharacteristic
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#431—44no—
101C2×C4⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C4×C8—22no—
106#391C2×C411yescharacteristic
107#391C2311yescharacteristic
108#390C2×C411yescharacteristic
109#390C2311yescharacteristic
110#389C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2311yescharacteristic
113#389C2311yescharacteristic
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#381—21no—
135#380C2×C411yescharacteristic
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138#380C2×C411yescharacteristic
139#379—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C22⋊C4—21no—
151C4×C22⋊C4—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164#376—21no—
165C43—21no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170#336C2314yes—
171#336C2314yes—
172C25C22⋊C411yescharacteristic
173C2×#41—22no—
174C2×#41—42no—
175C2×#41—42no—
176C2×#41C2×C814yescomplemented
177C22×Q8C22×C411yescharacteristic
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C22×C411yescharacteristic
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—21no—
187C23×C4—24no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C2×C814yes—
204C23×C4M1614yes—
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C22×C8—28no—
215C22×C8—28no—
216C22×C8—28no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—432no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—44no—
226#123—28no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8C2×C814yescomplemented
230C22⋊Q8C2×C814yescomplemented
231#121—22no—
232#121—22no—
233#121C2×C814yescomplemented
234#121C2×C814yescomplemented
235#120—28no—
236C4×Q8—42no—
237C4×Q8—42no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—42no—
241C4×D4—42no—
242C4×D4—44no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—44no—
247C4×D4—42no—
248C4×D4—42no—
249#117—22no—
250#117—44no—
251#117—42no—
252#117—42no—
253#117C2×C814yescomplemented
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
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×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22⋊C411yescharacteristic
311C2×C42C22⋊C411yescharacteristic
312C2×C42C22⋊C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2411yescharacteristic
323C4⋊C8—44no—
324C22⋊C8—44no—
325C22⋊C8—44no—
326C22⋊C8—44no—
327C4×C8—44no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24C22⋊C814yes—
352C24C2×C22⋊C411yescharacteristic
353C24C2×C22⋊C411yescharacteristic
354C24C2×C22⋊C411yescharacteristic
355#41—48no—
356#41—48no—
357C2×Q8—21no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8C22×C812yescomplemented
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4C22×C812yescomplemented
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—416no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4C22⋊C814yes—
442C22×C4C22⋊C814yes—
443C22×C4C22⋊C814yes—
444C22×C4#9914yes—
445C22×C4#10414yes—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C22×C812yes—
456C22×C4C22×C812yes—
457C22×C4C22×C812yes—
458C22×C4C2×M1612yes—
459C22×C4C2×M1612yes—
460C22×C4C2×M1612yes—
461C22×C4#13514yes—
462C22×C4AGL(1,ℤ/8)12yes—
463C22×C4#13712yes—
464C22×C4C23×C411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C2×C8—28no—
468C2×C8—416no—
469C2×C8—416no—
470C2×C8—44no—
471C2×C8—416no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C2×C8—44no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4C22×C812yescomplemented
492C4⋊C4C22×C812yescomplemented
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—42no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×C22⋊C812yes—
563C23C2×C22⋊C812yes—
564C23C2×C22⋊C812yes—
565C23C2×#9912yes—
566C23#41312yes—
567C23#41612yes—
568C23#41712yes—
569C23C2×#10412yes—
570C23#42112yes—
571C23C22×C22⋊C411yescharacteristic
572Q8—42no—
573Q8—42no—
574D4—42no—
575D4—42no—
576D4—48no—
577D4—48no—
578D4—48no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4C2×C22⋊C812yes—
652C2×C4C2×C22⋊C812yes—
653C2×C4C2×C22⋊C812yes—
654C2×C4C2×#9912yes—
655C2×C4C2×#9912yes—
656C2×C4#41612yes—
657C2×C4#41712yes—
658C2×C4C2×#10412yes—
659C2×C4C2×#10412yes—
660C2×C4C23×C811yescharacteristic
661C2×C4C22×M1611yescharacteristic
662C2×C4C2×#13512yes—
663C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
664C2×C4C2×#13711yescharacteristic
665C8—48no—
666C8—84no—
667C22—232no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22#145914yes—
694C22#147614yes—
695C22#147714yes—
696C22C22×C22⋊C811yescharacteristic
697C22C22×#9911yescharacteristic
698C22C2×#41311yescharacteristic
699C22C2×#41611yescharacteristic
700C22C2×#41711yescharacteristic
701C22C22×#10411yescharacteristic
702C22C2×#42111yescharacteristic
703C4—21no—
704C4—21no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—21no—
711C4—21no—
712C4—22no—
713C4—22no—
714C4—42no—
715C4—42no—
716C4—41no—
717C4—41no—
718C4—42no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4#150414yes—
724C2—24no—
725C2—24no—
726C2—24no—
727C2—82no—
728C2C2×#145911yescharacteristic
729C2C2×#147611yescharacteristic
730C2C2×#147711yescharacteristic
731C2C2×#150411yescharacteristic
732C2#1757211yescharacteristic
733C2#1764611yescharacteristic
734C2#1771911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#525C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C22×#336C2semidirect
4. #7589C2semidirect
5. #7586C2semidirect
6. #7579C2semidirect
7. #7578C2semidirect
8. #7566C2semidirect
9. #7565C2semidirect
47. #1272C22semidirect
48. #1271C22semidirect
49. #1271C22semidirect
50. #1270C22semidirect
62. #525C8semidirect
176. C2×#41C2×C8semidirect
229. C22⋊Q8C2×C8semidirect
230. C22⋊Q8C2×C8semidirect
233. #121C2×C8semidirect
234. #121C2×C8semidirect
253. #117C2×C8semidirect
360. C2×Q8C22×C8semidirect
373. C2×D4C22×C8semidirect
491. C4⋊C4C22×C8semidirect
492. C4⋊C4C22×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
#207142
C22×#3362
#75892
#75862
#75792
#75782
#75662
#75652