Finite Group Databasebeta

← #160730

Subgroups — #160730

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#525C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#7617C212yesmaximal, complemented
5#7616C212yesmaximal, complemented
6#7613C212yesmaximal, complemented
7#7612C212yesmaximal, complemented
8#7566C212yesmaximal, complemented
9#7565C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C2×C4×#41C411yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8—21no—
14C2×#407—21no—
15C22×C22⋊C8—21no—
16C2×C4×M16—21no—
17C4×C22⋊Q8C411yescharacteristic
18C4×C22⋊Q8C411yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21C4×#123—21no—
22C4×#123—21no—
23C4×#121C411yescharacteristic
24C4×#121C411yescharacteristic
25C4×#121C2211yescharacteristic
26C4×#121C2211yescharacteristic
27C4×#120—21no—
28C4×#120—21no—
29#2297C2211yescharacteristic
30#2296C2211yescharacteristic
31#2290—21no—
32#2290—21no—
33#2280C411yescharacteristic
34C42×D4—21no—
35C42×D4—21no—
36C4×#117C411yescharacteristic
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41#1749—21no—
42#1749C2212yes—
43C4×C22⋊C8—21no—
44C4×C22⋊C8C2212yes—
45#1728C2212yes—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57#525—24no—
58#525—24no—
59C2×#123—21no—
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×#121C2×C411yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×#120—21no—
65C4×#41—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×#117—21no—
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C22×C42—21no—
83C22×C42—44no—
84C22×C42C2×C411yescharacteristic
85C22×C42C2×C411yescharacteristic
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42D411yescharacteristic
92C22×C42D411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—44no—
99#407—28no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—44no—
105C4×M16—44no—
106C2×#97—22no—
107C2×C4×C8—22no—
108#391C2×C411yescharacteristic
109#391C2311yescharacteristic
110#390C2×C411yescharacteristic
111#390C2311yescharacteristic
112#389C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#389C2311yescharacteristic
115#389C2311yescharacteristic
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#381—21no—
137#380C2×C411yescharacteristic
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140#380C2×C411yescharacteristic
141#379—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4C2×C411yescharacteristic
150C4×C4⋊C4C2×C411yescharacteristic
151C4×C4⋊C4C2×C411yescharacteristic
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—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166#376—21no—
167C43—21no—
168#336—22no—
169#336—22no—
170#336C2314yes—
171#336C2314yes—
172C25C22⋊C411yescharacteristic
173C2×#41—22no—
174C2×#41—22no—
175C2×#41—42no—
176C2×#41—42no—
177C22×Q8C22×C411yescharacteristic
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C22×C411yescharacteristic
182C23×C4—24no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
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
214C2×M16—28no—
215C2×M16—44no—
216C2×M16—44no—
217C22×C8—24no—
218C22×C8—24no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8—44no—
222#123—28no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8M1614yescomplemented
227#121—22no—
228#121—22no—
229#121—22no—
230#121M1614yescomplemented
231#120—28no—
232C4×Q8—42no—
233C4×Q8—42no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—44no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—44no—
243C4×D4—42no—
244C4×D4—42no—
245#117—22no—
246#117—22no—
247#117—44no—
248#117—42no—
249#117—42no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
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×C42—21no—
282C2×C42—24no—
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×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—44no—
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2411yescharacteristic
319C4⋊C8—44no—
320C4⋊C8—44no—
321C22⋊C8—28no—
322C22⋊C8—28no—
323C22⋊C8—44no—
324C22⋊C8—44no—
325#97—44no—
326C4×C8—44no—
327#95—21no—
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—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—41no—
350C24—41no—
351C24C2×C22⋊C411yescharacteristic
352C24C2×C22⋊C411yescharacteristic
353C24C2×C22⋊C411yescharacteristic
354#41—48no—
355#41—48no—
356C2×Q8—21no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8C2×M1612yes—
360C2×D4—24no—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4C2×M1612yes—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
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—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
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—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×M1612yes—
459C22×C4C2×M1612yes—
460C22×C4#13514yes—
461C22×C4AGL(1,ℤ/8)12yes—
462C22×C4#13712yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466M16—416no—
467C2×C8—28no—
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—88no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4C2×M1612yes—
490C4⋊C4C2×M1612yes—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—42no—
497C22⋊C4—42no—
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—41no—
507C22⋊C4—41no—
508C22⋊C4—42no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42#9914yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—28no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23#40712yes—
565C23#41312yes—
566C23#41612yes—
567C23#41712yes—
568C23C2×#10412yes—
569C23C22×C22⋊C411yescharacteristic
570Q8—42no—
571Q8—42no—
572D4—42no—
573D4—42no—
574D4—48no—
575D4—48no—
576D4—48no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
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—84no—
648C2×C4—84no—
649C2×C4#40712yes—
650C2×C4C2×#9912yes—
651C2×C4C2×#9912yes—
652C2×C4C2×#9912yes—
653C2×C4#41612yes—
654C2×C4#41712yes—
655C2×C4C2×#10412yes—
656C2×C4C2×#10412yes—
657C2×C4#42112yes—
658C2×C4C22×M1611yescharacteristic
659C2×C4C22×M1611yescharacteristic
660C2×C4C2×#13512yes—
661C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
662C2×C4C2×#13711yescharacteristic
663C8—48no—
664C8—84no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—416no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22#148914yes—
694C22C2×#40711yescharacteristic
695C22C22×#9911yescharacteristic
696C22C2×#41311yescharacteristic
697C22C2×#41611yescharacteristic
698C22C2×#41711yescharacteristic
699C22C22×#10411yescharacteristic
700C22C2×#42111yescharacteristic
701C4—22no—
702C4—22no—
703C4—21no—
704C4—21no—
705C4—22no—
706C4—21no—
707C4—21no—
708C4—22no—
709C4—42no—
710C4—42no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—42no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4#150114yes—
721C4#151314yes—
722C2—24no—
723C2—42no—
724C2—42no—
725C2—82no—
726C2#1736211yescharacteristic
727C2#1744311yescharacteristic
728C2C2×#148911yescharacteristic
729C2C2×#150111yescharacteristic
730C2C2×#151311yescharacteristic
731C2#1765111yescharacteristic
732C2#1771911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#525C4
LCSG ▷ C22×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. C2×#1797C2semidirect
3. C4×#407C2semidirect
4. #7617C2semidirect
5. #7616C2semidirect
6. #7613C2semidirect
7. #7612C2semidirect
8. #7566C2semidirect
9. #7565C2semidirect
47. #1272C22semidirect
48. #1271C22semidirect
49. #1271C22semidirect
50. #1270C22semidirect
226. C22⋊Q8M16semidirect
230. #121M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5252
C2×#17972
C4×#4072
#76172
#76162
#76132
#76122
#75662
#75652