Finite Group Databasebeta

← #189471

Subgroups — #189471

Order 512.

1204 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
1C4×#529C211yesmaximal, characteristic
2C2×#1842C211yesmaximal, characteristic, complemented
3C4×#423C211yesmaximal, characteristic, complemented
4C2×#1483C212yesmaximal
5#8568C212yesmaximal, complemented
6#8567C212yesmaximal, complemented
7#8564C212yesmaximal, complemented
8#8563C212yesmaximal, complemented
9#7895C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C2×#423—21no—
12C22×C4⋊C8—21no—
13C22×C4⋊C8—21no—
14C22×C4⋊C8C2212yes—
15C2×C4×M16C2212yes—
16C4×#125C411yescharacteristic
17C4×#125C411yescharacteristic
18C4×#126—21no—
19C4×#126—21no—
20C4×#124C411yescharacteristic
21C4×#124C411yescharacteristic
22C4×C22⋊Q8C411yescharacteristic
23C4×C22⋊Q8C411yescharacteristic
24C4×#123—21no—
25C4×#123—21no—
26C4×#121C411yescharacteristic
27C4×#121C411yescharacteristic
28#2298—21no—
29#2295C2211yescharacteristic
30#2293C2211yescharacteristic
31#2290—21no—
32#2283C2211yescharacteristic
33C42×Q8C2211yescharacteristic
34C42×D4—21no—
35C42×D4C2211yescharacteristic
36C4×#117—21no—
37C2×C43C2211yescharacteristic
38#1842—21no—
39#1842—21no—
40#1842C2212yes—
41#1842C2212yes—
42#1767—21no—
43#1767—21no—
44C4×C4⋊C8C2212yes—
45C4×C4⋊C8C2212yes—
46#1728—21no—
47#1483—28no—
48C2×Q8⋊C8C2214yes—
49C2×D4⋊C8C2214yes—
50#1279C2214yes—
51#1279C2214yes—
52C22×M16—22no—
53C23×C8—22no—
54#529—22no—
55#529—22no—
56#529D414yes—
57#529Q814yes—
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75C22×C42—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83#423—28no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—44no—
94C2×C4⋊C8—44no—
95C2×C4⋊C8C2314yes—
96C4×M16—24no—
97C4×M16—24no—
98C2×#97—22no—
99C2×C4×C8C2314yes—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#389C2×C411yescharacteristic
105#389C2×C411yescharacteristic
106#389C2×C411yescharacteristic
107#389C2×C411yescharacteristic
108#388—21no—
109#388—21no—
110#388—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#386—21no—
125#386—21no—
126#385—21no—
127#385—21no—
128#383—21no—
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#381—21no—
132#380C2×C411yescharacteristic
133#380C2×C411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2×C411yescharacteristic
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2311yescharacteristic
149C4×C4⋊C4C2311yescharacteristic
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—
163#376—21no—
164#376—21no—
165C43—21no—
166C43C2×C411yescharacteristic
167C43C2×C411yescharacteristic
168C43D411yescharacteristic
169C43D411yescharacteristic
170C43D411yescharacteristic
171C43D411yescharacteristic
172C43C2311yescharacteristic
173C43C2311yescharacteristic
174#336—22no—
175#336—22no—
176Q8⋊C8—28no—
177D4⋊C8—28no—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C2×M16—28no—
192C2×M16—44no—
193C2×M16—44no—
194C22×C8—24no—
195C22×C8—24no—
196C22×C8—44no—
197C22×C8—44no—
198C22×C8—44no—
199#126—24no—
200#126—24no—
201#125—22no—
202#125C4⋊C414yes—
203#124—22no—
204#124C4⋊C414yes—
205#123—24no—
206#123—24no—
207C22⋊Q8—22no—
208C22⋊Q8C4⋊C414yes—
209#121—22no—
210#121C4⋊C414yes—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—24no—
214C4×Q8—42no—
215C4×Q8C2×D412yes—
216C4×Q8C2×Q812yes—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—42no—
223C4×D4—44no—
224C4×D4—44no—
225C4×D4C2×D412yes—
226C4×D4C2×Q812yes—
227#117—24no—
228#117—24no—
229#117—42no—
230#117—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C42—24no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×Q812yes—
308C2×C42#4112yes—
309C2×C42#4112yes—
310C2×C42C2411yescharacteristic
311C4⋊C8—28no—
312C4⋊C8—28no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—44no—
316C4⋊C8—44no—
317C4⋊C8—44no—
318C4⋊C8—44no—
319C4⋊C8—44no—
320#97—44no—
321C4×C8—24no—
322C4×C8—24no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
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—21no—
344C24C2×C22⋊C411yescharacteristic
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8C2×C4⋊C412yes—
349C2×D4—24no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4C2×C4⋊C412yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4#10414yes—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C4⋊C412yes—
407C22×C4#11712yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4C22⋊Q812yes—
411C22×C4C22⋊Q812yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415M16—416no—
416C2×C8—28no—
417C2×C8—44no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C2×C8—44no—
422C2×C8—88no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4C2×C4⋊C412yes—
441C4⋊C4C2×C4⋊C412yes—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—42no—
482C42—42no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42C2×C4⋊C412yes—
492C42#11712yes—
493C42#12112yes—
494C42#12112yes—
495C42C22⋊Q812yes—
496C42C22⋊Q812yes—
497C42#13114yes—
498C42AGL(1,ℤ/8)12yes—
499C42AGL(1,ℤ/8)12yes—
500C42#13712yes—
501C42#13712yes—
502C42C22×D411yescharacteristic
503C42C22×Q811yescharacteristic
504C42C2×#4111yescharacteristic
505C42C2×#4111yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23#38012yes—
518C23C2×#10412yes—
519C23#42112yes—
520C23C22×C22⋊C411yescharacteristic
521Q8—24no—
522Q8—42no—
523D4—24no—
524D4—48no—
525D4—42no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
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—44no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4#38012yes—
590C2×C4#38012yes—
591C2×C4#38012yes—
592C2×C4#40812yes—
593C2×C4#40812yes—
594C2×C4#41712yes—
595C2×C4#41712yes—
596C2×C4#41712yes—
597C2×C4#41712yes—
598C2×C4C2×#10412yes—
599C2×C4C2×#10412yes—
600C2×C4C22×C4⋊C411yescharacteristic
601C2×C4C2×#11711yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×C22⋊Q811yescharacteristic
605C2×C4C2×C22⋊Q811yescharacteristic
606C2×C4C2×#13112yes—
607C2×C4#56812yes—
608C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
609C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
610C2×C4C2×#13711yescharacteristic
611C2×C4C2×#13711yescharacteristic
612C8—48no—
613C8—84no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22#180614yes—
628C22C2×#38011yescharacteristic
629C22C2×#40811yescharacteristic
630C22#287911yescharacteristic
631C22C2×#41711yescharacteristic
632C22C2×#41711yescharacteristic
633C22C22×#10411yescharacteristic
634C22C2×#42111yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—21no—
643C4—21no—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4#148314yescomplemented
655C4#290612yes—
656C4#292512yes—
657C4#303811yescharacteristic
658C4#303911yescharacteristic
659C4#307911yescharacteristic
660C4#307911yescharacteristic
661C2—24no—
662C2—82no—
663C2C2×#148311yescharacteristic
664C2#1746611yescharacteristic
665C2#1749211yescharacteristic
666C2#1750411yescharacteristic
667C2#2062911yescharacteristic
668C2#2064811yescharacteristic
669C2C2×#180611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1842C2semidirect
3. C4×#423C2semidirect
5. #8568C2semidirect
6. #8567C2semidirect
7. #8564C2semidirect
8. #8563C2semidirect
9. #7895C2semidirect
654. C4#1483semidirect

Maximal subgroups

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

subgroupindex
C4×#5292
C2×#18422
C4×#4232
C2×#14832
#85682
#85672
#85642
#85632
#78952