Finite Group Databasebeta

← #204580

Subgroups — #204580

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1504C214yesmaximal, complemented
4C2×C4⋊#95C211yesmaximal, characteristic, complemented
5#9682C214yesmaximal, complemented
6#9680C212yesmaximal, complemented
7#9679C212yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525C2212yes—
10C23×C4⋊C4—21no—
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13C2×C4×M16—21no—
14#2857C411yescharacteristic, complemented
15#2857C2211yescharacteristic
16#2856C411yescharacteristic, complemented
17#2856C2211yescharacteristic
18#2854—21no—
19#2854—21no—
20#2850C412yescomplemented
21#2850C2212yes—
22#2847—21no—
23#2847—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28#2297C412yescomplemented
29#2285C411yescharacteristic, complemented
30C22×#95—21no—
31C22×#95C2212yes—
32#1749—21no—
33#1749C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yes—
36#1728—21no—
37#1504—216no—
38C2×#328C2218yes—
39C2×#327C2218yes—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22×#41C2×C412yes—
45C24×C4—22no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50#532—44no—
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×#127C2×C411yescharacteristic, complemented
53C2×#124C2×C412yescomplemented
54#525—28no—
55#525—28no—
56C2×#123—22no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2312yes—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121C2×C412yes—
66C2×#121C2312yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×#117C2×C412yes—
73C22×C4⋊C4—28no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—44no—
77C22×C4⋊C4—44no—
78C22×C42—44no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#407—24no—
92#407D418yescomplemented
93C2×C22⋊C8—22no—
94C2×C22⋊C8C2314yes—
95C4×M16—44no—
96C2×#97—22no—
97C2×C4×C8—22no—
98#391C2×C412yescomplemented
99#391C2×C412yescomplemented
100#390—21no—
101#390—21no—
102#390—21no—
103#390—21no—
104#390C2×C412yescomplemented
105#390C2×C412yescomplemented
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#384C2×C411yescharacteristic, complemented
121#384C2×C411yescharacteristic, complemented
122#382C2×C412yescomplemented
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C2×#95—28no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—44no—
138C2×#95C2314yes—
139#336—22no—
140#336—22no—
141#328—216no—
142#327—216no—
143C25C2×D411yescharacteristic
144C2×#41—28no—
145C2×#41—44no—
146C2×#41—44no—
147C22×Q8C22×C412yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22×C412yes—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D414yes—
191C23×C4#4114yes—
192C23×C4C2411yescharacteristic
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8—24no—
197C4⋊Q8—42no—
198#127—42no—
199#124—44no—
200#123—48no—
201C22⋊Q8—28no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204#121—28no—
205#121—24no—
206#121—24no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—44no—
211#121—42no—
212#121—42no—
213#121—42no—
214#121—42no—
215#120—44no—
216#120—44no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—48no—
220C4×D4—44no—
221C4×D4—44no—
222#117—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—432no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C22⋊C8—28no—
281C22⋊C8—24no—
282C22⋊C8C2×D418yes—
283#97—44no—
284C4×C8—44no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C24#12412yes—
307C24#12712yes—
308C24C22×D411yescharacteristic
309#41—816no—
310C2×Q8—24no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×D4—24no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C22×C4—216no—
335C22×C4—216no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—48no—
385C22×C4—41no—
386C22×C4—48no—
387C22×C4—41no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—416no—
393C22×C4—44no—
394C22×C4—416no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—416no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4#9914yes—
407C22×C4#10414yes—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C4×D414yes—
419C22×C4C4×D414yes—
420C22×C4#12114yes—
421C22×C4#12114yes—
422C22×C4#12412yes—
423C22×C4#12712yes—
424C22×C4AGL(1,ℤ/8)12yes—
425C22×C4AGL(1,ℤ/8)12yes—
426C22×C4#13712yes—
427C22×C4#13712yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C2×#4112yes—
432M16—416no—
433C2×C8—28no—
434C2×C8—44no—
435C2×C8—44no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—44no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—22no—
496C23—22no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23#39012yes—
513C23#39012yes—
514C23C2×#9912yes—
515C23#41012yes—
516C23C2×#10412yes—
517C23#42112yes—
518C23C22×C22⋊C411yescharacteristic
519C23C2×#12411yescharacteristic
520C23C2×#12711yescharacteristic
521C23C2×AGL(1,ℤ/8)11yescharacteristic
522C23C2×AGL(1,ℤ/8)11yescharacteristic
523C23C2×#13711yescharacteristic
524C23C2×#13711yescharacteristic
525Q8—84no—
526D4—84no—
527D4—84no—
528D4—82no—
529D4—82no—
530D4—82no—
531D4—82no—
532D4—88no—
533D4—88no—
534D4—84no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—28no—
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—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—88no—
596C2×C4—832no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4#39012yes—
602C2×C4#39012yes—
603C2×C4C2×#9912yes—
604C2×C4C2×#9912yes—
605C2×C4#41712yes—
606C2×C4#41712yes—
607C2×C4#41712yes—
608C2×C4#41712yes—
609C2×C4C2×#10412yes—
610C2×C4C2×#10412yes—
611C2×C4C2×C4×D412yes—
612C2×C4C2×#12112yes—
613C8—48no—
614C22—28no—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—21no—
619C22—21no—
620C22—22no—
621C22—21no—
622C22—21no—
623C22—22no—
624C22—22no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—22no—
629C22—21no—
630C22—21no—
631C22—22no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—21no—
640C22—21no—
641C22—22no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22#196414yes—
650C22#198014yes—
651C22C2×#39011yescharacteristic
652C22C22×#9911yescharacteristic
653C22C2×#41011yescharacteristic
654C22C2×#41711yescharacteristic
655C22C2×#41711yescharacteristic
656C22C22×#10411yescharacteristic
657C22C2×#42111yescharacteristic
658C22#314111yescharacteristic
659C22#314111yescharacteristic
660C22#315111yescharacteristic
661C22#315211yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—41no—
673C4—88no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#150414yescomplemented
678C2—21no—
679C2—21no—
680C2—21no—
681C2—21no—
682C2—24no—
683C2—24no—
684C2—162no—
685C2C2×#150411yescharacteristic
686C2#1754611yescharacteristic
687C2#1758811yescharacteristic
688C2#1759211yescharacteristic
689C2#2120011yescharacteristic
690C2C2×#196411yescharacteristic
691C2C2×#198011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. C2×#1504C2semidirect
4. C2×C4⋊#95C2semidirect
5. #9682C2semidirect
6. #9680C2semidirect
7. #9679C2semidirect
8. C2×#532C4semidirect
14. #2857C4semidirect
16. #2856C4semidirect
20. #2850C4semidirect
28. #2297C4semidirect
29. #2285C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×#127C2×C4semidirect
53. C2×#124C2×C4semidirect
92. #407D4semidirect
98. #391C2×C4semidirect
99. #391C2×C4semidirect
104. #390C2×C4semidirect
105. #390C2×C4semidirect
120. #384C2×C4semidirect
121. #384C2×C4semidirect
122. #382C2×C4semidirect
677. C4#1504semidirect

Maximal subgroups

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

subgroupindex
#366242
C4×#4072
C2×#15042
C2×C4⋊#952
#96822
#96802
#96792