Finite Group Databasebeta

← #204994

Subgroups — #204994

Order 512.

2040 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1504C214yesmaximal, complemented
4C2×C4×#95C211yesmaximal, characteristic, complemented
5#9713C214yesmaximal, complemented
6#9712C212yesmaximal, complemented
7#9711C212yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525C2212yes—
10C23×C42—21no—
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407C2212yes—
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#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
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×C42—28no—
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—44no—
77C22×C42—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#423—44no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94#407—28no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2314yes—
97#391C2×C412yescomplemented
98#391C2×C412yescomplemented
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#390C2×C412yescomplemented
104#390C2×C412yescomplemented
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#384C2×C411yescharacteristic, complemented
120#384C2×C411yescharacteristic, complemented
121#382C2×C412yescomplemented
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C2×#95—28no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95—22no—
136C2×#95—24no—
137C2×#95D418yescomplemented
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—28no—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D414yes—
190C23×C4#4114yes—
191C23×C4C2411yescharacteristic
192C2×M16—44no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8—24no—
196C4⋊Q8—42no—
197#127—42no—
198#124—44no—
199#123—44no—
200#123—44no—
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—44no—
212#121—42no—
213#121—42no—
214#120—48no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—48no—
218C4×D4—44no—
219C4×D4—44no—
220#117—28no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—28no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—432no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—44no—
279C4⋊C8—44no—
280C22⋊C8—28no—
281C22⋊C8—44no—
282#95—28no—
283#95—28no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—28no—
291#95—24no—
292#95C2×D418yes—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303C24#12114yes—
304C24C22×D411yescharacteristic
305#41—816no—
306C2×Q8—24no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×D4—24no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C22×C4—216no—
329C22×C4—216no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
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—28no—
351C22×C4—28no—
352C22×C4—232no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
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—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
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—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—416no—
387C22×C4—44no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4#9914yes—
399C22×C4#10414yes—
400C22×C4#10414yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C4×D414yes—
412C22×C4C4×D414yes—
413C22×C4#12114yes—
414C22×C4#12412yes—
415C22×C4#12412yes—
416C22×C4#12712yes—
417C22×C4#12712yes—
418C22×C4AGL(1,ℤ/8)12yes—
419C22×C4#13712yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C2×#4112yes—
424M16—88no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—44no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—416no—
468C42—416no—
469C42—44no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—28no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23#39012yes—
500C23#39012yes—
501C23C2×#9912yes—
502C23#41012yes—
503C23C2×#10412yes—
504C23C2×#10412yes—
505C23C22×C22⋊C411yescharacteristic
506C23C2×#12112yes—
507C23C2×AGL(1,ℤ/8)11yescharacteristic
508C23C2×#13711yescharacteristic
509C23#57512yes—
510Q8—84no—
511D4—84no—
512D4—84no—
513D4—88no—
514D4—88no—
515D4—84no—
516D4—84no—
517D4—84no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
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—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—216no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—464no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—48no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—88no—
583C2×C4#39012yes—
584C2×C4#39012yes—
585C2×C4C2×#9912yes—
586C2×C4C2×#9912yes—
587C2×C4#41712yes—
588C2×C4#41712yes—
589C2×C4C2×#10412yes—
590C2×C4C2×#10412yes—
591C2×C4C2×#10412yes—
592C2×C4C2×#10412yes—
593C2×C4C2×C4×D412yes—
594C2×C4C2×#12411yescharacteristic
595C2×C4C2×#12711yescharacteristic
596C8—84no—
597C22—216no—
598C22—216no—
599C22—22no—
600C22—22no—
601C22—21no—
602C22—21no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22#196214yes—
627C22#196414yes—
628C22C2×#39011yescharacteristic
629C22C22×#9911yescharacteristic
630C22C2×#41011yescharacteristic
631C22C2×#41711yescharacteristic
632C22#289611yescharacteristic
633C22C22×#10411yescharacteristic
634C22C22×#10411yescharacteristic
635C22#303612yes—
636C22#303712yes—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—21no—
643C4—22no—
644C4—22no—
645C4—21no—
646C4—22no—
647C4—416no—
648C4—41no—
649C4—41no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4#150414yescomplemented
654C2—22no—
655C2—24no—
656C2—24no—
657C2—22no—
658C2—162no—
659C2C2×#150411yescharacteristic
660C2#1754111yescharacteristic
661C2#1754211yescharacteristic
662C2#1754611yescharacteristic
663C2C2×#196211yescharacteristic
664C2C2×#196411yescharacteristic
665C2#2125811yescharacteristic
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. #20626C2semidirect
3. C2×#1504C2semidirect
4. C2×C4×#95C2semidirect
5. #9713C2semidirect
6. #9712C2semidirect
7. #9711C2semidirect
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
97. #391C2×C4semidirect
98. #391C2×C4semidirect
103. #390C2×C4semidirect
104. #390C2×C4semidirect
119. #384C2×C4semidirect
120. #384C2×C4semidirect
121. #382C2×C4semidirect
137. C2×#95D4semidirect
653. C4#1504semidirect

Maximal subgroups

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

subgroupindex
#366242
#206262
C2×#15042
C2×C4×#952
#97132
#97122
#97112