Finite Group Databasebeta

← #205002

Subgroups — #205002

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 658 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×#1434C211yesmaximal, characteristic, complemented
5#9716C214yesmaximal, complemented
6#9715C212yesmaximal, complemented
7#9714C212yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525C2212yes—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407C2212yes—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854—21no—
19#2850C412yescomplemented
20#2850C2212yes—
21#2847—21no—
22#2847—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27#2297C412yescomplemented
28#2285C411yescharacteristic, complemented
29C22×#95—21no—
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—
40#1434—21no—
41#1434—21no—
42#1434C2212yes—
43#1434C2212yes—
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—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#423—44no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89#407—28no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2314yes—
92#391C2×C412yescomplemented
93#391C2×C412yescomplemented
94#390—21no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390C2×C412yescomplemented
99#390C2×C412yescomplemented
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#384C2×C411yescharacteristic, complemented
115#384C2×C411yescharacteristic, complemented
116#382C2×C412yescomplemented
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—22no—
128C2×#95—24no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—44no—
134C2×#95—44no—
135C2×#95D418yescomplemented
136C2×#95C2314yes—
137#336—22no—
138#336—22no—
139#328—216no—
140#327—216no—
141C25C2×D411yescharacteristic
142C2×#41—28no—
143C2×#41—44no—
144C2×#41—44no—
145C22×Q8C22×C412yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—28no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D414yes—
188C23×C4#4114yes—
189C23×C4C2411yescharacteristic
190C2×M16—44no—
191C2×M16—44no—
192C2×M16—44no—
193C22×C8—24no—
194C4⋊Q8—42no—
195#127—42no—
196#124—44no—
197#123—44no—
198#123—44no—
199C22⋊Q8—28no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202#121—28no—
203#121—24no—
204#121—24no—
205#121—44no—
206#121—44no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—42no—
211#121—42no—
212#120—48no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—48no—
216C4×D4—44no—
217C4×D4—44no—
218#117—28no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
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×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C4⋊C8—44no—
270C4⋊C8—44no—
271C22⋊C8—28no—
272C22⋊C8—44no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—28no—
280#95—24no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95C2×D418yes—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24C2×C22⋊C411yescharacteristic
295C24C2×C22⋊C411yescharacteristic
296C24#12114yes—
297C24C22×D411yescharacteristic
298#41—816no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×D4—24no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4—42no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
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—28no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—41no—
372C22×C4—48no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—416no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—416no—
384C22×C4—416no—
385C22×C4—416no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4—88no—
392C22×C4—88no—
393C22×C4#9914yes—
394C22×C4C2×C22⋊C411yescharacteristic
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×C4C4×D414yes—
405C22×C4C4×D414yes—
406C22×C4#12114yes—
407C22×C4#12412yes—
408C22×C4#12412yes—
409C22×C4#12712yes—
410C22×C4#12712yes—
411C22×C4AGL(1,ℤ/8)12yes—
412C22×C4#13712yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D412yes—
416C22×C4C2×#4112yes—
417M16—88no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—44no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—44no—
461C23—24no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—24no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23#39012yes—
491C23#39012yes—
492C23C2×#9912yes—
493C23#41012yes—
494C23#42112yes—
495C23#42112yes—
496C23C22×C22⋊C411yescharacteristic
497C23C2×#12112yes—
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23C2×#13711yescharacteristic
500C23#57512yes—
501Q8—84no—
502D4—84no—
503D4—84no—
504D4—84no—
505D4—88no—
506D4—88no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
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—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—88no—
573C2×C4—832no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4#39012yes—
579C2×C4#39012yes—
580C2×C4C2×#9912yes—
581C2×C4C2×#9912yes—
582C2×C4#41712yes—
583C2×C4#41712yes—
584C2×C4C2×C4×D412yes—
585C2×C4C2×#12411yescharacteristic
586C2×C4C2×#12711yescharacteristic
587C8—84no—
588C22—216no—
589C22—216no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#196314yes—
618C22#196414yes—
619C22C2×#39011yescharacteristic
620C22C22×#9911yescharacteristic
621C22C2×#41011yescharacteristic
622C22C2×#41711yescharacteristic
623C22#289611yescharacteristic
624C22C2×#42111yescharacteristic
625C22C2×#42111yescharacteristic
626C22#303612yes—
627C22#303712yes—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—21no—
633C4—22no—
634C4—22no—
635C4—21no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—88no—
642C4—82no—
643C4—82no—
644C4—82no—
645C2—22no—
646C2—24no—
647C2—24no—
648C2—22no—
649C2—162no—
650C2#1758711yescharacteristic
651C2#1758811yescharacteristic
652C2#1759211yescharacteristic
653C2#1759311yescharacteristic
654C2C2×#196311yescharacteristic
655C2C2×#196411yescharacteristic
656C2#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×#1434C2semidirect
5. #9716C2semidirect
6. #9715C2semidirect
7. #9714C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
19. #2850C4semidirect
27. #2297C4semidirect
28. #2285C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×#127C2×C4semidirect
53. C2×#124C2×C4semidirect
92. #391C2×C4semidirect
93. #391C2×C4semidirect
98. #390C2×C4semidirect
99. #390C2×C4semidirect
114. #384C2×C4semidirect
115. #384C2×C4semidirect
116. #382C2×C4semidirect
135. C2×#95D4semidirect

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×#14342
#97162
#97152
#97142