Finite Group Databasebeta

← #205019

Subgroups — #205019

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1511C212yesmaximal, complemented
4C2×#1510C212yesmaximal
5C4×#381C211yesmaximal, characteristic, complemented
6#9716C214yesmaximal, complemented
7#9713C214yesmaximal, complemented
8C22×#124C411yescharacteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407C2212yes—
14#2850—21no—
15#2850—21no—
16#2850—21no—
17#2850—21no—
18#2850C412yescomplemented
19#2850C412yescomplemented
20#2850C2212yes—
21#2850C2212yes—
22C2×#391C411yescharacteristic, complemented
23C2×#390C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#388—21no—
26C2×#386—21no—
27C2×#386—21no—
28C2×#382C411yescharacteristic, complemented
29C2×#381C2212yes—
30C2×C4×C22⋊C4—21no—
31#1797—21no—
32#1797—21no—
33#1797C2212yescomplemented
34#1797C2212yescomplemented
35#1728—21no—
36#1511—28no—
37#1510—28no—
38C2×#328C2218yescomplemented
39C2×#327C2218yes—
40#1434—21no—
41#1434C2212yescomplemented
42C22⋊#95—21no—
43C4×#95—21no—
44C4×#95C2212yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—44no—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2312yes—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121C2×C412yes—
68C2×#121C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C42—22no—
80C22×C42—44no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#423—44no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96#407—24no—
97#407—24no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2314yes—
100#391—21no—
101#391—21no—
102#391C2×C412yes—
103#391C2×C412yes—
104#390—21no—
105#390—21no—
106#390C2×C412yes—
107#390C2×C412yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#382—21no—
125#382—21no—
126#382C2×C412yes—
127#382C2×C412yes—
128#381—24no—
129#381D418yes—
130C4×C22⋊C4—44no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95C2314yes—
140#336—22no—
141#336—22no—
142#328—216no—
143#327—216no—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×Q8—21no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C22×C412yes—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—24no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4C22×C412yes—
161C23×C4—24no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
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—44no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8—24no—
197#124—44no—
198#124—44no—
199#124—44no—
200#124—44no—
201C22⋊Q8—28no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—44no—
205C22⋊Q8—42no—
206C22⋊Q8—42no—
207#121—28no—
208#121—24no—
209#121—24no—
210#121—44no—
211#121—42no—
212#121—42no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
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—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C42—24no—
246C2×C42—24no—
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—44no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C4⋊C8—44no—
277C4⋊C8—44no—
278C22⋊C8—24no—
279C22⋊C8—24no—
280C22⋊C8—44no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—28no—
290#95—24no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95C2×D418yes—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24#12114yes—
312C24C22×D411yescharacteristic
313C2×Q8—24no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—88no—
320C2×D4—24no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—42no—
332C2×D4—88no—
333C2×D4—816no—
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—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
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—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—48no—
384C22×C4—41no—
385C22×C4—48no—
386C22×C4—41no—
387C22×C4—416no—
388C22×C4—416no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4#9914yes—
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×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C4×D414yes—
416C22×C4C4×D414yes—
417C22×C4#12114yes—
418C22×C4#12412yes—
419C22×C4#12412yes—
420C22×C4#12712yes—
421C22×C4#12712yes—
422C22×C4C23×C411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C2×#4112yes—
426M16—88no—
427C2×C8—44no—
428C2×C8—44no—
429C2×C8—44no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—416no—
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—88no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—28no—
483C23—28no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23#39012yes—
504C23#39012yes—
505C23C2×#9912yes—
506C23#41012yes—
507C23C22×C22⋊C411yescharacteristic
508C23C2×#12112yes—
509C23#57512yes—
510C23#57512yes—
511Q8—84no—
512Q8—84no—
513D4—84no—
514D4—84no—
515D4—88no—
516D4—88no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
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—24no—
535C2×C4—28no—
536C2×C4—24no—
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—21no—
548C2×C4—21no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4#39012yes—
586C2×C4#39012yes—
587C2×C4C2×#9912yes—
588C2×C4C2×#9912yes—
589C2×C4#42112yes—
590C2×C4#42112yes—
591C2×C4C2×C4×D412yes—
592C2×C4C2×#12411yescharacteristic
593C2×C4C2×#12711yescharacteristic
594C8—84no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—416no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—168no—
623C22#196414yes—
624C22C2×#39011yescharacteristic
625C22C22×#9911yescharacteristic
626C22C2×#41011yescharacteristic
627C22#289611yescharacteristic
628C22#289611yescharacteristic
629C22C2×#42111yescharacteristic
630C22C2×#42111yescharacteristic
631C22#304014yes—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—21no—
637C4—22no—
638C4—21no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—88no—
646C4—82no—
647C4—82no—
648C4—82no—
649C2—24no—
650C2—24no—
651C2—42no—
652C2—162no—
653C2#1755411yescharacteristic
654C2#1755911yescharacteristic
655C2#1758011yescharacteristic
656C2#1758711yescharacteristic
657C2#2120911yescharacteristic
658C2C2×#196411yescharacteristic
659C2#2126111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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×#1511C2semidirect
5. C4×#381C2semidirect
6. #9716C2semidirect
7. #9713C2semidirect
8. C22×#124C4semidirect
18. #2850C4semidirect
19. #2850C4semidirect
22. C2×#391C4semidirect
23. C2×#390C4semidirect
28. C2×#382C4semidirect
33. #1797C22semidirect
34. #1797C22semidirect
38. C2×#328C22semidirect
41. #1434C22semidirect
44. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#206262
C2×#15112
C2×#15102
C4×#3812
#97162
#97132