Finite Group Databasebeta

← #407976

Subgroups — #407976

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1956C211yesmaximal, characteristic, complemented
2C2×#1935C212yesmaximal, complemented
3C2×#1894C212yesmaximal
4C2×#1753C211yesmaximal, characteristic
5#15178C211yesmaximal, characteristic
6#15135C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11067C212yesmaximal, complemented
9#11010C211yesmaximal, characteristic, complemented
10#10970C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C23×M16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#399—21no—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1956—24no—
29#1935—28no—
30#1894—28no—
31#1753—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1445—21no—
39#1444—21no—
40#1444—21no—
41#1444—21no—
42#1439—21no—
43#1439—21no—
44#1439—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C23×Q8C2312yes—
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—24no—
52C22×M16—22no—
53C22×M16—42no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—42no—
67C2×C22⋊Q8—42no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×#97—24no—
99C2×#97—22no—
100C2×C4×C8—42no—
101C2×C4×C8D414yes—
102#399—44no—
103#391—24no—
104#391—42no—
105#389—24no—
106#389—42no—
107#384—22no—
108#384—22no—
109#380—24no—
110#380—42no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—42no—
126#340D418yes—
127#336—22no—
128#336—22no—
129C25C2×D411yescharacteristic
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8#4114yes—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—28no—
165C2×M16—48no—
166C2×M16—48no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C2×D412yes—
185C22×C8C2×D414yes—
186C22×C8C2×D412yes—
187C22×C8#4112yes—
188C22×C8#4112yes—
189C4⋊Q8—28no—
190C4⋊Q8—48no—
191C4⋊Q8—44no—
192C22⋊Q8—216no—
193C22⋊Q8—48no—
194C22⋊Q8—48no—
195C22⋊Q8—48no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D414yes—
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—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—82no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42C2×D412yes—
248C2×C42#4112yes—
249#103—216no—
250#103—48no—
251#103—48no—
252#103—48no—
253C22⋊C8—28no—
254#97—44no—
255C4×C8—44no—
256#95—22no—
257#95—22no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—42no—
264#95—42no—
265#95—42no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24#12412yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4#12112yes—
371C22×C4#12112yes—
372C22×C4#12412yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4#13712yes—
375C22×C4#13712yes—
376C22×C4#13712yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4H(5,𝔽2)12yes—
386C22×C4H(5,𝔽2)12yes—
387M16—416no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8#12714yes—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—416no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—84no—
424C4⋊C4—88no—
425C4⋊C4—84no—
426C22⋊C4—416no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—88no—
430C42—22no—
431C42—22no—
432C42—24no—
433C42—41no—
434C42—41no—
435C42—42no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23C2×#12111yescharacteristic
460C23C2×#12411yescharacteristic
461C23#53211yescharacteristic
462C23D4212yes—
463C23#54612yes—
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C23C2×#13711yescharacteristic
467C23C2×#13711yescharacteristic
468C23#57712yes—
469C23#57812yes—
470Q8—44no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474Q8—82no—
475Q8—82no—
476Q8—82no—
477Q8—82no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#46814yes—
536C2×C4#47014yes—
537C2×C4#52511yescharacteristic
538C2×C4#52511yescharacteristic
539C2×C4#52511yescharacteristic
540C2×C4#52512yes—
541C2×C4#52512yes—
542C2×C4#52911yescharacteristic
543C2×C4#52911yescharacteristic
544C2×C4C2×#12711yescharacteristic
545C2×C4#53411yescharacteristic
546C2×C4#53511yescharacteristic
547C2×C4#53812yes—
548C2×C4#54012yes—
549C2×C4#54612yes—
550C2×C4#54612yes—
551C2×C4#55012yes—
552C2×C4#55112yes—
553C2×C4#55112yes—
554C2×C4#55312yes—
555C8—24no—
556C8—42no—
557C8—42no—
558C8—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—81no—
565C22—81no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22#246412yes—
571C22#247212yes—
572C22#253411yescharacteristic
573C22#254111yescharacteristic
574C22#285011yescharacteristic
575C22C2×#46811yescharacteristic
576C22C2×#47011yescharacteristic
577C22#305711yescharacteristic
578C22#305911yescharacteristic
579C22#313712yes—
580C22#313911yescharacteristic
581C22#314111yescharacteristic
582C22#317812yes—
583C22#318211yescharacteristic
584C22#318511yescharacteristic
585C22#327212yes—
586C22#327612yes—
587C22#327612yes—
588C22#327812yes—
589C22#330812yes—
590C22#331212yes—
591C22#331612yes—
592C22#332012yes—
593C4—24no—
594C4—24no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—84no—
599C4—84no—
600C4—84no—
601C4—162no—
602C4#239711yescharacteristic
603C4#246712yes—
604C4#247112yes—
605C4#253911yescharacteristic
606C4#259811yescharacteristic
607C4#263511yescharacteristic
608C2—82no—
609C2#1497711yescharacteristic
610C2#2698312yes—
611C2#2698812yes—
612C2#2727812yes—
613C2#2728312yes—
614C2#2810511yescharacteristic
615C2#2810811yescharacteristic
616C2#2848911yescharacteristic
617C2#2849011yescharacteristic
618C2#3304611yescharacteristic
619C2#3304911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1956C2semidirect
2. C2×#1935C2semidirect
7. #11357C2semidirect
8. #11067C2semidirect
9. #11010C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19562
C2×#19352
C2×#18942
C2×#17532
#151782
#151352
#113572
#110672
#110102
#109702
#108202