Finite Group Databasebeta

← #429942

Subgroups — #429942

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic, complemented
3C2×#2001C212yesmaximal
4C2×#1960C211yesmaximal, characteristic, complemented
5C2×#1872C212yesmaximal
6#15178C211yesmaximal, characteristic
7#11372C211yesmaximal, characteristic, complemented
8#11052C212yesmaximal, complemented
9#11024C212yesmaximal, complemented
10#10928C211yesmaximal, characteristic, complemented
11#10922C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C22×#97C2211yescharacteristic
23#2503—24no—
24#2503—24no—
25C2×#400—22no—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#381—22no—
32C2×C4×C22⋊C4C2211yescharacteristic
33#2016—24no—
34#2001—28no—
35#1960—24no—
36#1872—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1444—22no—
43#1439—22no—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×Q8C2312yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—42no—
53C22×Q16—42no—
54C22×Q16—42no—
55C22×Q16—42no—
56C23×C8C2312yes—
57C23×C8C2312yes—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—42no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—24no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—42no—
70C2×C22⋊Q8—42no—
71C2×C22⋊Q8—42no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C22⋊C4D412yes—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×C22⋊C8—42no—
102C2×C22⋊C8—42no—
103C2×C22⋊C8D418yes—
104C2×#97—22no—
105C2×#97—42no—
106#400—216no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391—24no—
111#391—42no—
112#391D414yes—
113#382—22no—
114#382—22no—
115#382—22no—
116#382D414yes—
117#381—216no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—24no—
126C2×#95D412yes—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—22no—
131#336—42no—
132C25C2×D411yescharacteristic
133C22×Q8—21no—
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—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C2×D412yes—
154C22×Q8#4114yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—42no—
170C23×C4—44no—
171C23×C4—42no—
172C23×C4—44no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D414yes—
177C23×C4C2411yescharacteristic
178C2×Q16—48no—
179C2×Q16—48no—
180C2×Q16—48no—
181C2×Q16—48no—
182C2×Q16—48no—
183C2×Q16—48no—
184C2×Q16—48no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C2×D414yes—
198C22×C8C2×D414yes—
199C4⋊Q8—28no—
200C4⋊Q8—48no—
201C4⋊Q8—44no—
202C22⋊Q8—216no—
203C22⋊Q8—48no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206C22⋊Q8—48no—
207C22⋊Q8—48no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256#103—48no—
257#103—48no—
258#103—48no—
259#103—48no—
260C22⋊C8—48no—
261#97—44no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—24no—
267#95—48no—
268#95—44no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—24no—
275C24#12712yes—
276C24#13714yes—
277C24C22×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
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—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—48no—
358C22×C4—416no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—84no—
385C22×C4—88no—
386C22×C4—84no—
387C22×C4#12014yes—
388C22×C4#12712yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C4H(5,𝔽2)12yes—
397Q16—432no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—416no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—416no—
429C42—22no—
430C42—22no—
431C42—24no—
432C42—41no—
433C42—41no—
434C42—42no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—28no—
442C23—28no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23#49714yes—
451C23C2×#12011yescharacteristic
452C23C2×#12711yescharacteristic
453C23#53511yescharacteristic
454C23#54612yes—
455C23#54612yes—
456C23C2×#13712yes—
457C23#57512yes—
458C23#57814yes—
459Q8—44no—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—44no—
464Q8—44no—
465Q8—82no—
466Q8—82no—
467Q8—82no—
468Q8—82no—
469Q8—82no—
470Q8—82no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—816no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4#52511yescharacteristic
523C2×C4#52512yes—
524C2×C4#53211yescharacteristic
525C2×C4#53412yes—
526C2×C4#53511yescharacteristic
527C2×C4#53511yescharacteristic
528C2×C4#53812yes—
529C2×C4#54012yes—
530C2×C4D4212yes—
531C2×C4D4212yes—
532C2×C4#54612yes—
533C2×C4#54612yes—
534C2×C4#54612yes—
535C2×C4#55012yes—
536C2×C4#55312yes—
537C2×C4#56012yes—
538C8—44no—
539C8—44no—
540C22—24no—
541C22—28no—
542C22—42no—
543C22—48no—
544C22—42no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22#240411yescharacteristic
549C22#243111yescharacteristic
550C22#243311yescharacteristic
551C22#253912yes—
552C22#270312yes—
553C22#300712yes—
554C22#301112yes—
555C22C2×#49711yescharacteristic
556C22#315311yescharacteristic
557C22#315612yes—
558C22#321512yes—
559C22#322312yes—
560C22#331214yes—
561C22#331214yes—
562C22#331514yes—
563C22#332714yes—
564C4—42no—
565C4—44no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4#245612yes—
574C4#246412yes—
575C4#267312yes—
576C4#267812yes—
577C2—44no—
578C2#1541211yescharacteristic
579C2#2398312yes—
580C2#2558412yes—
581C2#2573011yescharacteristic
582C2#2573511yescharacteristic
583C2#2834512yes—
584C2#2835612yes—
585C2#3185714yes—
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
2. C2×#2016C2semidirect
4. C2×#1960C2semidirect
7. #11372C2semidirect
8. #11052C2semidirect
9. #11024C2semidirect
10. #10928C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#20162
C2×#20012
C2×#19602
C2×#18722
#151782
#113722
#110522
#110242
#109282
#109222