Finite Group Databasebeta

← #415798

Subgroups — #415798

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3C2×#2538C211yesmaximal, characteristic
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1954C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic, complemented
7#11096C214yesmaximal, complemented
8#10970C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#477—24no—
15C2×#477—24no—
16C2×#477—24no—
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2214yes—
22C22×C4×C8C2211yescharacteristic
23#2563—24no—
24#2563—24no—
25#2538—24no—
26#2538—24no—
27C2×#396—22no—
28C2×#393—22no—
29C2×#390—21no—
30C2×#390—21no—
31C2×#389—21no—
32C2×#384C2211yescharacteristic
33C2×#382—21no—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990—24no—
38#1954—24no—
39C4×#103—24no—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#340C2214yes—
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×D4D411yescharacteristic
46C24×C4C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D414yes—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—28no—
58C2×C22⋊Q8—44no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8D414yes—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4D412yes—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#477—232no—
89C2×#107—24no—
90C2×#107—24no—
91C2×#107—24no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8D414yes—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—28no—
103C2×C22⋊C8—44no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107#396—216no—
108#393—216no—
109#390—28no—
110#390—28no—
111#389—28no—
112#384—22no—
113#384D414yes—
114#382—28no—
115#380—28no—
116#380—28no—
117#380—24no—
118#380—24no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4D414yes—
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—22no—
124#340—28no—
125#340—24no—
126#340—44no—
127C25C2×D412yes—
128C22×Q8—24no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—82no—
140C22×Q8—82no—
141C22×Q8C2×D412yes—
142C22×Q8#4114yes—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4Q1614yes—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—24no—
175C22×C8—22no—
176C22×C8—24no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C2×D414yes—
180C22×C8#4114yes—
181C4⋊Q8—28no—
182C4⋊Q8—28no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185C22⋊Q8—416no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C4×Q8—28no—
189C4×Q8—28no—
190C4×Q8—44no—
191C4×Q8—44no—
192C4×D4—44no—
193C4×D4—44no—
194C4×D4—44no—
195C4×D4—44no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—48no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—48no—
241C2×C22⋊C4—84no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4—84no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256#107—216no—
257C4⋊C8—28no—
258#103—216no—
259#103—216no—
260C22⋊C8—416no—
261C4×C8—44no—
262#95—216no—
263#95—28no—
264#95—24no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—24no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—24no—
274C24—24no—
275C24C2×Q1614yes—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—48no—
295C2×Q8—88no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×Q8#12414yes—
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4#12012yes—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4C2×Q1612yes—
366C22×C4#13512yes—
367C22×C4#13712yes—
368C22×C4#13712yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—28no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—44no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—416no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—84no—
412C22⋊C4—416no—
413C22⋊C4—48no—
414C22⋊C4—88no—
415C22⋊C4—816no—
416C42—48no—
417C42—48no—
418C42#12014yes—
419C42#12114yes—
420C42#12114yes—
421C42#12714yes—
422C23—28no—
423C23—24no—
424C23—24no—
425C23—28no—
426C23—24no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—44no—
445C23C2×#12011yescharacteristic
446C23C2×#12011yescharacteristic
447C23C2×#12111yescharacteristic
448C23#54612yes—
449C23#54712yes—
450C23C22×Q1611yescharacteristic
451C23C2×#13511yescharacteristic
452C23C2×#13711yescharacteristic
453C23C2×#13711yescharacteristic
454C23#57612yes—
455C23#57712yes—
456Q8—24no—
457Q8—24no—
458Q8—42no—
459Q8—44no—
460Q8—42no—
461Q8—44no—
462Q8—816no—
463D4—416no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—82no—
490C2×C4—84no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—84no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—84no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4#39214yes—
507C2×C4#39214yes—
508C2×C4#45114yes—
509C2×C4#45114yes—
510C2×C4#45214yes—
511C2×C4#45214yes—
512C2×C4#46714yes—
513C2×C4#47014yes—
514C2×C4C2×#12011yescharacteristic
515C2×C4C2×#12111yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4#52512yes—
518C2×C4C2×#12411yescharacteristic
519C2×C4#52912yes—
520C2×C4C2×#12711yescharacteristic
521C2×C4#53211yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53611yescharacteristic
524C2×C4#53712yes—
525C2×C4#54212yes—
526C2×C4#54612yes—
527C2×C4#54612yes—
528C2×C4#54812yes—
529C2×C4#54812yes—
530C2×C4#54812yes—
531C2×C4#55012yes—
532C8—44no—
533C8—44no—
534C22—44no—
535C22—48no—
536C22—44no—
537C22—44no—
538C22—48no—
539C22—84no—
540C22—82no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×#39211yescharacteristic
546C22#240712yes—
547C22#242712yes—
548C22#243211yescharacteristic
549C22#243511yescharacteristic
550C22C2×#45111yescharacteristic
551C22C2×#45111yescharacteristic
552C22C2×#45211yescharacteristic
553C22C2×#45211yescharacteristic
554C22#300811yescharacteristic
555C22#301011yescharacteristic
556C22#301212yes—
557C22C2×#46711yescharacteristic
558C22C2×#47011yescharacteristic
559C22#306212yes—
560C22#329912yes—
561C22#330512yes—
562C22#332912yes—
563C22#333312yes—
564C22#333812yes—
565C22#334012yes—
566C22#335612yes—
567C22#335712yes—
568C4—24no—
569C4—44no—
570C4—84no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4#200118yes—
576C4#242612yes—
577C4#243111yescharacteristic
578C4#243611yescharacteristic
579C4#254012yes—
580C4#262011yescharacteristic
581C4#285011yescharacteristic
582C2—84no—
583C2#1521511yescharacteristic
584C2C2×#200111yescharacteristic
585C2#2339311yescharacteristic
586C2#2412112yes—
587C2#2412212yes—
588C2#2534012yes—
589C2#2534912yes—
590C2#2570111yescharacteristic
591C2#2570611yescharacteristic
592C2#2588311yescharacteristic
593C2#2588811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#477C2semidirect
4. C2×#1990C2semidirect
6. C2×C4×#103C2semidirect
7. #11096C2semidirect
8. #10970C2semidirect
9. #10849C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#25632
C2×#25382
C2×#19902
C2×#19542
C2×C4×#1032
#110962
#109702
#108492