Finite Group Databasebeta

← #398041

Subgroups — #398041

Order 512.

1724 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×#2026C211yesmaximal, characteristic, complemented
3C2×#1892C212yesmaximal
4C22×#336C211yesmaximal, characteristic
5#14702C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11268C211yesmaximal, characteristic, complemented
9#11067C212yesmaximal, complemented
10#11052C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2212yes—
13C22×#125C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18#2851—28no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#385—22no—
24C2×#382C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27#2026—24no—
28#1892—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336—24no—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1436—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×Q8C2311yescharacteristic
44C24×C4D412yes—
45C24×C4C2311yescharacteristic
46C23×C8—24no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—42no—
52C23×C8—42no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×#125—24no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—42no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8Q814yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C42—24no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—42no—
78C22×C42D412yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87#391—22no—
88#391—22no—
89#391—42no—
90#389—24no—
91#389—24no—
92#387—216no—
93#385—216no—
94#382—24no—
95#380—24no—
96#380—24no—
97#380D418yes—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—24no—
113#336—22no—
114#336D414yes—
115C25C2×D411yescharacteristic
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8C2×Q812yes—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D414yes—
149C23×C4C2×Q812yes—
150C23×C4#4114yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C22×C8—216no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—48no—
167C22×C8—48no—
168C22×C8—48no—
169C22×C8—48no—
170C22×C8—48no—
171C22×C8—48no—
172C22×C8C2×D414yes—
173C22×C8#4114yes—
174#125—48no—
175#125—48no—
176C22⋊Q8—216no—
177C22⋊Q8—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×Q812yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—48no—
228C2×C42—48no—
229C2×C42—42no—
230C2×C42C2×D412yes—
231C2×C42#4112yes—
232#103—48no—
233#103—48no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—42no—
241#95—44no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24#12012yes—
249C24C22⋊Q812yes—
250C24#12412yes—
251C24#13514yes—
252C24#13714yes—
253C24C22×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—42no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—216no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—48no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4#12012yes—
344C22×C4#12114yes—
345C22×C4C22⋊Q818yes—
346C22×C4C22⋊Q812yes—
347C22×C4#12418yes—
348C22×C4#12412yes—
349C22×C4#13514yes—
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4112yes—
354C22×C4C2×#4112yes—
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+411yescharacteristic
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—432no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—416no—
381C4⋊C4—416no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C22⋊C4—48no—
393C22⋊C4—416no—
394C22⋊C4—84no—
395C42—24no—
396C42—24no—
397C42—48no—
398C42—42no—
399C42—42no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23#39314yes—
419C23#45218yes—
420C23#47918yes—
421C23#48814yes—
422C23#49114yes—
423C23C2×#12011yescharacteristic
424C23C2×#12111yescharacteristic
425C23C2×C22⋊Q811yescharacteristic
426C23C2×#12411yescharacteristic
427C23#53211yescharacteristic
428C23#53411yescharacteristic
429C23#53711yescharacteristic
430C23C2×#13512yes—
431C23C2×#13512yes—
432C23C2×#13712yes—
433C23#57512yes—
434Q8—44no—
435Q8—44no—
436Q8—82no—
437Q8—82no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—28no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—432no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#39314yes—
500C2×C4C2×C22⋊Q812yes—
501C2×C4#52512yes—
502C2×C4C2×#12412yes—
503C2×C4#52912yes—
504C2×C4#53812yes—
505C2×C4#53912yes—
506C2×C4D4212yes—
507C2×C4#54612yes—
508C2×C4#54612yes—
509C2×C4#54611yescharacteristic
510C2×C4#54712yes—
511C2×C4D4×Q811yescharacteristic
512C2×C4#55112yes—
513C2×C4#55211yescharacteristic
514C2×C4#55412yes—
515C2×C4#55411yescharacteristic
516C8—44no—
517C8—44no—
518C22—28no—
519C22—24no—
520C22—24no—
521C22—28no—
522C22—24no—
523C22—28no—
524C22—24no—
525C22—28no—
526C22—24no—
527C22—24no—
528C22#202618yescomplemented
529C22C2×#39311yescharacteristic
530C22#241011yescharacteristic
531C22#241711yescharacteristic
532C22#243011yescharacteristic
533C22#256411yescharacteristic
534C22#263911yescharacteristic
535C22#285011yescharacteristic
536C22C2×#45212yes—
537C22#300212yes—
538C22#305012yes—
539C22#305612yes—
540C22C2×#47912yes—
541C22#307612yes—
542C22C2×#48811yescharacteristic
543C22#313111yescharacteristic
544C22C2×#49111yescharacteristic
545C22#313511yescharacteristic
546C22#316812yes—
547C22#318112yes—
548C22#319012yes—
549C22#319112yes—
550C22#324812yes—
551C22#325012yes—
552C4—24no—
553C4—44no—
554C4—44no—
555C4—84no—
556C4—82no—
557C4—84no—
558C4—82no—
559C4—161no—
560C4—161no—
561C4#244912yes—
562C4#245012yes—
563C4#247212yes—
564C4#248812yes—
565C2—28no—
566C2#1475911yescharacteristic
567C2C2×#202611yescharacteristic
568C2#2345111yescharacteristic
569C2#2438312yes—
570C2#2484012yes—
571C2#2549912yes—
572C2#2915812yes—
573C2#3018611yescharacteristic
574C2#3019011yescharacteristic
575C2#3304411yescharacteristic
576C2#3305311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2026C2semidirect
6. #11385C2semidirect
7. #11292C2semidirect
8. #11268C2semidirect
9. #11067C2semidirect
10. #11052C2semidirect
528. C22#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#20262
C2×#18922
C22×#3362
#147022
#113852
#112922
#112682
#110672
#110522
#107212