Finite Group Databasebeta

← #398042

Subgroups — #398042

Order 512.

1882 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2025C211yesmaximal, characteristic, complemented
3C2×#1893C212yesmaximal
4C22×#336C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6#11385C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11268C211yesmaximal, characteristic, complemented
9#11066C212yesmaximal, complemented
10#11051C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal, complemented
12C24×C8C2212yes—
13C22×#125C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2851—28no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#387—22no—
23C2×#386—22no—
24C2×#385—22no—
25C2×#382C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28#2025—24no—
29#1893—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336—24no—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1437—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4D412yes—
46C24×C4C2311yescharacteristic
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—24no—
52C23×C8—42no—
53C23×C8—42no—
54C23×C8C2312yes—
55C23×C8C2312yes—
56C2×#125—24no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—42no—
60C2×#121—22no—
61C2×#121—24no—
62C2×#121—24no—
63C2×#121Q814yes—
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—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—24no—
77C22×C42—22no—
78C22×C42—22no—
79C22×C42—42no—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—42no—
88C2×#102—42no—
89#390—22no—
90#390—22no—
91#390—42no—
92#389—24no—
93#389—24no—
94#387—216no—
95#386—48no—
96#385—216no—
97#382—24no—
98#380—24no—
99#380—24no—
100#380D418yes—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—22no—
115#336D414yescomplemented
116C25—21no—
117C25C2×D411yescharacteristic
118C22×D4—24no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—44no—
131C22×D4—44no—
132C22×D4C2×Q812yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D414yes—
155C23×C4C2×Q812yes—
156C23×C4#4114yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—216no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—48no—
173C22×C8—48no—
174C22×C8—48no—
175C22×C8—48no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8C2×D414yes—
179C22×C8#4114yes—
180#125—48no—
181#125—48no—
182#121—216no—
183#121—48no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×Q812yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—22no—
236C2×C42—48no—
237C2×C42—48no—
238C2×C42—42no—
239C2×C42C2×D412yes—
240C2×C42#4112yes—
241#102—48no—
242#102—48no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—44no—
247#95—44no—
248#95—42no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24#12012yes—
267C24C22⋊Q812yes—
268C24#12412yes—
269C24#13514yes—
270C24AGL(1,ℤ/8)14yes—
271C24C22×D411yescharacteristic
272C2×D4—24no—
273C2×D4—22no—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—48no—
277C2×D4—416no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—88no—
291C2×D4—84no—
292C2×D4—84no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—216no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
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—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4#12012yes—
360C22×C4#12114yes—
361C22×C4C22⋊Q818yes—
362C22×C4C22⋊Q812yes—
363C22×C4#12418yes—
364C22×C4#12412yes—
365C22×C4#13514yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+411yescharacteristic
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—432no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—416no—
397C4⋊C4—416no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C22⋊C4—416no—
408C22⋊C4—48no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C22⋊C4—84no—
412C42—24no—
413C42—24no—
414C42—48no—
415C42—42no—
416C42—42no—
417C23—22no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—24no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—81no—
444C23—81no—
445C23—81no—
446C23—82no—
447C23—81no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—81no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23#39314yes—
459C23#44918yes—
460C23#47818yes—
461C23#48814yes—
462C23#49014yes—
463C23C2×#12011yescharacteristic
464C23C2×#12111yescharacteristic
465C23C2×C22⋊Q811yescharacteristic
466C23C2×#12411yescharacteristic
467C23#53211yescharacteristic
468C23#53411yescharacteristic
469C23#53711yescharacteristic
470C23C2×#13512yes—
471C23C2×#13512yes—
472C23C2×AGL(1,ℤ/8)12yes—
473C23#57512yes—
474D4—44no—
475D4—44no—
476D4—816no—
477D4—82no—
478D4—82no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—432no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#39314yes—
534C2×C4C2×C22⋊Q812yes—
535C2×C4#52512yes—
536C2×C4C2×#12412yes—
537C2×C4#52912yes—
538C2×C4#53812yes—
539C2×C4#53912yes—
540C2×C4D4212yes—
541C2×C4#54612yes—
542C2×C4#54612yes—
543C2×C4#54611yescharacteristic
544C2×C4#54712yes—
545C2×C4D4×Q811yescharacteristic
546C2×C4#55112yes—
547C2×C4#55211yescharacteristic
548C2×C4#55412yes—
549C2×C4#55411yescharacteristic
550C8—44no—
551C8—44no—
552C22—28no—
553C22—24no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—28no—
558C22—24no—
559C22—28no—
560C22—24no—
561C22—24no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—84no—
567C22—161no—
568C22—161no—
569C22—162no—
570C22—162no—
571C22—161no—
572C22—161no—
573C22#202518yescomplemented
574C22C2×#39311yescharacteristic
575C22#241011yescharacteristic
576C22#241711yescharacteristic
577C22#243011yescharacteristic
578C22#256411yescharacteristic
579C22#263911yescharacteristic
580C22#285011yescharacteristic
581C22C2×#44912yes—
582C22#300212yes—
583C22#305012yes—
584C22#305512yes—
585C22C2×#47812yes—
586C22#307612yes—
587C22C2×#48811yescharacteristic
588C22#313111yescharacteristic
589C22C2×#49011yescharacteristic
590C22#313511yescharacteristic
591C22#316812yes—
592C22#318012yes—
593C22#318812yes—
594C22#319112yes—
595C22#324712yes—
596C22#325012yes—
597C4—24no—
598C4—44no—
599C4—44no—
600C4—84no—
601C4—82no—
602C4—84no—
603C4—161no—
604C4—161no—
605C4#244912yes—
606C4#245012yes—
607C4#247212yes—
608C4#248812yes—
609C2—28no—
610C2—162no—
611C2#1475911yescharacteristic
612C2C2×#202511yescharacteristic
613C2#2345111yescharacteristic
614C2#2438212yes—
615C2#2483912yes—
616C2#2550012yes—
617C2#2916112yes—
618C2#3018611yescharacteristic
619C2#3018911yescharacteristic
620C2#3304411yescharacteristic
621C2#3305211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
1. C2×#2851C2semidirect
2. C2×#2025C2semidirect
4. C22×#336C2semidirect
6. #11385C2semidirect
7. #11293C2semidirect
8. #11268C2semidirect
9. #11066C2semidirect
10. #11051C2semidirect
11. #10721C2semidirect
115. #336D4semidirect
573. C22#2025semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#20252
C2×#18932
C22×#3362
#146982
#113852
#112932
#112682
#110662
#110512
#107212