Finite Group Databasebeta

← #366786

Subgroups — #366786

Order 512.

2016 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1874C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic, complemented
5C2×#1752C211yesmaximal, characteristic, complemented
6C22×#340C211yesmaximal, characteristic, complemented
7#13881C211yesmaximal, characteristic, complemented
8#11371C212yesmaximal, complemented
9#11051C212yesmaximal, complemented
10#10927C212yesmaximal, complemented
11#10843C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×#123—22no—
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2212yes—
22#2501—24no—
23#2501C418yescomplemented
24C2×#399—22no—
25C2×#390C2212yes—
26C2×#384C2212yes—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1893—24no—
31#1874—24no—
32#1823—24no—
33#1752—24no—
34C2×#340—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1442—22no—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×D4—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—42no—
55C23×C8D414yes—
56C23×C8C2312yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×#123—216no—
60C2×#123—48no—
61C2×#123—48no—
62C2×#121—22no—
63C2×#121—22no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2×C414yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—24no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2×C414yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D412yes—
90C22×C42C2312yes—
91C2×#102—24no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97C2×#102—42no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8D414yes—
101C2×#97—24no—
102C2×#97—24no—
103C2×#97—42no—
104#399—216no—
105#390—24no—
106#390C2×C418yescomplemented
107#384—24no—
108#384C2×C418yescomplemented
109#382—24no—
110#382—42no—
111#380—22no—
112#380—22no—
113C4×C22⋊C4C2×C414yes—
114C4×C22⋊C4D414yes—
115C2×#95—24no—
116C2×#95—21no—
117C2×#95—24no—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—24no—
125#340—24no—
126#336—24no—
127C25—21no—
128C25C2×D411yescharacteristic
129C22×D4—24no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4C22×C412yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—82no—
162C23×C4C22×C412yes—
163C23×C4C2×D414yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—28no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—28no—
186C22×C8—44no—
187C22×C8—48no—
188C22×C8—48no—
189C22×C8C2×D412yes—
190C22×C8C2×D414yes—
191C22×C8C2×D412yes—
192C22×C8#4114yes—
193#123—464no—
194#121—28no—
195#121—28no—
196#121—44no—
197#121—44no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—216no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—48no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4C22⋊C418yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42C22×C414yes—
255C2×C42C2×D414yes—
256#102—48no—
257#102—44no—
258#102—44no—
259C22⋊C8—28no—
260#97—48no—
261#95—48no—
262#95—42no—
263#95—84no—
264#95C22⋊C418yes—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24C2×C22⋊C412yes—
279C24#12412yes—
280C24#12712yes—
281C24AGL(1,ℤ/8)14yes—
282C24AGL(1,ℤ/8)14yes—
283C24C22×D411yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—48no—
289C2×D4—416no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—88no—
305C2×D4—84no—
306C2×D4—84no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—48no—
346C22×C4—416no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—816no—
356C22×C4—84no—
357C22×C4—88no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4C2×C22⋊C418yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C4×D414yes—
364C22×C4C4×D414yes—
365C22×C4#12114yes—
366C22×C4#12114yes—
367C22×C4#12412yes—
368C22×C4#12712yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+412yes—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—416no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8#12314yescomplemented
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—48no—
404C4⋊C4—432no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—48no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C22⋊C4—216no—
419C22⋊C4—432no—
420C22⋊C4—48no—
421C22⋊C4—84no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C23—28no—
427C23—28no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—22no—
435C23—28no—
436C23—22no—
437C23—28no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—84no—
443C23—84no—
444C23—82no—
445C23—82no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—82no—
451C23—81no—
452C23—84no—
453C23—81no—
454C23—84no—
455C23#39014yes—
456C23#41818yes—
457C23#49018yes—
458C23#49614yes—
459C23#49614yes—
460C23C22×C22⋊C411yescharacteristic
461C23C2×#12411yescharacteristic
462C23C2×#12711yescharacteristic
463C23#54612yes—
464C23#54712yes—
465C23C2×AGL(1,ℤ/8)12yes—
466C23C2×AGL(1,ℤ/8)12yes—
467C23#57712yes—
468C23#57812yes—
469D4—42no—
470D4—44no—
471D4—42no—
472D4—816no—
473D4—81no—
474D4—81no—
475D4—82no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—816no—
500C2×C4—84no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—816no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4#39014yes—
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C4×#4112yes—
522C2×C4#52012yes—
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4#52511yescharacteristic
527C2×C4#52912yes—
528C2×C4#53411yescharacteristic
529C2×C4#53611yescharacteristic
530C2×C4#53912yes—
531C2×C4#54612yes—
532C2×C4#54612yes—
533C2×C4#54712yes—
534C2×C4#54712yes—
535C2×C4#55112yes—
536C2×C4#55212yes—
537C8—24no—
538C8—42no—
539C8—44no—
540C22—24no—
541C22—24no—
542C22—216no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—216no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—88no—
552C22—161no—
553C22—161no—
554C22—164no—
555C22—161no—
556C22—161no—
557C22#196118yes—
558C22C2×#39011yescharacteristic
559C22#233912yes—
560C22#248812yes—
561C22#250111yescharacteristic
562C22#253911yescharacteristic
563C22C2×#41812yes—
564C22#289812yes—
565C22C2×#49012yes—
566C22#313712yes—
567C22C2×#49611yescharacteristic
568C22C2×#49611yescharacteristic
569C22#315411yescharacteristic
570C22#315611yescharacteristic
571C22#330614yes—
572C22#330814yes—
573C22#334314yes—
574C22#334514yes—
575C4—24no—
576C4—48no—
577C4—84no—
578C4—84no—
579C4—82no—
580C4—161no—
581C4—161no—
582C4C4×#12311yescharacteristic
583C4#232211yescharacteristic
584C4#235912yes—
585C4#241211yescharacteristic
586C4#241311yescharacteristic
587C4#242512yes—
588C2—28no—
589C2—162no—
590C2#1399611yescharacteristic
591C2C2×#196111yescharacteristic
592C2#2120511yescharacteristic
593C2#2249514yes—
594C2#2771714yes—
595C2#2793411yescharacteristic
596C2#2793911yescharacteristic
597C2#2834311yescharacteristic
598C2#2834611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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
4. C2×#1823C2semidirect
5. C2×#1752C2semidirect
6. C22×#340C2semidirect
7. #13881C2semidirect
8. #11371C2semidirect
9. #11051C2semidirect
10. #10927C2semidirect
11. #10843C2semidirect
23. #2501C4semidirect
106. #390C2×C4semidirect
108. #384C2×C4semidirect
396. C2×C8#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#18932
C2×#18742
C2×#18232
C2×#17522
C22×#3402
#138812
#113712
#110512
#109272
#108432