Finite Group Databasebeta

← #366779

Subgroups — #366779

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 604 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×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6C22×#340C211yesmaximal, characteristic, complemented
7#13880C211yesmaximal, characteristic, complemented
8#11373C212yesmaximal, complemented
9#11049C212yesmaximal, complemented
10#10925C212yesmaximal, complemented
11#10841C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×#123—22no—
14C22×#121C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2212yes—
22#2501—24no—
23#2501C418yescomplemented
24C2×#399—22no—
25C2×#390C2212yes—
26C2×#385—22no—
27C2×#384C2212yes—
28C2×#384C2212yes—
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#1893—24no—
32#1874—24no—
33#1823—24no—
34C8×C22⋊C4—24no—
35C2×#340—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—22no—
45C4⋊#95C2212yes—
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—24no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2×C414yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D412yes—
91C22×C42C2312yes—
92C2×#102—24no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8D414yes—
102C2×C4×C8—24no—
103C2×C4×C8—24no—
104C2×C4×C8—42no—
105#399—216no—
106#390—24no—
107#390C2×C418yescomplemented
108#385—48no—
109#384—24no—
110#384—24no—
111#384—42no—
112#384C2×C418yescomplemented
113#380—22no—
114#380—22no—
115C4×C22⋊C4C2×C414yes—
116C4×C22⋊C4D414yes—
117C2×#95—24no—
118C2×#95—21no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#340—24no—
126#340—24no—
127#336—24no—
128C25—21no—
129C25C2×D411yescharacteristic
130C22×D4—24no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4C22×C412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—44no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—82no—
163C23×C4C22×C412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D414yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—28no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—28no—
187C22×C8—44no—
188C22×C8—48no—
189C22×C8—48no—
190C22×C8C2×D412yes—
191C22×C8C2×D414yes—
192C22×C8C2×D412yes—
193C22×C8#4114yes—
194#123—464no—
195#121—28no—
196#121—28no—
197#121—44no—
198#121—44no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—216no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—48no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—84no—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4#4114yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C22⋊C418yes—
252C2×C22⋊C4D818yes—
253C2×C22⋊C4SD1618yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C22×C414yes—
259C2×C42C2×D414yes—
260#102—48no—
261#102—44no—
262#102—44no—
263C22⋊C8—28no—
264C4×C8—48no—
265#95—42no—
266#95—48no—
267#95C22⋊C418yes—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24C2×C22⋊C412yes—
282C24#12412yes—
283C24#12712yes—
284C24C2×D814yes—
285C24C2×SD1614yes—
286C24C22×D411yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—48no—
292C2×D4—416no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—88no—
308C2×D4—84no—
309C2×D4—84no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—416no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—816no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C418yes—
364C22×C4C4×D414yes—
365C22×C4C4×D414yes—
366C22×C4#12114yes—
367C22×C4#12114yes—
368C22×C4#12412yes—
369C22×C4#12712yes—
370C22×C4C2×D818yes—
371C22×C4C2×SD1618yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—416no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8#12314yescomplemented
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—48no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—48no—
414C4⋊C4—432no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—816no—
422C22⋊C4—48no—
423C22⋊C4—432no—
424C22⋊C4—84no—
425C22⋊C4#102132yescomplemented
426C42—24no—
427C42—24no—
428C42—24no—
429C42—24no—
430C23—28no—
431C23—28no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—28no—
440C23—28no—
441C23—22no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—84no—
447C23—84no—
448C23—82no—
449C23—82no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—82no—
455C23—81no—
456C23—84no—
457C23—84no—
458C23—81no—
459C23#39014yes—
460C23C2×#10218yes—
461C23#48618yes—
462C23#49214yes—
463C23#49314yes—
464C23C22×C22⋊C411yescharacteristic
465C23C2×#12411yescharacteristic
466C23C2×#12711yescharacteristic
467C23#54612yes—
468C23#54712yes—
469C23C22×D812yes—
470C23C22×SD1612yes—
471C23#57712yes—
472C23#57812yes—
473D4—42no—
474D4—44no—
475D4—42no—
476D4—816no—
477D4—81no—
478D4—81no—
479D4—82no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—816no—
503C2×C4—84no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—816no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—84no—
520C2×C4—81no—
521C2×C4#39014yes—
522C2×C4C2×#102116yes—
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C4×#4112yes—
526C2×C4#52012yes—
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4#52511yescharacteristic
531C2×C4#52912yes—
532C2×C4#53411yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#53912yes—
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4#54712yes—
538C2×C4#54712yes—
539C2×C4#55112yes—
540C2×C4#55212yes—
541C8—24no—
542C8—42no—
543C8—44no—
544C22—24no—
545C22—24no—
546C22—216no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—216no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—88no—
556C22—161no—
557C22—161no—
558C22—164no—
559C22—161no—
560C22—161no—
561C22#195718yes—
562C22C2×#39011yescharacteristic
563C22#233912yes—
564C22#248812yes—
565C22#250111yescharacteristic
566C22#253911yescharacteristic
567C22C22×#10212yes—
568C22#289812yes—
569C22C2×#48612yes—
570C22#313712yes—
571C22C2×#49211yescharacteristic
572C22C2×#49311yescharacteristic
573C22#315411yescharacteristic
574C22#315611yescharacteristic
575C22#329414yes—
576C22#329814yes—
577C22#333414yes—
578C22#333514yes—
579C4—24no—
580C4—216no—
581C4—84no—
582C4—84no—
583C4—82no—
584C4—161no—
585C4—161no—
586C4C4×#12311yescharacteristic
587C4#232211yescharacteristic
588C4#235912yes—
589C4#241211yescharacteristic
590C4#241311yescharacteristic
591C4#242512yes—
592C2—28no—
593C2—162no—
594C2#1399611yescharacteristic
595C2C2×#195711yescharacteristic
596C2#2120511yescharacteristic
597C2#2248514yes—
598C2#2769514yes—
599C2#2792411yescharacteristic
600C2#2792911yescharacteristic
601C2#2832911yescharacteristic
602C2#2833611yescharacteristic
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×C8×C22⋊C4C2semidirect
6. C22×#340C2semidirect
7. #13880C2semidirect
8. #11373C2semidirect
9. #11049C2semidirect
10. #10925C2semidirect
11. #10841C2semidirect
23. #2501C4semidirect
107. #390C2×C4semidirect
112. #384C2×C4semidirect
399. C2×C8#123semidirect
425. C22⋊C4#102semidirect

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×C8×C22⋊C42
C22×#3402
#138802
#113732
#110492
#109252
#108412