Finite Group Databasebeta

← #393043

Subgroups — #393043

Order 512.

2612 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1983C212yesmaximal, complemented
4C2×#1874C212yesmaximal
5C2×#1872C212yesmaximal, complemented
6C2×#1797C211yesmaximal, characteristic, complemented
7#10932C212yesmaximal, complemented
8#10931C212yesmaximal, complemented
9#10850C212yesmaximal, complemented
10C22×#124C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2212yes—
14C23×C42—21no—
15C23×C42C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#103C2214yes—
18C22×#102C2214yes—
19C22×C22⋊C8C2212yes—
20#2850—24no—
21#2850C418yescomplemented
22C2×#391C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×#386—21no—
27C2×#386—21no—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—28no—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C22×#95—21no—
35#1983—28no—
36#1874—28no—
37#1872—28no—
38#1797—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#336C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—22no—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C23×C8C2314yes—
50C2×#124—22no—
51C2×#124—24no—
52C2×#124—24no—
53C2×#124C2×C414yescomplemented
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8C2×C414yes—
58C2×#121—22no—
59C2×#121—24no—
60C2×#121—24no—
61C2×#121C2×C414yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—28no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D418yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2314yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42—28no—
80C22×C42—22no—
81C22×C42—22no—
82C22×C42—44no—
83C22×C42—44no—
84C22×C42C2×C414yes—
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D414yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×#103—28no—
94C2×#103—28no—
95C2×#103—28no—
96C2×#102—28no—
97C2×#102—28no—
98C2×#102—28no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102#391—22no—
103#391C2×C414yescomplemented
104#390—22no—
105#390C2×C414yescomplemented
106#388—28no—
107#388—28no—
108#386—28no—
109#386—28no—
110#382—22no—
111#382C2×C414yescomplemented
112C4×C4⋊C4—28no—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4D418yescomplemented
117C2×#95—28no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124#340—28no—
125#340—28no—
126#336—24no—
127C25—21no—
128C25C2×D411yescharacteristic
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—24no—
135C22×Q8C22×C412yes—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4C22×C412yes—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4C22⋊C414yes—
173C23×C4C22⋊C414yes—
174C23×C4C22×C412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D414yes—
182C23×C4#4114yes—
183C23×C4C2411yescharacteristic
184C22×C8—24no—
185C22×C8—24no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—24no—
191#124—216no—
192#124—48no—
193C22⋊Q8—216no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196#121—216no—
197#121—44no—
198#121—44no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—232no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—232no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C2×D418yes—
218C2×C4⋊C4C2×D418yes—
219C2×C4⋊C4#4118yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—216no—
242C2×C42—216no—
243C2×C42—28no—
244C2×C42—28no—
245C2×C42—432no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C22⋊C414yes—
251C2×C42C22⋊C414yes—
252C2×C42C22×C412yes—
253C2×C42C22×C412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D414yes—
256C2×C42C2×D412yes—
257C2×C42#4114yes—
258C4⋊C8—48no—
259#103—416no—
260#102—416no—
261C22⋊C8—48no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—28no—
268C24—21no—
269C24—21no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—22no—
274C24—22no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24—42no—
279C24—42no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—42no—
285C24C2×C22⋊C412yes—
286C24#12114yes—
287C24#13514yes—
288C24AGL(1,ℤ/8)12yes—
289C24#13712yes—
290C24C22×D411yescharacteristic
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—41no—
312C2×D4—44no—
313C2×D4—41no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—88no—
317C2×D4—88no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—232no—
347C22×C4—21no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—44no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—416no—
363C22×C4—416no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—416no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—416no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C412yes—
388C22×C4C2×C22⋊C412yes—
389C22×C4C4×D418yes—
390C22×C4#12012yes—
391C22×C4#12012yes—
392C22×C4#12114yes—
393C22×C4#12312yes—
394C22×C4#12312yes—
395C22×C4#12414yes—
396C22×C4#12714yes—
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4#13712yes—
399C22×C4C23×C411yescharacteristic
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4114yes—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—44no—
413C4⋊C4—232no—
414C4⋊C4—28no—
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—48no—
424C4⋊C4#121116yes—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C42—24no—
432C42—24no—
433C42—28no—
434C42—28no—
435C42—416no—
436C42—416no—
437C42C4×D418yes—
438C42#12118yes—
439C23—24no—
440C23—22no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—22no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—81no—
475C23—81no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23#39014yes—
481C23#41614yes—
482C23#41714yes—
483C23#46318yes—
484C23#46418yes—
485C23C22×C22⋊C411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×#12112yes—
488C23C2×#12311yescharacteristic
489C23#52512yes—
490C23C2×#13512yes—
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×#13711yescharacteristic
494C23C2×#13711yescharacteristic
495C23#57512yes—
496Q8—48no—
497Q8—82no—
498Q8—82no—
499D4—48no—
500D4—816no—
501D4—82no—
502D4—82no—
503C2×C4—232no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—216no—
527C2×C4—216no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—416no—
532C2×C4—464no—
533C2×C4—42no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#38614yes—
557C2×C4#38614yes—
558C2×C4#39014yes—
559C2×C4#41714yes—
560C2×C4C2×C4×D412yes—
561C2×C4C2×C4×D412yes—
562C2×C4C4×#4112yes—
563C2×C4C2×#12112yes—
564C2×C4C2×#12114yes—
565C2×C4#52514yes—
566C2×C4C2×#12411yescharacteristic
567C2×C4#52914yes—
568C2×C4C2×#12711yescharacteristic
569C2×C4#53212yes—
570C2×C4#53214yes—
571C8—84no—
572C22—24no—
573C22—28no—
574C22—28no—
575C22—24no—
576C22—24no—
577C22—28no—
578C22—28no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—162no—
590C22—162no—
591C22—162no—
592C22—162no—
593C22#198318yescomplemented
594C22C2×#38611yescharacteristic
595C22C2×#39011yescharacteristic
596C22#229711yescharacteristic
597C22#284712yes—
598C22#285412yes—
599C22C2×#41611yescharacteristic
600C22C2×#41711yescharacteristic
601C22C2×#41711yescharacteristic
602C22#289611yescharacteristic
603C22#300212yes—
604C22#300311yescharacteristic
605C22#300411yescharacteristic
606C22C2×#46312yes—
607C22C2×#46412yes—
608C22#303612yes—
609C22#303712yes—
610C22#309112yes—
611C22#309211yescharacteristic
612C22#309211yescharacteristic
613C22#310512yes—
614C22#310812yes—
615C22#311412yes—
616C22#311712yes—
617C4—24no—
618C4—24no—
619C4—24no—
620C4—24no—
621C4—416no—
622C4—42no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4C4×#12114yes—
627C4#285012yes—
628C4#285612yes—
629C2—28no—
630C2—162no—
631C2#1469811yescharacteristic
632C2#2090811yescharacteristic
633C2#2090911yescharacteristic
634C2C2×#198311yescharacteristic
635C2#2125811yescharacteristic
636C2#2138811yescharacteristic
637C2#2139111yescharacteristic
638C2#3290812yes—
639C2#3290912yes—
640C2#3311012yes—
641C2#3311112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×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. C22×C4×C4⋊C4C2semidirect
3. C2×#1983C2semidirect
5. C2×#1872C2semidirect
6. C2×#1797C2semidirect
7. #10932C2semidirect
8. #10931C2semidirect
9. #10850C2semidirect
21. #2850C4semidirect
53. C2×#124C2×C4semidirect
103. #391C2×C4semidirect
105. #390C2×C4semidirect
111. #382C2×C4semidirect
116. C4×C4⋊C4D4semidirect
593. C22#1983semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C22×C4×C4⋊C42
C2×#19832
C2×#18742
C2×#18722
C2×#17972
#109322
#109312
#108502