Finite Group Databasebeta

← #414729

Subgroups — #414729

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic, complemented
3C2×#2580C211yesmaximal, characteristic
4C2×#1991C211yesmaximal, characteristic, complemented
5C2×#1984C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic
7#11328C212yesmaximal, complemented
8#11327C212yesmaximal, complemented
9#11066C212yesmaximal
10#11066C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#478—24no—
16C2×#478—24no—
17C2×#478—24no—
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2584—24no—
26#2584—24no—
27#2580—24no—
28#2580—24no—
29C2×#399—21no—
30C2×#398—21no—
31C2×#395—21no—
32C2×#393—21no—
33C2×#390C2212yes—
34C2×#389—22no—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#385—22no—
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41#1991—24no—
42#1984—24no—
43C4×#102—24no—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C2×#336C2212yes—
51C23×Q8D411yescharacteristic
52C23×D4C2311yescharacteristic
53C24×C4—21no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125D414yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4D414yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D412yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93#478—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#106—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8D414yes—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—24no—
110C2×#102—24no—
111C2×#102Q818yescomplemented
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115#399—28no—
116#398—28no—
117#395—28no—
118#393—28no—
119#390—24no—
120#390—24no—
121#389—216no—
122#389—24no—
123#389—24no—
124#389—24no—
125#389—24no—
126#389—24no—
127#389—24no—
128#385—216no—
129#382—22no—
130#382D414yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4D414yes—
133C2×#95—21no—
134C2×#95—22no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139#340—24no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—24no—
144#336—24no—
145#336—24no—
146C25—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—24no—
151C22×Q8—42no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4C2×D412yes—
161C22×D4C2×Q814yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—82no—
171C23×C4—82no—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×Q814yes—
194#125—28no—
195#125—28no—
196#125—44no—
197#125—44no—
198C4×Q8—44no—
199C4×Q8—44no—
200C4×Q8—44no—
201C4×Q8—44no—
202C4×D4—28no—
203C4×D4—28no—
204C4×D4—44no—
205C4×D4—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4—82no—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×Q814yes—
256C2×C4⋊C4#4114yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4—82no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—82no—
274C2×C42C2×D412yes—
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42#4112yes—
280#107—28no—
281#106—28no—
282C4⋊C8—28no—
283#102—28no—
284#102—28no—
285#102—216no—
286C4×C8—44no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293C24—22no—
294C24—24no—
295C24—41no—
296C24—41no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C22×D411yescharacteristic
303C2×Q8—28no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4C4⋊Q814yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—48no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12012yes—
380C22×C4#12012yes—
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4#13514yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q812yes—
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4112yes—
392C22×C42−1+412yes—
393C22×C42−1+412yes—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—42no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—416no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C4⋊C4—88no—
437C4⋊C4C22⋊Q818yes—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—88no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—84no—
446C42#12014yes—
447C42#12114yes—
448C42#12114yes—
449C42#12714yes—
450C23—24no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—82no—
471C23C2×#12011yescharacteristic
472C23C2×#12011yescharacteristic
473C23C2×#12111yescharacteristic
474C23#54812yes—
475C23D4×Q812yes—
476C23C2×#13512yes—
477C23#57512yes—
478C23#57612yes—
479C23#57712yes—
480Q8—416no—
481D4—24no—
482D4—24no—
483D4—42no—
484D4—44no—
485D4—42no—
486D4—44no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—816no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4#39214yes—
540C2×C4#39214yes—
541C2×C4C2×#12011yescharacteristic
542C2×C4C2×#12111yescharacteristic
543C2×C4C2×#12111yescharacteristic
544C2×C4C2×C22⋊Q812yes—
545C2×C4C2×#12711yescharacteristic
546C2×C4C2×C4⋊Q811yescharacteristic
547C2×C4#53211yescharacteristic
548C2×C4#53312yes—
549C2×C4#53611yescharacteristic
550C2×C4#53611yescharacteristic
551C2×C4#53612yes—
552C2×C4#54112yes—
553C2×C4#54712yes—
554C2×C4#54712yes—
555C2×C4#54812yes—
556C2×C4D4×Q812yes—
557C2×C4D4×Q812yes—
558C2×C4D4×Q812yes—
559C8—42no—
560C8—42no—
561C8—44no—
562C22—44no—
563C22—48no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—82no—
569C22—84no—
570C22—84no—
571C22C2×#39211yescharacteristic
572C22#240912yes—
573C22#242912yes—
574C22#243211yescharacteristic
575C22#243411yescharacteristic
576C22#300212yes—
577C22#300212yes—
578C22#300811yescharacteristic
579C22#301011yescharacteristic
580C22#301212yes—
581C22#305011yescharacteristic
582C22#305411yescharacteristic
583C22#306212yes—
584C22#336112yes—
585C22#336512yes—
586C22#337012yes—
587C22#337112yes—
588C22#338012yes—
589C22#338112yes—
590C22#338812yes—
591C22#338912yes—
592C4—24no—
593C4—24no—
594C4—48no—
595C4—44no—
596C4—84no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4#242912yes—
602C4#243211yescharacteristic
603C4#243411yescharacteristic
604C4#254212yes—
605C4#262111yescharacteristic
606C4#285711yescharacteristic
607C2—84no—
608C2#1517811yescharacteristic
609C2#2338611yescharacteristic
610C2#2339311yescharacteristic
611C2#2433412yes—
612C2#2433512yes—
613C2#2547312yes—
614C2#2547512yes—
615C2#2565312yes—
616C2#2584312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2584C2semidirect
4. C2×#1991C2semidirect
5. C2×#1984C2semidirect
7. #11328C2semidirect
8. #11327C2semidirect
11. #10850C2semidirect
111. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#25842
C2×#25802
C2×#19912
C2×#19842
C2×C4×#1022
#113282
#113272
#110662
#108502