Finite Group Databasebeta

← #406267

Subgroups — #406267

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C22×#470C211yesmaximal, characteristic
3C2×#2503C211yesmaximal, characteristic
4C2×#1873C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#14138C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#11052C212yesmaximal, complemented
10#11024C212yesmaximal, complemented
11#10715C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#484—24no—
19C2×#484—24no—
20C2×#484—24no—
21C2×#470—24no—
22C2×#470—24no—
23C2×#470—24no—
24C22×#107C2212yes—
25C22×#106C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2212yes—
30C22×C22⋊C8C2211yescharacteristic
31#2503—24no—
32#2503—24no—
33C2×#400—22no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#382C2212yes—
37C2×#381—22no—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1873—24no—
40#1736—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1440—22no—
47#1436—22no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—24no—
57C22×Q16—42no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—24no—
61C22×M16—42no—
62C22×M16D414yes—
63C23×C8C2312yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8D414yes—
77C2×C22⋊Q8D414yes—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—24no—
81C2×#117—42no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4D414yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4D412yes—
99C22×C42—21no—
100C22×C42C2312yes—
101#484—232no—
102#470—232no—
103C2×#107—24no—
104C2×#107—24no—
105C2×#107—24no—
106C2×#106—22no—
107C2×#106—22no—
108C2×#106—22no—
109C2×#103—24no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—22no—
114C2×#103—22no—
115C2×#103—24no—
116C2×#103—22no—
117C2×#103—22no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—42no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8D414yes—
124#400—216no—
125#391—24no—
126#391—24no—
127#391—24no—
128#391—42no—
129#382—24no—
130#382—24no—
131#381—216no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95D412yes—
139#340—24no—
140#340—24no—
141#340—24no—
142#336—24no—
143#336—42no—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8C2×D412yes—
164C22×Q8C2×D412yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—48no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2×D414yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4#4112yes—
187C23×C4C2411yescharacteristic
188C2×Q16—28no—
189C2×Q16—48no—
190C2×Q16—48no—
191C2×Q16—44no—
192C2×Q16—44no—
193C2×M16—28no—
194C2×M16—28no—
195C2×M16—44no—
196C2×M16—48no—
197C2×M16—48no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8C2×D412yes—
212C22×C8C2×D414yes—
213C22×C8C2×D412yes—
214C22×C8#4114yes—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221C22⋊Q8—44no—
222C22⋊Q8—44no—
223#117—28no—
224#117—48no—
225#117—44no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4#4114yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—44no—
280C2×C42—42no—
281#107—216no—
282#106—28no—
283#103—28no—
284#103—28no—
285#103—216no—
286#103—48no—
287C22⋊C8—28no—
288#95—216no—
289#95—216no—
290#95—24no—
291#95—48no—
292#95—48no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24#12012yes—
298C24#12112yes—
299C24#12712yes—
300C24C22×D411yescharacteristic
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—416no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—88no—
390C22×C4—88no—
391C22×C4#12014yes—
392C22×C4#12014yes—
393C22×C4#12012yes—
394C22×C4#12114yes—
395C22×C4#12114yes—
396C22×C4#12112yes—
397C22×C4#12712yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4H(5,𝔽2)12yes—
407C22×C42−1+412yes—
408Q16—416no—
409M16—416no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8#12314yes—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—42no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C42—48no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23#39214yes—
470C23C2×#12011yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×#12711yescharacteristic
473C23#54612yes—
474C23#54712yes—
475C23#57514yes—
476C23#57814yes—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—44no—
482Q8—44no—
483Q8—81no—
484Q8—81no—
485Q8—81no—
486Q8—81no—
487Q8—82no—
488Q8—82no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—44no—
518C2×C4—816no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—84no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—84no—
534C2×C4#39214yes—
535C2×C4C2×#12011yescharacteristic
536C2×C4C2×#12011yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4#52912yes—
542C2×C4#53212yes—
543C2×C4#53411yescharacteristic
544C2×C4#53611yescharacteristic
545C2×C4#53712yes—
546C2×C4#53912yes—
547C2×C4#54612yes—
548C2×C4#54612yes—
549C2×C4#54612yes—
550C2×C4#54712yes—
551C2×C4#54712yes—
552C2×C4#54712yes—
553C8—24no—
554C8—42no—
555C8—44no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—44no—
560C22—48no—
561C22—44no—
562C22—48no—
563C22C2×#39211yescharacteristic
564C22#242512yes—
565C22#243012yes—
566C22#250111yescharacteristic
567C22#253911yescharacteristic
568C22#300512yes—
569C22#301112yes—
570C22#304012yes—
571C22#304412yes—
572C22#315312yes—
573C22#315612yes—
574C22#331514yes—
575C22#332714yes—
576C22#334914yes—
577C22#335214yes—
578C4—24no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—84no—
585C4—84no—
586C4#240511yescharacteristic
587C4#240611yescharacteristic
588C4#242512yes—
589C4#243012yes—
590C4#261811yescharacteristic
591C4#285411yescharacteristic
592C2—44no—
593C2#1492611yescharacteristic
594C2#2342511yescharacteristic
595C2#2343111yescharacteristic
596C2#2522914yes—
597C2#2555714yes—
598C2#2794312yes—
599C2#2835612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
7. #11325C2semidirect
8. #11124C2semidirect
9. #11052C2semidirect
10. #11024C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4702
C2×#25032
C2×#18732
C2×#17362
#141382
#113252
#111242
#110522
#110242
#107152