Finite Group Databasebeta

← #406254

Subgroups — #406254

Order 512.

1964 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#483C211yesmaximal, characteristic
2C22×#467C211yesmaximal, characteristic
3C2×#2503C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#14139C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11122C212yesmaximal, complemented
9#11052C212yesmaximal, complemented
10#11024C212yesmaximal, complemented
11#10714C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#483—24no—
19C2×#483—24no—
20C2×#483—24no—
21C2×#467—24no—
22C2×#467—24no—
23C2×#467—24no—
24C22×#107C2211yescharacteristic
25C22×#106C2212yes—
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#1872—24no—
40C2×#340—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47C4⋊#95C2212yes—
48C22⋊#95—22no—
49C23×Q8C2311yescharacteristic
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16—22no—
57C22×Q16—24no—
58C22×Q16—42no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—24no—
62C23×C8—42no—
63C23×C8D414yes—
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
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⋊Q8—22no—
77C2×C22⋊Q8D414yes—
78C2×C22⋊Q8D414yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4D414yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—21no—
99C22×C22⋊C4D412yes—
100C22×C42C2312yes—
101#483—232no—
102#467—232no—
103C2×#107—22no—
104C2×#107—22no—
105C2×#107—22no—
106C2×#106—24no—
107C2×#106—24no—
108C2×#106—24no—
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—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—82no—
184C23×C4C2×D414yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4#4112yes—
192C23×C4C2411yescharacteristic
193C2×Q16—28no—
194C2×Q16—48no—
195C2×Q16—48no—
196C2×Q16—44no—
197C2×Q16—44no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—28no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—28no—
213C22×C8—44no—
214C22×C8—48no—
215C22×C8—48no—
216C22×C8C2×D412yes—
217C22×C8C2×D414yes—
218C22×C8C2×D412yes—
219C22×C8#4114yes—
220C22⋊Q8—28no—
221C22⋊Q8—28no—
222C22⋊Q8—28no—
223C22⋊Q8—28no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4#4114yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285#107—28no—
286#106—216no—
287#103—28no—
288#103—28no—
289#103—216no—
290#103—48no—
291C22⋊C8—28no—
292#95—216no—
293#95—216no—
294#95—24no—
295#95—48no—
296#95—84no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24#12012yes—
302C24#12112yes—
303C24#12712yes—
304C24#13514yes—
305C24#13714yes—
306C24C22×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—42no—
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—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—42no—
339C2×Q8—42no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—816no—
400C22×C4—84no—
401C22×C4—88no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4#12014yes—
406C22×C4#12014yes—
407C22×C4#12012yes—
408C22×C4#12114yes—
409C22×C4#12114yes—
410C22×C4#12112yes—
411C22×C4#12712yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C42−1+412yes—
422Q16—416no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—416no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8#12314yes—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C23—28no—
469C23—28no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—28no—
477C23—24no—
478C23—22no—
479C23—28no—
480C23#39214yes—
481C23#45218yes—
482C23#46418yes—
483C23#49514yes—
484C23#49714yes—
485C23C2×#12011yescharacteristic
486C23C2×#12111yescharacteristic
487C23C2×#12711yescharacteristic
488C23#54612yes—
489C23#54712yes—
490C23C2×#13512yes—
491C23C2×#13712yes—
492C23#57712yes—
493C23#57812yes—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—44no—
499Q8—44no—
500Q8—81no—
501Q8—81no—
502Q8—81no—
503Q8—81no—
504Q8—82no—
505Q8—82no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—84no—
550C2×C4—816no—
551C2×C4—84no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4#39214yes—
556C2×C4C2×#12011yescharacteristic
557C2×C4C2×#12011yescharacteristic
558C2×C4C2×#12111yescharacteristic
559C2×C4C2×#12111yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52912yes—
563C2×C4#53212yes—
564C2×C4#53411yescharacteristic
565C2×C4#53611yescharacteristic
566C2×C4#53712yes—
567C2×C4#53912yes—
568C2×C4#54612yes—
569C2×C4#54612yes—
570C2×C4#54612yes—
571C2×C4#54712yes—
572C2×C4#54712yes—
573C2×C4#54712yes—
574C8—24no—
575C8—42no—
576C8—44no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—216no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—216no—
585C22#201518yes—
586C22C2×#39211yescharacteristic
587C22#242512yes—
588C22#243012yes—
589C22#250111yescharacteristic
590C22#253911yescharacteristic
591C22C2×#45212yes—
592C22#301312yes—
593C22C2×#46412yes—
594C22#304612yes—
595C22C2×#49511yescharacteristic
596C22C2×#49711yescharacteristic
597C22#315411yescharacteristic
598C22#315611yescharacteristic
599C22#329614yes—
600C22#331214yes—
601C22#333614yes—
602C22#334414yes—
603C4—24no—
604C4—82no—
605C4—82no—
606C4—84no—
607C4—82no—
608C4—82no—
609C4—84no—
610C4—84no—
611C4#240511yescharacteristic
612C4#240611yescharacteristic
613C4#242512yes—
614C4#243012yes—
615C4#261811yescharacteristic
616C4#285411yescharacteristic
617C2—28no—
618C2#1492611yescharacteristic
619C2C2×#201511yescharacteristic
620C2#2342711yescharacteristic
621C2#2520814yes—
622C2#2554114yes—
623C2#2793111yescharacteristic
624C2#2793311yescharacteristic
625C2#2833011yescharacteristic
626C2#2834511yescharacteristic
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. #11122C2semidirect
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×#4832
C22×#4672
C2×#25032
C2×#18722
C22×#3402
#141392
#113252
#111222
#110522
#110242
#107142