Finite Group Databasebeta

← #406305

Subgroups — #406305

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2625C211yesmaximal, characteristic
2C2×#1894C211yesmaximal, characteristic
3C2×#1892C211yesmaximal, characteristic
4C2×#1807C211yesmaximal, characteristic
5C2×#1729C211yesmaximal, characteristic
6#14343C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#106C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20#2625—24no—
21#2625—24no—
22C2×#400—21no—
23C2×#398—21no—
24C2×#397—22no—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30#1894—24no—
31#1892—24no—
32#1807—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39#1729—24no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1445—21no—
43#1445—21no—
44#1440—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—42no—
54C23×C8—22no—
55C23×C8—22no—
56C23×C8—24no—
57C23×C8—42no—
58C23×C8D414yes—
59C23×C8C2311yescharacteristic
60C23×C8C2312yes—
61C23×C8C2311yescharacteristic
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8D414yes—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4D412yes—
88C22×C42C2312yes—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—42no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99#400—28no—
100#398—28no—
101#397—216no—
102#391—24no—
103#391—24no—
104#389—24no—
105#389—24no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380D414yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122#340—22no—
123#340—22no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—42no—
128#336—24no—
129#336—24no—
130C25C2×D411yescharacteristic
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C2×D412yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4C2411yescharacteristic
161C2×M16—28no—
162C2×M16—48no—
163C2×M16—48no—
164C2×M16—44no—
165C2×M16—44no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—28no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—28no—
181C22×C8—44no—
182C22×C8—48no—
183C22×C8—48no—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8#4114yes—
187C22×C8#4114yes—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192#117—44no—
193#117—44no—
194#117—44no—
195#117—44no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—81no—
244C2×C22⋊C4—81no—
245C2×C22⋊C4—81no—
246C2×C22⋊C4—81no—
247C2×C22⋊C4—82no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254#106—48no—
255#103—48no—
256#103—44no—
257#103—44no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—42no—
263#95—42no—
264#95—82no—
265#95—82no—
266#95—82no—
267#95—82no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—22no—
276C24#12012yes—
277C24#12112yes—
278C24#12312yes—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4#12012yes—
365C22×C4#12114yes—
366C22×C4#12114yes—
367C22×C4#12112yes—
368C22×C4#12314yes—
369C22×C4#12312yes—
370C22×C4#12414yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C42−1+412yes—
381M16—416no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
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#12314yes—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—88no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23#39414yes—
454C23C2×#12011yescharacteristic
455C23C2×#12111yescharacteristic
456C23C2×#12311yescharacteristic
457C23#53212yes—
458C23#53712yes—
459C23#57712yes—
460C23#57711yescharacteristic
461C23#57711yescharacteristic
462C23#57711yescharacteristic
463C23#57711yescharacteristic
464C23#57812yes—
465Q8—44no—
466Q8—42no—
467Q8—42no—
468Q8—81no—
469Q8—81no—
470Q8—82no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4#39414yes—
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×#12111yescharacteristic
534C2×C4C2×#12311yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4#52912yes—
539C2×C4#53411yescharacteristic
540C2×C4#53611yescharacteristic
541C2×C4#53912yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54712yes—
545C2×C4#54712yes—
546C2×C4#55112yes—
547C2×C4#55112yes—
548C2×C4#55212yes—
549C2×C4#55212yes—
550C8—24no—
551C8—42no—
552C8—44no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22C2×#39411yescharacteristic
565C22#243012yes—
566C22#249812yes—
567C22#261811yescharacteristic
568C22#285411yescharacteristic
569C22#300911yescharacteristic
570C22#301011yescharacteristic
571C22#301312yes—
572C22#305711yescharacteristic
573C22#305911yescharacteristic
574C22#306011yescharacteristic
575C22#306011yescharacteristic
576C22#309712yes—
577C22#309911yescharacteristic
578C22#309911yescharacteristic
579C22#317312yes—
580C22#317412yes—
581C22#317812yes—
582C22#317812yes—
583C22#325812yes—
584C22#325812yes—
585C22#326012yes—
586C22#326312yes—
587C4—24no—
588C4—82no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—161no—
594C4—161no—
595C4—162no—
596C4#241211yescharacteristic
597C4#241311yescharacteristic
598C4#241912yes—
599C4#248611yescharacteristic
600C4#248812yes—
601C4#253411yescharacteristic
602C2—44no—
603C2#1492711yescharacteristic
604C2#2348711yescharacteristic
605C2#2348911yescharacteristic
606C2#2550912yes—
607C2#2551112yes—
608C2#2786712yes—
609C2#2787312yes—
610C2#2977711yescharacteristic
611C2#2978111yescharacteristic
612C2#3313311yescharacteristic
613C2#3313711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11293C2semidirect
8. #11220C2semidirect
9. #11190C2semidirect
10. #11122C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26252
C2×#18942
C2×#18922
C2×#18072
C2×#17292
#143432
#112932
#112202
#111902
#111222
#107222