Finite Group Databasebeta

← #409951

Subgroups — #409951

Order 512.

1746 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×#483C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4C2×#1824C211yesmaximal, characteristic
5C2×#1825C211yesmaximal, characteristic
6#14258C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11323C212yesmaximal, complemented
9#11052C212yesmaximal, complemented
10#11050C212yesmaximal, complemented
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#485—24no—
17C2×#485—24no—
18C2×#485—24no—
19C2×#483—24no—
20C2×#483—24no—
21C2×#483—24no—
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2212yes—
25C22×#103C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30#2317—24no—
31#2317—24no—
32C2×#391C2212yes—
33C2×#389—22no—
34C2×#385—22no—
35C2×#384C2212yes—
36C2×#382C2212yes—
37C2×#380C2211yescharacteristic
38#1825—24no—
39#1824—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47#1440—22no—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8D414yes—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—24no—
65C2×#117—24no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117D414yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4D412yes—
80C22×C4⋊C4D414yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4D412yes—
90C22×C42—21no—
91C22×C42C2312yes—
92#485—232no—
93#483—232no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#106—24no—
98C2×#106—22no—
99C2×#106—24no—
100C2×#106—24no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#103—24no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—22no—
111C2×#103—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8D414yes—
117C2×C22⋊C8D414yes—
118#391—24no—
119#391—24no—
120#389—216no—
121#385—216no—
122#384—24no—
123#384—24no—
124#382—24no—
125#382—24no—
126#380—22no—
127#380—22no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95—21no—
131C2×#95—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137#336—24no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C2×D412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—84no—
163C23×C4C2×D414yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8C2×D412yes—
187C22×C8C2×D414yes—
188C22×C8C2×D412yes—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193#117—28no—
194#117—28no—
195#117—216no—
196#117—28no—
197#117—28no—
198#117—44no—
199#117—44no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
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—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—216no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—84no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—82no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42#4114yes—
263C2×C42#4114yes—
264#107—28no—
265#106—216no—
266#106—28no—
267#103—216no—
268#103—28no—
269#103—28no—
270C22⋊C8—28no—
271C22⋊C8—28no—
272#95—216no—
273#95—48no—
274#95—42no—
275#95—84no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—22no—
281C24#12012yes—
282C24#12112yes—
283C24#12712yes—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4—88no—
358C22×C4#12014yes—
359C22×C4#12014yes—
360C22×C4#12012yes—
361C22×C4#12114yes—
362C22×C4#12114yes—
363C22×C4#12112yes—
364C22×C4#12318yes—
365C22×C4#12712yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4112yes—
373C22×C4C2×#4112yes—
374C22×C42−1+412yes—
375C22×C42−1+412yes—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—216no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—816no—
417C22⋊C4—216no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—84no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—48no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23#39214yes—
442C23C2×#12011yescharacteristic
443C23C2×#12111yescharacteristic
444C23C2×#12312yes—
445C23C2×#12711yescharacteristic
446C23#53612yes—
447C23#57714yes—
448C23#57712yes—
449C23#57812yes—
450Q8—44no—
451Q8—42no—
452Q8—42no—
453Q8—82no—
454Q8—81no—
455Q8—81no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—82no—
503C2×C4#39214yes—
504C2×C4C2×#12011yescharacteristic
505C2×C4C2×#12011yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×#12111yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4#52512yes—
510C2×C4#52511yescharacteristic
511C2×C4#53611yescharacteristic
512C2×C4#53612yes—
513C2×C4#53611yescharacteristic
514C2×C4#54712yes—
515C2×C4#54712yes—
516C2×C4#54712yes—
517C2×C4#54712yes—
518C2×C4#54712yes—
519C2×C4#54712yes—
520C2×C4#55212yes—
521C2×C4#55212yes—
522C8—44no—
523C8—42no—
524C8—42no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—48no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22C2×#39211yescharacteristic
534C22#248612yes—
535C22#250112yes—
536C22#261411yescharacteristic
537C22#284911yescharacteristic
538C22#300912yes—
539C22#301312yes—
540C22#304312yes—
541C22#304612yes—
542C22#309714yes—
543C22#309914yes—
544C22#315411yescharacteristic
545C22#315412yes—
546C22#315611yescharacteristic
547C22#324214yes—
548C22#324414yes—
549C4—24no—
550C4—48no—
551C4—82no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—162no—
557C4#240612yes—
558C4#240611yescharacteristic
559C4#240611yescharacteristic
560C4#241312yes—
561C4#241311yescharacteristic
562C4#241311yescharacteristic
563C2—44no—
564C2#1506911yescharacteristic
565C2#2342311yescharacteristic
566C2#2342711yescharacteristic
567C2#2765714yes—
568C2#2792012yes—
569C2#2792212yes—
570C2#2964812yes—
571C2#3298812yes—
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. #11325C2semidirect
8. #11323C2semidirect
9. #11052C2semidirect
10. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C22×#4832
C2×#23172
C2×#18242
C2×#18252
#142582
#113252
#113232
#110522
#110502
#107132