Finite Group Databasebeta

← #410007

Subgroups — #410007

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C22×#428C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4C2×#1894C211yesmaximal, characteristic
5C2×#1825C211yesmaximal, characteristic
6#13904C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11324C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#484—24no—
17C2×#484—24no—
18C2×#484—24no—
19C2×#428—24no—
20C2×#428—24no—
21C2×#428—24no—
22C22×#107C2212yes—
23C22×#107C2212yes—
24C22×#106C2212yes—
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×C22⋊C8C2211yescharacteristic
28#2499—24no—
29#2499—24no—
30C2×#397—22no—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#387—22no—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#381—22no—
37C2×#380C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1894—24no—
40#1825—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1445—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16D414yes—
57C23×C8C2312yes—
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8D414yes—
65C2×#117—22no—
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—22no—
79C22×C4⋊C4D414yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#484—232no—
93#428—232no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#107—24no—
97C2×#107—24no—
98C2×#107—24no—
99C2×#107—24no—
100C2×#106—24no—
101C2×#106—24no—
102C2×#106—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8D414yes—
112#397—216no—
113#391—22no—
114#391—22no—
115#391—22no—
116#391—22no—
117#387—216no—
118#384—22no—
119#384—22no—
120#382—22no—
121#382—22no—
122#381—48no—
123#380—22no—
124#380—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95—21no—
133C2×#95D412yes—
134#340—24no—
135#340—24no—
136#340—24no—
137#336—24no—
138#336—24no—
139C25C2×D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C2×D412yes—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—24no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4#4112yes—
174C23×C4C2411yescharacteristic
175C2×M16—216no—
176C2×M16—28no—
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—216no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×D414yes—
191C22×C8C2×D414yes—
192C22×C8#4114yes—
193C22×C8#4114yes—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198#117—28no—
199#117—28no—
200#117—44no—
201#117—44no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
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—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
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—22no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
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—44no—
255C2×C22⋊C4—42no—
256C2×C42—24no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—42no—
267#107—216no—
268#107—216no—
269#106—216no—
270#103—216no—
271#103—48no—
272C22⋊C8—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—48no—
278#95—42no—
279#95—48no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24#12012yes—
286C24#12112yes—
287C24#12712yes—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
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—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—416no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—88no—
366C22×C4#12014yes—
367C22×C4#12014yes—
368C22×C4#12012yes—
369C22×C4#12114yes—
370C22×C4#12114yes—
371C22×C4#12112yes—
372C22×C4#12712yes—
373C22×C4AGL(1,ℤ/8)14yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C42−1+412yes—
383C22×C42−1+412yes—
384M16—232no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8#12318yes—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—416no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—84no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—42no—
431C42—42no—
432C42—48no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23#39214yes—
449C23C2×#12011yescharacteristic
450C23C2×#12111yescharacteristic
451C23C2×#12711yescharacteristic
452C23#54712yes—
453C23#54712yes—
454C23C2×AGL(1,ℤ/8)12yes—
455C23#57512yes—
456C23#57814yes—
457Q8—44no—
458Q8—44no—
459Q8—81no—
460Q8—81no—
461Q8—81no—
462Q8—81no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—48no—
494C2×C4—816no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#39214yes—
514C2×C4C2×#12011yescharacteristic
515C2×C4C2×#12011yescharacteristic
516C2×C4C2×#12111yescharacteristic
517C2×C4C2×#12111yescharacteristic
518C2×C4C2×#12312yes—
519C2×C4#52512yes—
520C2×C4#52511yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4#53612yes—
523C2×C4#53611yescharacteristic
524C2×C4#53611yescharacteristic
525C2×C4#53612yes—
526C2×C4#54712yes—
527C2×C4#54712yes—
528C2×C4#54712yes—
529C2×C4#54712yes—
530C2×C4#55212yes—
531C2×C4#55212yes—
532C8—28no—
533C8—44no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—48no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22C2×#39211yescharacteristic
543C22#240611yescharacteristic
544C22#240611yescharacteristic
545C22#241312yes—
546C22#250112yes—
547C22#300612yes—
548C22#301112yes—
549C22#303612yes—
550C22#304412yes—
551C22#315111yescharacteristic
552C22#315311yescharacteristic
553C22#315612yes—
554C22#334314yes—
555C22#334914yes—
556C22#335214yes—
557C22#335314yes—
558C4—24no—
559C4—44no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—84no—
565C4—82no—
566C4—162no—
567C4#240612yes—
568C4#241311yescharacteristic
569C4#241311yescharacteristic
570C4#248612yes—
571C4#261411yescharacteristic
572C4#284911yescharacteristic
573C2—44no—
574C2#1506911yescharacteristic
575C2#2342511yescharacteristic
576C2#2342811yescharacteristic
577C2#2410712yes—
578C2#2411012yes—
579C2#2458814yes—
580C2#2794312yes—
581C2#2794412yes—
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. #11324C2semidirect
9. #11124C2semidirect
10. #11122C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C22×#4282
C2×#24992
C2×#18942
C2×#18252
#139042
#113252
#113242
#111242
#111222
#107152