Finite Group Databasebeta

← #410008

Subgroups — #410008

Order 512.

1742 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C22×#428C211yesmaximal, characteristic
3C2×#2503C211yesmaximal, characteristic
4C2×#1894C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic
6#13939C211yesmaximal, characteristic
7#11325C212yesmaximal, complemented
8#11325C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#485—24no—
17C2×#485—24no—
18C2×#485—24no—
19C2×#428—24no—
20C2×#428—24no—
21C2×#428—24no—
22C22×#107C2211yescharacteristic
23C22×#107C2212yes—
24C22×#106C2211yescharacteristic
25C22×#106C2212yes—
26C22×#103C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2212yes—
29C22×C22⋊C8C2211yescharacteristic
30#2503—24no—
31#2503—24no—
32C2×#400—22no—
33C2×#391C2212yes—
34C2×#382C2212yes—
35C2×#381—22no—
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1894—24no—
39#1824—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1440—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48#1434—22no—
49C23×Q8C2311yescharacteristic
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16D414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2312yes—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8D414yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117D414yes—
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⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2312yes—
91#485—232no—
92#428—232no—
93C2×#107—22no—
94C2×#107—24no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—24no—
98C2×#107—24no—
99C2×#106—22no—
100C2×#106—24no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—24no—
104C2×#106—24no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8D414yes—
117#400—216no—
118#391—24no—
119#391—24no—
120#382—24no—
121#382—24no—
122#381—216no—
123#380—22no—
124#380—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95—21no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95D412yes—
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—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—48no—
161C23×C4—42no—
162C23×C4—42no—
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
172C2×M16—28no—
173C2×M16—28no—
174C2×M16—216no—
175C2×M16—28no—
176C2×M16—216no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C2×D412yes—
190C22×C8C2×D412yes—
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—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—21no—
218C2×C4⋊C4—22no—
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—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—82no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—42no—
260C2×C42—42no—
261#107—216no—
262#107—28no—
263#106—216no—
264#106—28no—
265#103—28no—
266#103—28no—
267#103—48no—
268C22⋊C8—28no—
269#95—216no—
270#95—216no—
271#95—24no—
272#95—48no—
273#95—42no—
274#95—84no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—22no—
279C24—22no—
280C24#12012yes—
281C24#12112yes—
282C24#12712yes—
283C24C22×D411yescharacteristic
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
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—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—416no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
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—48no—
345C22×C4—42no—
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#12014yes—
358C22×C4#12014yes—
359C22×C4#12012yes—
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4#12112yes—
363C22×C4#12712yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C42−1+412yes—
373C22×C42−1+412yes—
374M16—232no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8#12318yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—84no—
418C42—42no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—48no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23#39214yes—
439C23C2×#12011yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×#12711yescharacteristic
442C23#54712yes—
443C23#54712yes—
444C23#57514yes—
445C23#57712yes—
446C23#57812yes—
447Q8—42no—
448Q8—42no—
449Q8—44no—
450Q8—81no—
451Q8—81no—
452Q8—82no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—48no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—816no—
481C2×C4—82no—
482C2×C4—84no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4#39214yes—
500C2×C4C2×#12011yescharacteristic
501C2×C4C2×#12011yescharacteristic
502C2×C4C2×#12111yescharacteristic
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×#12312yes—
505C2×C4#52511yescharacteristic
506C2×C4#52511yescharacteristic
507C2×C4#52512yes—
508C2×C4#53611yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#53612yes—
511C2×C4#53612yes—
512C2×C4#54712yes—
513C2×C4#54712yes—
514C2×C4#54712yes—
515C2×C4#54712yes—
516C2×C4#55212yes—
517C2×C4#55212yes—
518C8—28no—
519C8—42no—
520C8—42no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—48no—
528C22—44no—
529C22C2×#39211yescharacteristic
530C22#240612yes—
531C22#241311yescharacteristic
532C22#241311yescharacteristic
533C22#250112yes—
534C22#300512yes—
535C22#301312yes—
536C22#304012yes—
537C22#304612yes—
538C22#315312yes—
539C22#315411yescharacteristic
540C22#315611yescharacteristic
541C22#334814yes—
542C22#334914yes—
543C22#335114yes—
544C22#335214yes—
545C4—24no—
546C4—84no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4—84no—
551C4—82no—
552C4—162no—
553C4#240611yescharacteristic
554C4#240611yescharacteristic
555C4#241312yes—
556C4#248612yes—
557C4#261411yescharacteristic
558C4#284911yescharacteristic
559C2—44no—
560C2#1506911yescharacteristic
561C2#2342711yescharacteristic
562C2#2343111yescharacteristic
563C2#2410614yes—
564C2#2459312yes—
565C2#2459512yes—
566C2#2794312yes—
567C2#2794612yes—
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. #11325C2semidirect
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×#4852
C22×#4282
C2×#25032
C2×#18942
C2×#18242
#139392
#113252
#111242
#111222
#107152