Finite Group Databasebeta

← #618456

Subgroups — #618456

Order 512.

5306 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2482C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#15670C214yesmaximal, complemented
6#15430C214yesmaximal, complemented
7#15354C218yesmaximal, complemented
8#14918C212yesmaximal
9#14874C212yesmaximal, complemented
10#14693C212yesmaximal
11#14342C211yesmaximal, characteristic, complemented
12#14263C212yesmaximal, complemented
13#13870C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—28no—
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8C2214yes—
21C22×C22⋊Q8C2214yes—
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29#2484—28no—
30#2482—28no—
31C2×#400C2212yescomplemented
32C2×#399C2212yes—
33C2×#398C2212yescomplemented
34C2×#397C2212yes—
35C2×#395C2218yescomplemented
36#2317—28no—
37C2×#389C2214yescomplemented
38C2×#388C2214yes—
39C2×#387C2214yescomplemented
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2218yes—
45C2×#384C2212yescomplemented
46C2×#382C2212yescomplemented
47C2×#381C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#381C2212yes—
50C2×#381C2212yes—
51C2×#379C2212yes—
52C2×#379C2212yes—
53#1446C2218yescomplemented
54#1445C2214yescomplemented
55#1445C2214yescomplemented
56#1444C2218yescomplemented
57#1442C2218yescomplemented
58#1441C2214yescomplemented
59#1440C2218yes—
60#1439C2214yes—
61#1439C2218yes—
62#1437C2214yes—
63#1436C2214yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2218yes—
69C26C2311yescharacteristic
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—48no—
74C23×Q8C2314yes—
75C24×C4—24no—
76C24×C4—24no—
77C24×C4—24no—
78C24×C4—48no—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×C22⋊Q8—2128no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—216no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4D418yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—24no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4D418yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#400—28no—
138#399—28no—
139#398—28no—
140#397—28no—
141#395—416no—
142#389—216no—
143#388—216no—
144#387—216no—
145#385—28no—
146#385—28no—
147#385—28no—
148#385—28no—
149#385—416no—
150#384—28no—
151#382—28no—
152#381—24no—
153#381—24no—
154#381—28no—
155#381—28no—
156#379—28no—
157#379—28no—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C25—22no—
174C25—22no—
175C25—28no—
176C25—24no—
177C25—24no—
178C25—44no—
179C25C2×D414yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—232no—
183C22×Q8—28no—
184C22×Q8—28no—
185C22×Q8—28no—
186C22×Q8—216no—
187C22×Q8—48no—
188C22×Q8—432no—
189C22×Q8—432no—
190C22×Q8—432no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—232no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—216no—
211C23×C4—48no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—432no—
216C23×C4—432no—
217C23×C4—432no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C22⋊Q8—2128no—
239C22⋊Q8—2128no—
240#117—232no—
241#117—416no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—232no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—232no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—216no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4C2×D418yes—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4#4118yes—
306C2×C42—24no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—216no—
322#95—216no—
323#95—48no—
324#95—48no—
325#95—48no—
326C24—24no—
327C24—24no—
328C24—216no—
329C24—216no—
330C24—28no—
331C24—216no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—28no—
336C24—24no—
337C24—24no—
338C24—216no—
339C24—48no—
340C24—432no—
341C24—432no—
342C24—48no—
343C24—48no—
344C24—44no—
345C24—44no—
346C24—48no—
347C24C22×D414yes—
348C24C22×D412yes—
349C24C22×D412yes—
350C24C22×D412yes—
351C24C2×#4114yes—
352C242−1+418yes—
353C242−1+414yes—
354C24C2511yescharacteristic
355C2×Q8—232no—
356C2×Q8—432no—
357C2×Q8—432no—
358C2×Q8—4128no—
359C2×Q8—416no—
360C2×Q8—416no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—232no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—216no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—432no—
421C22×C4—432no—
422C22×C4—4128no—
423C22×C4—416no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4#120116yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C2×#4118yes—
447C22×C4H(5,𝔽2)18yes—
448C4⋊C4—232no—
449C4⋊C4—216no—
450C4⋊C4—216no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—416no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C22⋊C4—232no—
461C22⋊C4—216no—
462C22⋊C4—216no—
463C22⋊C4—216no—
464C22⋊C4—432no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—88no—
470C42—216no—
471C42—48no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—216no—
479C23—216no—
480C23—232no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—216no—
488C23—216no—
489C23—464no—
490C23—416no—
491C23—464no—
492C23—432no—
493C23—432no—
494C23—416no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—48no—
501C23—48no—
502C23C2×#12014yes—
503C23#52514yes—
504C23#53618yes—
505C23#53614yes—
506C23#54718yes—
507C23#54718yes—
508C23C23×D411yescharacteristic
509C23C23×D411yescharacteristic
510C23C23×D411yescharacteristic
511C23C22×#4112yes—
512C23C2×H(5,𝔽2)14yes—
513C23C2×2−1+412yes—
514C23C2×2−1+414yes—
515Q8—464no—
516C2×C4—232no—
517C2×C4—232no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—216no—
540C2×C4—464no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4C2×#12014yes—
576C2×C4C2×#12014yes—
577C2×C4#53518yes—
578C22—216no—
579C22—28no—
580C22—28no—
581C22—216no—
582C22—24no—
583C22—24no—
584C22—216no—
585C22—28no—
586C22—464no—
587C22—416no—
588C22—48no—
589C22—48no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22#240618yes—
594C22C22×#12011yescharacteristic
595C22C2×#52511yescharacteristic
596C22C2×#53512yes—
597C22C2×#53611yescharacteristic
598C22C2×#53612yes—
599C22C2×#54712yes—
600C22C2×#54712yes—
601C22#348412yes—
602C22#348412yes—
603C22#348612yes—
604C22#348612yes—
605C22#352018yes—
606C22#353818yes—
607C4—28no—
608C4—24no—
609C4—24no—
610C4—48no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—44no—
615C4—84no—
616C4—84no—
617C4—82no—
618C4—82no—
619C2—28no—
620C2—48no—
621C2C2×#240611yescharacteristic
622C2#3522211yescharacteristic
623C2#3524211yescharacteristic
624C2#3535712yes—
625C2#3670812yes—
626C2#3803714yes—
627C2#3891714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2484C2semidirect
4. C2×#2317C2semidirect
5. #15670C2semidirect
6. #15430C2semidirect
7. #15354C2semidirect
9. #14874C2semidirect
11. #14342C2semidirect
12. #14263C2semidirect
13. #13870C2semidirect
31. C2×#400C22semidirect
33. C2×#398C22semidirect
35. C2×#395C22semidirect
37. C2×#389C22semidirect
39. C2×#387C22semidirect
45. C2×#384C22semidirect
46. C2×#382C22semidirect
53. #1446C22semidirect
54. #1445C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
63. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#24842
C2×#24822
C2×#23172
#156702
#154302
#153542
#149182
#148742
#146932
#143422
#142632
#138702
#138682