Finite Group Databasebeta

← #618457

Subgroups — #618457

Order 512.

4170 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#126C211yesmaximal, characteristic
2C2×#2483C211yesmaximal, characteristic
3C2×#2483C211yesmaximal, characteristic
4C2×#2315C211yesmaximal, characteristic
5#16024C214yesmaximal, complemented
6#16024C214yesmaximal, complemented
7#15911C218yesmaximal, complemented
8#15092C212yesmaximal, complemented
9#15073C212yesmaximal, complemented
10#14705C212yesmaximal, complemented
11#14341C211yesmaximal, characteristic
12#14263C212yesmaximal, complemented
13#13869C211yesmaximal, characteristic
14#13869C211yesmaximal, characteristic
15C22×#126—28no—
16C22×#126—28no—
17C22×#126—28no—
18C22×#126C2214yes—
19C22×#126C2214yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2483—28no—
30#2483—28no—
31C2×#397C2214yes—
32C2×#397C2214yes—
33#2315—28no—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#380C2214yes—
49C2×#379C2211yescharacteristic
50C2×#379C2211yescharacteristic
51#1444C2218yescomplemented
52#1444C2218yescomplemented
53#1442C2218yescomplemented
54#1442C2218yescomplemented
55#1441C2218yescomplemented
56#1441C2218yescomplemented
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1436C2218yescomplemented
62#1436C2218yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×#126—2128no—
80C2×#126—216no—
81C2×#126—216no—
82C2×#126—216no—
83C2×#126—216no—
84C2×#126—216no—
85C2×#126—216no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4D418yes—
114C22×C22⋊C4D418yes—
115C22×C22⋊C4D418yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C42—24no—
133C22×C42—24no—
134C22×C42—24no—
135C22×C42C2314yes—
136#397—216no—
137#397—216no—
138#385—28no—
139#385—28no—
140#385—28no—
141#385—28no—
142#385—28no—
143#385—28no—
144#385—28no—
145#385—28no—
146#381—28no—
147#381—28no—
148#381—28no—
149#381—28no—
150#381—28no—
151#381—28no—
152#380—216no—
153#379—24no—
154#379—24no—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C25—24no—
170C25—24no—
171C25—44no—
172C25—44no—
173C25C2×D414yes—
174C25C2×D414yes—
175C25C2×D414yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228#126—2128no—
229#126—2128no—
230C2×C4⋊C4—232no—
231C2×C4⋊C4—232no—
232C2×C4⋊C4—232no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—48no—
250C2×C22⋊C4—232no—
251C2×C22⋊C4—232no—
252C2×C22⋊C4—232no—
253C2×C22⋊C4—232no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—416no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D418yes—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×D418yes—
294C2×C22⋊C4C2×D418yes—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D418yes—
297C2×C22⋊C4#4118yes—
298C2×C42—232no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—216no—
311#95—216no—
312#95—48no—
313#95—48no—
314#95—48no—
315#95—48no—
316C24—24no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—28no—
327C24—28no—
328C24—216no—
329C24—216no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—816no—
337C24—816no—
338C24#120116yes—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C22×D412yes—
347C24C22×D412yes—
348C24C2×#4114yes—
349C242−1+414yes—
350C24C2511yescharacteristic
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—816no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D414yes—
437C22×C4C2×#4118yes—
438C4⋊C4—232no—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—232no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—216no—
469C23—216no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—232no—
491C23—416no—
492C23—416no—
493C23—48no—
494C23—48no—
495C23—48no—
496C23—48no—
497C23—832no—
498C23—816no—
499C23—816no—
500C23—832no—
501C23C2×#12014yes—
502C23C2×#12014yes—
503C23C2×#12014yes—
504C23#52514yes—
505C23#53614yes—
506C23#54718yes—
507C23#54718yes—
508C23C23×D411yescharacteristic
509C23C23×D411yescharacteristic
510C23C23×D411yescharacteristic
511C23C22×#4112yes—
512C23C2×2−1+412yes—
513C23#58514yes—
514C23#58514yes—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—216no—
529C2×C4—416no—
530C2×C4—416no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—416no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—816no—
571C2×C4—88no—
572C2×C4—88no—
573C22—216no—
574C22—24no—
575C22—24no—
576C22—28no—
577C22—28no—
578C22—216no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—48no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—832no—
594C22#240618yes—
595C22C22×#12011yescharacteristic
596C22C2×#52511yescharacteristic
597C22C2×#53611yescharacteristic
598C22#344912yes—
599C22#344912yes—
600C22C2×#54712yes—
601C22C2×#54712yes—
602C22#348512yes—
603C22#348512yes—
604C22#348512yes—
605C22#348512yes—
606C22#352118yes—
607C22#356418yes—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—48no—
613C4—44no—
614C4—44no—
615C4—84no—
616C4—82no—
617C4—82no—
618C4—84no—
619C2—28no—
620C2—44no—
621C2—44no—
622C2C2×#240611yescharacteristic
623C2#3508811yescharacteristic
624C2#3508811yescharacteristic
625C2#3545212yes—
626C2#3697012yes—
627C2#3804614yes—
628C2#3891314yes—
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
5. #16024C2semidirect
6. #16024C2semidirect
7. #15911C2semidirect
8. #15092C2semidirect
9. #15073C2semidirect
10. #14705C2semidirect
12. #14263C2semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1262
C2×#24832
C2×#23152
#160242
#159112
#150922
#150732
#147052
#143412
#142632
#138692