Finite Group Databasebeta

← #616971

Subgroups — #616971

Order 512.

6233 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#123C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic, complemented
3C2×#2412C212yesmaximal, complemented
4C2×#2613C211yesmaximal, characteristic
5C2×#2308C211yesmaximal, characteristic, complemented
6#15270C218yesmaximal, complemented
7#15269C218yesmaximal, complemented
8#15073C212yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14681C211yesmaximal, characteristic, complemented
11#14262C211yesmaximal, characteristic, complemented
12#14255C211yesmaximal, characteristic, complemented
13#13869C212yesmaximal, complemented
14C24×D4C2211yescharacteristic
15C25×C4C2211yescharacteristic
16C22×#123—28no—
17C22×#123—216no—
18C22×#123C2218yes—
19C22×#123C2212yes—
20C22×#123C2211yescharacteristic
21C22×#120C2211yescharacteristic
22C22×#120C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27#2849—28no—
28#2613—28no—
29#2412—216no—
30C2×#397C2214yescomplemented
31C2×#394C2214yes—
32C2×#394C2214yescomplemented
33C2×#392C2214yes—
34#2308—28no—
35C2×#386C2212yes—
36C2×#386C2218yes—
37C2×#386C2214yescomplemented
38C2×#386C2214yescomplemented
39C2×#385C2212yes—
40C2×#385C2214yescomplemented
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#380C2214yescomplemented
47C2×#379C2211yescharacteristic
48C2×#379C2211yescharacteristic
49C2×#379C2211yescharacteristic
50C2×#379C2211yescharacteristic
51#1444C2218yescomplemented
52#1442C2218yescomplemented
53#1441C2218yescomplemented
54#1440C2218yescomplemented
55#1440C2214yescomplemented
56#1439C2218yescomplemented
57#1439C2218yescomplemented
58#1438C2214yescomplemented
59C22⋊#95C2212yes—
60C22⋊#95C2212yes—
61C22⋊#95C2218yescomplemented
62C22⋊#95C2218yescomplemented
63C26C2311yescharacteristic
64C26C2311yescharacteristic
65C23×D4—24no—
66C23×D4—28no—
67C23×D4—24no—
68C23×D4—44no—
69C23×D4D418yes—
70C23×D4C2311yescharacteristic
71C23×D4C2314yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4—24no—
75C24×C4—28no—
76C24×C4—24no—
77C24×C4—24no—
78C24×C4—44no—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2314yes—
82C24×C4C2314yes—
83C24×C4C2312yes—
84C2×#123—2128no—
85C2×#123—232no—
86C2×#123—24no—
87C2×#123—28no—
88C2×#123—232no—
89C2×#123—232no—
90C2×#123—28no—
91C2×#123—44no—
92C2×#123—44no—
93C2×#120—24no—
94C2×#120—24no—
95C2×#120—24no—
96C2×#120—28no—
97C2×#120—24no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4D4116yescomplemented
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131#397—216no—
132#394—216no—
133#394—216no—
134#392—216no—
135#386—28no—
136#386—232no—
137#386—216no—
138#386—216no—
139#385—28no—
140#385—216no—
141#381—24no—
142#381—28no—
143#381—28no—
144#381—28no—
145#381—44no—
146#380—216no—
147#379—24no—
148#379—24no—
149#379—24no—
150#379D418yescomplemented
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C25—24no—
162C25—24no—
163C25—22no—
164C25—22no—
165C25—24no—
166C25—48no—
167C25—48no—
168C25C2×D418yes—
169C25C2×D414yes—
170C25C2×D414yes—
171C25C2×D414yes—
172C25#4114yes—
173C25C2411yescharacteristic
174C25C2412yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C22×D4—232no—
178C22×D4—232no—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—28no—
182C22×D4—28no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—416no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—48no—
198C22×D4—48no—
199C22×D4—432no—
200C22×D4—432no—
201C22×D4C2×D4116yes—
202C23×C4—232no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—216no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—48no—
222C23×C4—48no—
223C23×C4—44no—
224C23×C4—48no—
225C23×C4—48no—
226C23×C4—432no—
227C23×C4—432no—
228C23×C4—416no—
229C23×C4—416no—
230C23×C4C2×D418yes—
231C23×C4C2×D414yes—
232C23×C4C2×D418yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242#123—2256no—
243#123—432no—
244#123—416no—
245#120—416no—
246#120—416no—
247C2×C4⋊C4—264no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—216no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—48no—
258C2×C22⋊C4—232no—
259C2×C22⋊C4—264no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—432no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4C2×D4116yes—
304C2×C22⋊C4C2×D4116yes—
305C2×C22⋊C4C2×D4116yes—
306#95—28no—
307#95—28no—
308#95—216no—
309#95—216no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—48no—
314#95—48no—
315#95—48no—
316C24—28no—
317C24—28no—
318C24—28no—
319C24—216no—
320C24—216no—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—28no—
328C24—28no—
329C24—28no—
330C24—28no—
331C24—28no—
332C24—28no—
333C24—216no—
334C24—216no—
335C24—22no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—28no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—416no—
349C24—432no—
350C24—88no—
351C24—88no—
352C24—816no—
353C24#120116yes—
354C24#121116yes—
355C24C22×D414yes—
356C24C22×D414yes—
357C24C22×D414yes—
358C24C22×D414yes—
359C24C22×D414yes—
360C24C22×D412yes—
361C24C22×D414yes—
362C24C22×D412yes—
363C24C2×#4114yes—
364C24C2×#4112yes—
365C24H(5,𝔽2)14yes—
366C24C2511yescharacteristic
367C2×D4—232no—
368C2×D4—232no—
369C2×D4—416no—
370C2×D4—4128no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—88no—
385C2×D4—88no—
386C2×D4—88no—
387C2×D4—88no—
388C22×C4—232no—
389C22×C4—216no—
390C22×C4—216no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—416no—
436C22×C4—432no—
437C22×C4—432no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—4128no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—416no—
451C22×C4—816no—
452C22×C4—88no—
453C22×C4—816no—
454C22×C4C22×D418yes—
455C22×C4C22×D418yes—
456C22×C4C22×D418yes—
457C22×C4C22×D414yes—
458C4⋊C4—264no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—88no—
462C22⋊C4—232no—
463C22⋊C4—232no—
464C22⋊C4—232no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—216no—
486C23—24no—
487C23—24no—
488C23—232no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—28no—
495C23—24no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—232no—
510C23—432no—
511C23—432no—
512C23—432no—
513C23—464no—
514C23—432no—
515C23—464no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—48no—
521C23—416no—
522C23—48no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—48no—
534C23—832no—
535C23—816no—
536C23—816no—
537C23—832no—
538C23C2×#12018yes—
539C23C2×#12014yes—
540C23C2×#12118yes—
541C23C2×#12114yes—
542C23#53414yes—
543C23D4218yes—
544C23#54618yes—
545C23C23×D411yescharacteristic
546C23C23×D412yes—
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23#58518yes—
551D4—464no—
552D4—816no—
553D4—816no—
554D4—88no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—216no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—432no—
567C2×C4—464no—
568C2×C4—44no—
569C2×C4—432no—
570C2×C4—416no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—832no—
603C22—28no—
604C22—216no—
605C22—28no—
606C22—28no—
607C22—216no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—464no—
615C22—48no—
616C22—48no—
617C22—416no—
618C22—48no—
619C22—48no—
620C22—48no—
621C22—48no—
622C22—416no—
623C22—48no—
624C22—48no—
625C22—832no—
626C22—82no—
627C22—84no—
628C22—81no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22#241018yes—
633C22C22×#12011yescharacteristic
634C22C22×#12111yescharacteristic
635C22C2×#53411yescharacteristic
636C22#344914yes—
637C22C2×D4212yes—
638C22C2×#54612yes—
639C22#348514yes—
640C22#348514yes—
641C22#348514yes—
642C22#349114yes—
643C22#353318yes—
644C4—48no—
645C4—48no—
646C4—48no—
647C4—44no—
648C4—82no—
649C4—84no—
650C4—81no—
651C4—82no—
652C4—82no—
653C2—28no—
654C2—48no—
655C2—48no—
656C2—161no—
657C2C2×#241011yescharacteristic
658C2#3508812yes—
659C2#3544712yes—
660C2#3695512yes—
661C2#3765414yes—
662C2#3784214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#123C2semidirect
2. C2×#2849C2semidirect
3. C2×#2412C2semidirect
5. C2×#2308C2semidirect
6. #15270C2semidirect
7. #15269C2semidirect
8. #15073C2semidirect
9. #15069C2semidirect
10. #14681C2semidirect
11. #14262C2semidirect
12. #14255C2semidirect
13. #13869C2semidirect
30. C2×#397C22semidirect
32. C2×#394C22semidirect
37. C2×#386C22semidirect
38. C2×#386C22semidirect
40. C2×#385C22semidirect
46. C2×#380C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1438C22semidirect
61. C22⋊#95C22semidirect
62. C22⋊#95C22semidirect
112. C22×C22⋊C4D4semidirect
150. #379D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1232
C2×#28492
C2×#24122
C2×#26132
C2×#23082
#152702
#152692
#150732
#150692
#146812
#142622
#142552
#138692