Finite Group Databasebeta

← #4453315

Subgroups — #4453315

Order 512.

3156 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic, complemented
2C2×#2321C211yesmaximal, characteristic
3#16591C214yesmaximal, complemented
4#16335C214yesmaximal, complemented
5#16277C214yesmaximal, complemented
6#16269C214yesmaximal, complemented
7#15092C211yesmaximal, characteristic, complemented
8#15091C211yesmaximal, characteristic, complemented
9#14955C211yesmaximal, characteristic, complemented
10#14945C211yesmaximal, characteristic, complemented
11#14727C212yesmaximal, complemented
12#14718C212yesmaximal
13#14341C211yesmaximal, characteristic, complemented
14#14039C212yesmaximal, complemented
15#14015C212yesmaximal, complemented
16C22×#120C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C2×#399—28no—
23C2×#399C2214yes—
24C2×#399C2214yes—
25C2×#397C2214yes—
26C2×#394C2214yes—
27#2321—28no—
28C2×#390C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#387C2214yescomplemented
32C2×#386C2211yescharacteristic
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#385C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yescomplemented
40C2×#379C2211yescharacteristic
41C2×#379C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1444C2214yescomplemented
45#1444C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yescomplemented
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1438C2214yes—
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C26C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#120—24no—
82C2×#120—24no—
83C2×#120—24no—
84C2×C4×D4—24no—
85C2×C4×D4—44no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—48no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—48no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4D418yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#399—216no—
122#399—216no—
123#397—48no—
124#394—48no—
125#390—28no—
126#388—28no—
127#388—28no—
128#387—48no—
129#386—28no—
130#386—24no—
131#386—28no—
132#386—44no—
133#385—24no—
134#385—28no—
135#385—48no—
136#385—44no—
137#379—24no—
138#379—44no—
139C4×C22⋊C4—24no—
140C2×#95—24no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—24no—
162C25—22no—
163C25—22no—
164C25—24no—
165C25—44no—
166C25—44no—
167C25C2×D414yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—22no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4#4114yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—44no—
198C23×C4C2×D414yes—
199C23×C4C2×D414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212#120—232no—
213C4×D4—416no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—432no—
230C2×C4⋊C4—416no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—416no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D418yes—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4#4114yes—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—42no—
288C2×C42—44no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309C24—24no—
310C24—24no—
311C24—28no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—28no—
329C24—28no—
330C24—48no—
331C24—48no—
332C24—416no—
333C24—416no—
334C24—48no—
335C24—48no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—88no—
340C24C22×D414yes—
341C24C22×D412yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2×#4112yes—
345C24H(5,𝔽2)14yes—
346C242−1+414yes—
347C24C2511yescharacteristic
348C2×D4—28no—
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—88no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—88no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—416no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4H(5,𝔽2)18yes—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—44no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C22⋊C4—216no—
445C22⋊C4—28no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C42—44no—
463C42—84no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—24no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—28no—
482C23—432no—
483C23—416no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—816no—
511C23—88no—
512C23—88no—
513C23—832no—
514C23—832no—
515C23#52514yes—
516C23#53414yes—
517C23#53614yes—
518C23#53918yes—
519C23#54218yes—
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C22×#4111yescharacteristic
523C23C2×H(5,𝔽2)12yes—
524C23C2×H(5,𝔽2)14yes—
525C23C2×2−1+412yes—
526C23#58514yes—
527D4—416no—
528D4—816no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—432no—
544C2×C4—416no—
545C2×C4—44no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4—88no—
593C2×C4—816no—
594C22—28no—
595C22—24no—
596C22—28no—
597C22—24no—
598C22—216no—
599C22—216no—
600C22—42no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—48no—
610C22—832no—
611C22—81no—
612C22—82no—
613C22—81no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22#263818yes—
620C22C2×#52511yescharacteristic
621C22C2×#53411yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#53912yes—
624C22C2×#54212yes—
625C22#348412yes—
626C22#348512yes—
627C22#351314yes—
628C22#351314yes—
629C22#353314yes—
630C22#354114yes—
631C22#354114yes—
632C22#354314yes—
633C4—42no—
634C4—48no—
635C4—44no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—81no—
640C4—82no—
641C4—81no—
642C4—84no—
643C4—84no—
644C4—84no—
645C2—28no—
646C2—82no—
647C2—84no—
648C2C2×#263811yescharacteristic
649C2#3535412yes—
650C2#3671612yes—
651C2#3695511yescharacteristic
652C2#3723211yescharacteristic
653C2#4069714yes—
654C2#4069914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#399C2semidirect
3. #16591C2semidirect
4. #16335C2semidirect
5. #16277C2semidirect
6. #16269C2semidirect
7. #15092C2semidirect
8. #15091C2semidirect
9. #14955C2semidirect
10. #14945C2semidirect
11. #14727C2semidirect
13. #14341C2semidirect
14. #14039C2semidirect
15. #14015C2semidirect
31. C2×#387C22semidirect
39. C2×#385C22semidirect
44. #1444C22semidirect
45. #1444C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
58. #1440C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23212
#165912
#163352
#162772
#162692
#150922
#150912
#149552
#149452
#147272
#147182
#143412
#140392
#140152