Finite Group Databasebeta

← #4416992

Subgroups — #4416992

Order 512.

4342 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2847C212yesmaximal, complemented
2#16797C218yesmaximal, complemented
3#16249C214yesmaximal, complemented
4#15233C212yesmaximal, complemented
5#15219C212yesmaximal, complemented
6#15219C212yesmaximal, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15130C212yesmaximal, complemented
9#15124C211yesmaximal, characteristic, complemented
10#15124C212yesmaximal, complemented
11#14621C214yesmaximal, complemented
12C4×#379C211yesmaximal, characteristic, complemented
13C25×C4C2211yescharacteristic
14C22×#127C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×#120C2214yes—
19#2847—216no—
20C2×#394C2218yes—
21C2×#392C2214yes—
22C2×#392C2218yes—
23C2×#390C2212yescomplemented
24C2×#390C2212yescomplemented
25C2×#390C2212yescomplemented
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#384C2211yescharacteristic, complemented
30C2×#382C2212yescomplemented
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#379C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1443C2218yescomplemented
40#1442C2214yescomplemented
41#1440C2218yes—
42#1440C2218yescomplemented
43#1440C2218yescomplemented
44#1439C2214yes—
45#1439C2218yescomplemented
46#1438C2218yes—
47#1438C2218yes—
48#1438C2218yescomplemented
49#1438C2214yescomplemented
50C4⋊#95C2212yescomplemented
51C4⋊#95C2211yescharacteristic, complemented
52C4⋊#95C2212yescomplemented
53C4⋊#95C2211yescharacteristic, complemented
54C22⋊#95C2212yes—
55C22⋊#95C2214yes—
56C26C2311yescharacteristic
57C23×D4C2312yes—
58C23×D4C2314yes—
59C23×D4C2314yes—
60C23×D4C2314yes—
61C23×D4C2314yes—
62C24×C4—42no—
63C24×C4—44no—
64C24×C4—42no—
65C24×C4—44no—
66C24×C4D418yescomplemented
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C2×#127—42no—
72C2×#127—44no—
73C2×#121—28no—
74C2×#121—28no—
75C2×#121—28no—
76C2×#121—44no—
77C2×#121—44no—
78C2×#121—44no—
79C2×#121—44no—
80C2×#121—44no—
81C2×#121—44no—
82C2×#120—216no—
83C2×#120—416no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2318yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2318yes—
100C22×C22⋊C4C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#394—416no—
105#392—216no—
106#392—416no—
107#390—44no—
108#390—44no—
109#390—44no—
110#386—216no—
111#386—216no—
112#386—216no—
113#384—42no—
114#382—44no—
115#380—28no—
116#380—28no—
117#380—28no—
118#379—28no—
119C4×C22⋊C4—44no—
120C4×C22⋊C4—42no—
121C4×C22⋊C4—44no—
122C4×C22⋊C4—42no—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2318yes—
128C2×#95C2318yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C25—24no—
136C25—28no—
137C25—41no—
138C25—41no—
139C25—42no—
140C25—42no—
141C25—42no—
142C25C2×D414yes—
143C25C2414yes—
144C25C2411yescharacteristic
145C25C2412yes—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—28no—
149C22×D4—216no—
150C22×D4—216no—
151C22×D4—216no—
152C22×D4—24no—
153C22×D4—28no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—48no—
161C22×D4—48no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—48no—
170C22×D4—48no—
171C22×D4—44no—
172C22×D4—48no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—432no—
184C23×C4—416no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—41no—
192C23×C4—42no—
193C23×C4—41no—
194C23×C4—42no—
195C23×C4—416no—
196C23×C4—432no—
197C23×C4—416no—
198C23×C4C2×D414yes—
199C23×C4C2×D418yes—
200C23×C4C2×D414yes—
201C23×C4C2×D418yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211#127—416no—
212#121—432no—
213#121—432no—
214#121—432no—
215#120—464no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—48no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—216no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—216no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—48no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—48no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4#4118yes—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—44no—
278#95—42no—
279#95—48no—
280#95—48no—
281#95—48no—
282C24—24no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—24no—
297C24—28no—
298C24—216no—
299C24—24no—
300C24—28no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—28no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—28no—
309C24—28no—
310C24—48no—
311C24—44no—
312C24—44no—
313C24—416no—
314C24—416no—
315C24—48no—
316C24—48no—
317C24—48no—
318C24—48no—
319C24—416no—
320C24—416no—
321C24—44no—
322C24—48no—
323C24—44no—
324C24C22×D412yes—
325C24C22×D414yes—
326C24C2×#4114yes—
327C24H(5,𝔽2)18yes—
328C24C2511yescharacteristic
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—416no—
332C2×D4—416no—
333C2×D4—416no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—88no—
347C2×D4—816no—
348C2×D4—816no—
349C2×D4—816no—
350C2×D4—816no—
351C2×D4—816no—
352C2×D4—816no—
353C2×D4—816no—
354C2×D4—88no—
355C2×D4—816no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—4128no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)14yes—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—88no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—816no—
445C22⋊C4—816no—
446C22⋊C4—816no—
447C42—44no—
448C42—48no—
449C42—44no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—432no—
471C23—416no—
472C23—432no—
473C23—416no—
474C23—416no—
475C23—432no—
476C23—464no—
477C23—432no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23#53214yes—
519C23#53518yes—
520C23C23×D411yescharacteristic
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)14yes—
527C23#58512yes—
528D4—832no—
529D4—832no—
530D4—832no—
531D4—832no—
532D4—816no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—432no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—416no—
558C2×C4—432no—
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—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—48no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4#53418yes—
582C2×C4#53414yes—
583C2×C4#56014yes—
584C2×C4#56018yes—
585C2×C4H(3,𝔽4)14yes—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—28no—
590C22—24no—
591C22—464no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—48no—
597C22—48no—
598C22—42no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—44no—
604C22—48no—
605C22—44no—
606C22—84no—
607C22—84no—
608C22—88no—
609C22—88no—
610C22—88no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—88no—
616C22—84no—
617C22C2×#53211yescharacteristic
618C22C2×#53412yes—
619C22C2×#53411yescharacteristic
620C22C2×#53512yes—
621C22#344911yescharacteristic
622C22#350512yes—
623C22#350612yes—
624C22#351914yes—
625C22#351914yes—
626C22C2×#56011yescharacteristic
627C22C2×#56012yes—
628C22C2×H(3,𝔽4)11yescharacteristic
629C22#353712yes—
630C22#353712yes—
631C22#354214yes—
632C22#354214yes—
633C4—44no—
634C4—42no—
635C4—44no—
636C4—44no—
637C4—84no—
638C4—84no—
639C4—84no—
640C4—84no—
641C4#281218yes—
642C2—44no—
643C2—48no—
644C2—164no—
645C2#3503211yescharacteristic
646C2#3507812yes—
647C2C2×#281211yescharacteristic
648C2#3744812yes—
649C2#3745011yescharacteristic
650C2#4559912yes—
651C2#4560014yes—
652C2#4564512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2847C2semidirect
2. #16797C2semidirect
3. #16249C2semidirect
4. #15233C2semidirect
5. #15219C2semidirect
6. #15219C2semidirect
7. #15195C2semidirect
8. #15130C2semidirect
9. #15124C2semidirect
10. #15124C2semidirect
11. #14621C2semidirect
12. C4×#379C2semidirect
14. C22×#127C22semidirect
23. C2×#390C22semidirect
24. C2×#390C22semidirect
25. C2×#390C22semidirect
29. C2×#384C22semidirect
30. C2×#382C22semidirect
39. #1443C22semidirect
40. #1442C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
45. #1439C22semidirect
48. #1438C22semidirect
49. #1438C22semidirect
50. C4⋊#95C22semidirect
51. C4⋊#95C22semidirect
52. C4⋊#95C22semidirect
53. C4⋊#95C22semidirect
66. C24×C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28472
#167972
#162492
#152332
#152192
#151952
#151302
#151242
#146212
C4×#3792